Examkin

Elementary Algebra

302 soru

Tüm türler302Çoktan seçmeli155Sayısal125Eşleştirme15Boşluk doldurma7
Soru 161Soru →

A square garden plot has a side length of 3s−23s - 2 yards. A walkway of uniform width s+3s + 3 yards is built around the entire garden. Which of the following expressions represents the area, in square yards, of the walkway?

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Cevap: 16s^2 + 52s + 12

Cevap

16s^2 + 52s + 12
The correct answer is found by first calculating the outer side length, which is the inner side length plus twice the walkway width: (3s−2)+2(s+3)=5s+4(3s - 2) + 2(s + 3) = 5s + 4. Squaring this yields the outer area of 25s2+40s+1625s^2 + 40s + 16. The inner area is (3s−2)2=9s2−12s+4(3s - 2)^2 = 9s^2 - 12s + 4. Subtracting the inner area from the outer area and distributing the negative sign properly gives (25s2+40s+16)−(9s2−12s+4)=16s2+52s+12(25s^2 + 40s + 16) - (9s^2 - 12s + 4) = 16s^2 + 52s + 12.

Adım Adım Çözüm

1
Determine the outer side length of the square including the walkway.
The outer side length is (3s−2)+2(s+3)=3s−2+2s+6=5s+4(3s - 2) + 2(s + 3) = 3s - 2 + 2s + 6 = 5s + 4 yards.
Since the walkway surrounds the garden on all sides, the width of the walkway must be added to both ends of the garden's side length.
2
Calculate the area of the outer square and the inner square garden by squaring their respective side lengths.
Outer Area = (5s+4)2=25s2+40s+16(5s + 4)^2 = 25s^2 + 40s + 16 and Inner Area = (3s−2)2=9s2−12s+4(3s - 2)^2 = 9s^2 - 12s + 4.
The area of a square is equal to the square of its side length.
3
Subtract the inner garden area from the outer area to find the walkway area, distributing the negative sign to all terms of the inner area.
Walkway Area = (25s2+40s+16)−(9s2−12s+4)=25s2+40s+16−9s2+12s−4=16s2+52s+12(25s^2 + 40s + 16) - (9s^2 - 12s + 4) = 25s^2 + 40s + 16 - 9s^2 + 12s - 4 = 16s^2 + 52s + 12 square yards.
The area of the walkway is the difference between the total outer area and the inner garden area.

Anahtar Kavram

Operations on Polynomials
Soru 162Soru →

If the equation 3x2+5x=23x^2 + 5x = 2 is solved for xx, what is the positive difference between the two solutions?

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Cevap: 73\frac{7}{3}

Cevap

The positive difference between the two solutions is 73\frac{7}{3}.
Rearranging the equation 3x2+5x=23x^2 + 5x = 2 by subtracting 22 from both sides gives the standard quadratic equation 3x2+5x−2=03x^2 + 5x - 2 = 0. Factoring this expression yields (3x−1)(x+2)=0(3x - 1)(x + 2) = 0. Setting each factor to zero gives the solutions x=13x = \frac{1}{3} and x=−2x = -2. The positive difference between these solutions is 13−(−2)=13+2=73\frac{1}{3} - (-2) = \frac{1}{3} + 2 = \frac{7}{3}.

Adım Adım Çözüm

1
Rearrange the quadratic equation into standard form ax2+bx+c=0ax^2 + bx + c = 0.
3x2+5x−2=03x^2 + 5x - 2 = 0
Before factoring a quadratic equation, all terms must be moved to one side so that the equation is set to zero.
2
Factor the quadratic expression by finding two binomials that multiply to 3x2+5x−23x^2 + 5x - 2.
(3x−1)(x+2)=0(3x - 1)(x + 2) = 0
Since the product of 33 and −2-2 is −6-6, we look for factors of −6-6 that sum to 55, which are 66 and −1-1. Splitting the middle term yields 3x2+6x−x−2=3x(x+2)−1(x+2)=(3x−1)(x+2)3x^2 + 6x - x - 2 = 3x(x + 2) - 1(x + 2) = (3x - 1)(x + 2).
3
Use the zero-product property to solve for xx by setting each binomial factor to zero.
3x−1=0⇒x=133x - 1 = 0 \Rightarrow x = \frac{1}{3} and x+2=0⇒x=−2x + 2 = 0 \Rightarrow x = -2
If the product of two quantities is zero, at least one of the quantities must be zero.
4
Calculate the positive difference between the two solutions.
∣13−(−2)∣=∣13+2∣=73|\frac{1}{3} - (-2)| = |\frac{1}{3} + 2| = \frac{7}{3}
The positive difference is the absolute value of the subtraction of one root from the other.

Anahtar Kavram

Solving a quadratic equation by factoring over the integers after setting the equation equal to zero.

Alternatif Yöntem

Alternatively, the quadratic formula can be used. Once the equation is rewritten as 3x2+5x−2=03x^2 + 5x - 2 = 0, substitute a=3a=3, b=5b=5, and c=−2c=-2 into x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} to get x=−5±25−4(3)(−2)6=−5±76x = \frac{-5 \pm \sqrt{25 - 4(3)(-2)}}{6} = \frac{-5 \pm 7}{6}. This simplifies to x=13x = \frac{1}{3} and x=−2x = -2. The positive difference between these values is 13−(−2)=73\frac{1}{3} - (-2) = \frac{7}{3}.
Tahmini Süre:1m 30s
Soru 163Soru →

For what greatest integer value of yy is the inequality 9−2y≥169 - 2y \geq 16 true?

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Cevap: -4

Cevap

The greatest integer value of yy that satisfies the inequality is −4-4.
Subtracting 9 from both sides of 9−2y≥169 - 2y \geq 16 gives −2y≥7-2y \geq 7. Dividing both sides by −2-2 and reversing the inequality sign yields y≤−3.5y \leq -3.5. The greatest integer less than or equal to −3.5-3.5 is −4-4.

Adım Adım Çözüm

1
Subtract 9 from both sides of the inequality.
−2y≥7-2y \geq 7
This isolates the term containing yy on the left side.
2
Divide both sides of the inequality by −2-2 and reverse the direction of the inequality sign.
y≤−3.5y \leq -3.5
Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.
3
Identify the greatest integer that is less than or equal to −3.5-3.5.
−4-4
The value of yy must be less than or equal to −3.5-3.5. The integers satisfying this condition are −4,−5,−6,…-4, -5, -6, \dots, and the greatest of these is −4-4.

Anahtar Kavram

Solving linear inequalities by isolating the variable and reversing the inequality sign when dividing by a negative number.
Soru 164Soru →

An agricultural cooperative packages a premium seed mixture containing rye grass, fescue, and bluegrass. The weight of the fescue in the mixture is 1010 pounds less than twice the weight of the rye grass. The weight of the bluegrass is 1515 pounds more than half the weight of the fescue. If the total weight of the mixture is 120120 pounds, how many pounds of bluegrass are in the mixture?

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Cevap: 40

Cevap

The weight of the bluegrass in the mixture is 40 pounds.
By translating the given relationships into equations (f=2r−10f = 2r - 10, b=12f+15b = \frac{1}{2}f + 15, and r+f+b=120r + f + b = 120), we can express all variables in terms of rr, yielding r+(2r−10)+(r+10)=120r + (2r - 10) + (r + 10) = 120. Solving this gives r=30r = 30. Substituting this back gives the weight of bluegrass as 30+10=4030 + 10 = 40 pounds.

Adım Adım Çözüm

1
Define variables for each type of grass in the mixture.
Let rr represent the weight of rye grass, ff represent the weight of fescue, and bb represent the weight of bluegrass.
Establishing variables is necessary to translate the verbal descriptions into algebraic terms.
2
Translate the given relationships into equations.
f=2r−10f = 2r - 10 and b=12f+15b = \frac{1}{2}f + 15
The problem states the fescue is 10 pounds less than twice the rye grass, and the bluegrass is 15 pounds more than half the fescue.
3
Substitute the expression for f into the equation for b to express b solely in terms of r.
b=12(2r−10)+15=r−5+15=r+10b = \frac{1}{2}(2r - 10) + 15 = r - 5 + 15 = r + 10
Reducing the number of variables simplifies the system of equations.
4
Set up the total weight equation and solve for r.
r+(2r−10)+(r+10)=120  ⟹  4r=120  ⟹  r=30r + (2r - 10) + (r + 10) = 120 \implies 4r = 120 \implies r = 30
The sum of the three grass weights is given as 120 pounds.
5
Calculate the weight of the bluegrass using the value of r.
b=30+10=40b = 30 + 10 = 40
The question asks specifically for the weight of the bluegrass.

Anahtar Kavram

Translating verbal relationships into linear equations and solving a system of equations
Soru 165Soru →

A manufacturing company determines that its weekly profit from producing xx batches of a product is constrained by resource availability. To meet these resource constraints, the number of batches xx must satisfy the inequality:

5(6−x)3−3(x+2)4>2x−11\frac{5(6 - x)}{3} - \frac{3(x + 2)}{4} > 2x - 11

What is the greatest number of whole batches the company can produce while satisfying this constraint?

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Cevap: 4

Cevap

The greatest number of whole batches the company can produce is 4.
Solving the inequality yields x<23453≈4.415x < \frac{234}{53} \approx 4.415. The greatest integer value that satisfies this condition is 4.

Adım Adım Çözüm

1
Multiply both sides of the inequality by the least common multiple of the denominators, which is 12.
20(6−x)−9(x+2)>24x−13220(6 - x) - 9(x + 2) > 24x - 132
Multiplying by 12 eliminates the fractions, making the inequality easier to solve.
2
Distribute the constants on the left side of the inequality.
120−20x−9x−18>24x−132120 - 20x - 9x - 18 > 24x - 132
Distributing 20 to (6−x)(6 - x) yields 120−20x120 - 20x, and distributing −9-9 to (x+2)(x + 2) yields −9x−18-9x - 18.
3
Combine like terms on the left side of the inequality.
102−29x>24x−132102 - 29x > 24x - 132
Combining 120−18120 - 18 gives 102102, and combining −20x−9x-20x - 9x gives −29x-29x.
4
Subtract 24x24x from both sides to group the variable terms on the left side.
102−53x>−132102 - 53x > -132
This groups all terms containing the variable xx on one side of the inequality.
5
Subtract 102 from both sides to isolate the variable term.
−53x>−234-53x > -234
This isolates the term containing xx on the left side of the inequality.
6
Divide both sides by −53-53 and reverse the inequality sign.
x<23453x < \frac{234}{53}
Dividing by a negative number requires reversing the direction of the inequality sign.
7
Evaluate the fraction as a decimal and determine the greatest integer value of xx that satisfies the inequality.
x<4.415x < 4.415, which means the greatest integer is 4.
Since the company must produce a whole number of batches, we find the largest integer less than 4.415.

Anahtar Kavram

Solving multi-step linear inequalities involving fractional coefficients, distributing negative numbers, and reversing the inequality sign when multiplying or dividing by a negative number.

Alternatif Yöntem

Instead of solving algebraically, you can test integer values for xx directly in the inequality. Testing x=4x = 4 gives 103−184=3.33−4.5=−1.17\frac{10}{3} - \frac{18}{4} = 3.33 - 4.5 = -1.17, which is greater than 2(4)−11=−32(4) - 11 = -3 (True). Testing x=5x = 5 gives 53−214=1.67−5.25=−3.58\frac{5}{3} - \frac{21}{4} = 1.67 - 5.25 = -3.58, which is not greater than 2(5)−11=−12(5) - 11 = -1 (False). This confirms 4 is the largest integer satisfying the inequality.
Tahmini Süre:2m 30s
Soru 166Soru →

Which of the following is the completely factored form of the expression 3x(x−2)2−12x3x(x - 2)^2 - 12x?

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Cevap: 3x2(x−4)3x^2(x - 4)

Cevap

The completely factored form of the expression is 3x2(x−4)3x^2(x - 4).
The correct answer is 3x2(x−4)3x^2(x-4). Factoring out the greatest common factor 3x3x from the terms 3x(x−2)23x(x-2)^2 and −12x-12x gives 3x[(x−2)2−4]3x[(x-2)^2-4]. The expression within the brackets is a difference of squares that can be factored as [(x−2)−2][(x−2)+2][(x-2)-2][(x-2)+2], which simplifies to x(x−4)x(x-4). Multiplying this by the GCF 3x3x yields 3x2(x−4)3x^2(x-4).

Adım Adım Çözüm

1
Factor out the greatest common factor (GCF), 3x3x, from both terms of the expression 3x(x−2)2−12x3x(x - 2)^2 - 12x.
3x[(x−2)2−4]3x[(x - 2)^2 - 4]
Both terms 3x(x−2)23x(x-2)^2 and 12x12x share the common factors 33 and xx.
2
Factor the difference of squares inside the bracket, (x−2)2−4(x-2)^2 - 4, using the formula a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b) where a=x−2a = x-2 and b=2b = 2.
[(x−2)−2][(x−2)+2]=(x−4)(x)[(x-2)-2][(x-2)+2] = (x-4)(x)
Since 4=224 = 2^2, the terms inside the brackets form a difference of squares.
3
Combine the factored parts and simplify the expression by multiplying the variable terms.
3x⋅x(x−4)=3x2(x−4)3x \cdot x(x-4) = 3x^2(x-4)
Multiplying 3x3x by xx requires adding their exponents (1+1=21 + 1 = 2).

Anahtar Kavram

Factoring polynomials using the greatest common factor (GCF) and difference of squares.
Soru 167Soru →

When the expression 4x(x−2y)−(2x−3y)2+5y(2x−y)4x(x - 2y) - (2x - 3y)^2 + 5y(2x - y) is simplified to the form Ax2+Bxy+Cy2Ax^2 + Bxy + Cy^2, where AA, BB, and CC are constants, what is the value of BB?

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Cevap: 14

Cevap

The value of the coefficient BB is 14.
Expanding the entire expression yields 4x2−8xy−4x2+12xy−9y2+10xy−5y24x^2 - 8xy - 4x^2 + 12xy - 9y^2 + 10xy - 5y^2. Grouping and combining the xyxy terms gives (−8+12+10)xy=14xy(-8 + 12 + 10)xy = 14xy. Therefore, the coefficient BB is 14.

Adım Adım Çözüm

1
Expand the first term
4x2−8xy4x^2 - 8xy
Distribute 4x4x to both terms inside the parentheses: 4x(x)−4x(2y)=4x2−8xy4x(x) - 4x(2y) = 4x^2 - 8xy.
2
Expand the squared binomial and apply the negative sign
−4x2+12xy−9y2-4x^2 + 12xy - 9y^2
Use the binomial expansion formula (2x−3y)2=4x2−12xy+9y2(2x - 3y)^2 = 4x^2 - 12xy + 9y^2, then multiply each term by −1-1.
3
Expand the third term
10xy−5y210xy - 5y^2
Distribute 5y5y to both terms inside the parentheses: 5y(2x)−5y(y)=10xy−5y25y(2x) - 5y(y) = 10xy - 5y^2.
4
Combine the coefficients of the like terms
0x2+14xy−14y20x^2 + 14xy - 14y^2
Sum the coefficients for each corresponding variable group: (4−4)x2+(−8+12+10)xy+(−9−5)y2(4 - 4)x^2 + (-8 + 12 + 10)xy + (-9 - 5)y^2.

Anahtar Kavram

Simplifying Algebraic Expressions and Combining Like Terms
Soru 168Soru →

For a real number xx, 12\frac{1}{2} minus 23\frac{2}{3} of xx is greater than 56\frac{5}{6}. Which of the following is the complete set of solutions for xx?

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Cevap: x<−12x < -\frac{1}{2}

Cevap

x<−12x < -\frac{1}{2}
Subtracting 12\frac{1}{2} from both sides of the inequality 12−23x>56\frac{1}{2} - \frac{2}{3}x > \frac{5}{6} gives −23x>26-\frac{2}{3}x > \frac{2}{6}, which simplifies to −23x>13-\frac{2}{3}x > \frac{1}{3}. Multiplying both sides by the negative fraction −32-\frac{3}{2} isolates xx on the left and reverses the inequality sign from greater than (>>) to less than (<<). Performing the multiplication on the right yields 13⋅(−32)=−12\frac{1}{3} \cdot \left(-\frac{3}{2}\right) = -\frac{1}{2}. Thus, the solution set is x<−12x < -\frac{1}{2}.

Adım Adım Çözüm

1
Translate the verbal description into an algebraic inequality.
12−23x>56\frac{1}{2} - \frac{2}{3}x > \frac{5}{6}
The phrase '12\frac{1}{2} minus 23\frac{2}{3} of xx' represents the expression 12−23x\frac{1}{2} - \frac{2}{3}x, and 'is greater than' translates to the inequality symbol >>.
2
Subtract 12\frac{1}{2} from both sides of the inequality.
−23x>13-\frac{2}{3}x > \frac{1}{3}
To isolate the variable term on the left, we subtract 12\frac{1}{2} (which is equivalent to 36\frac{3}{6}) from 56\frac{5}{6} to get 26=13\frac{2}{6} = \frac{1}{3}.
3
Multiply both sides of the inequality by −32-\frac{3}{2} and reverse the inequality sign.
x<−12x < -\frac{1}{2}
Multiplying both sides of an inequality by a negative number requires reversing the direction of the inequality sign from >> to << to keep the inequality true.

Anahtar Kavram

Solving multi-step linear inequalities involving multiplication or division by a negative number.
Soru 169Soru →

The polynomial P(x)P(x) is defined by P(x)=(2x2−3x+5)2−4x(x3−2x2−7x+1)P(x) = (2x^2 - 3x + 5)^2 - 4x(x^3 - 2x^2 - 7x + 1). When P(x)P(x) is written in standard form, what is the coefficient of the x2x^2 term?

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Cevap: 57

Cevap

The coefficient of the x2x^2 term is 57.
Expanding (2x2−3x+5)2(2x^2 - 3x + 5)^2 yields (2x2−3x+5)(2x2−3x+5)=4x4−12x3+29x2−30x+25(2x^2 - 3x + 5)(2x^2 - 3x + 5) = 4x^4 - 12x^3 + 29x^2 - 30x + 25. Distributing the −4x-4x term yields −4x(x3−2x2−7x+1)=−4x4+8x3+28x2−4x-4x(x^3 - 2x^2 - 7x + 1) = -4x^4 + 8x^3 + 28x^2 - 4x. Combining the x2x^2 terms from both expressions gives 29x2+28x2=57x229x^2 + 28x^2 = 57x^2. Thus, the coefficient of the x2x^2 term is 57.

Adım Adım Çözüm

1
Expand the squared trinomial (2x2−3x+5)2(2x^2 - 3x + 5)^2
4x4−12x3+29x2−30x+254x^4 - 12x^3 + 29x^2 - 30x + 25
Expanding the first part of the expression by multiplying the trinomial by itself.
2
Distribute the term −4x-4x to the trinomial (x3−2x2−7x+1)(x^3 - 2x^2 - 7x + 1)
−4x4+8x3+28x2−4x-4x^4 + 8x^3 + 28x^2 - 4x
Expanding the second part of the polynomial expression while distributing the negative sign to all terms.
3
Combine the expanded expressions and isolate the x2x^2 terms
29x2+28x2=57x229x^2 + 28x^2 = 57x^2
Adding the coefficients of the terms of degree 2 to find the combined coefficient.

Anahtar Kavram

Operations on Polynomials
Soru 170Soru →
For all non-zero real numbers yy, the expression
(y3)2⋅(y−4)ay5\frac{(y^3)^2 \cdot (y^{-4})^a}{y^5}
is equivalent to y−7y^{-7}. What is the value of aa?
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Cevap: 22

Cevap

The correct value of aa is 2.
Applying the exponent rules systematically allows us to simplify the expression. First, (y3)2(y^3)^2 becomes y6y^6 and (y−4)a(y^{-4})^a becomes y−4ay^{-4a} using the power of a power rule. Second, we combine the terms in the numerator using the product rule to get y6−4ay^{6-4a}. Third, we divide by y5y^5 using the quotient rule to obtain y6−4a−5=y1−4ay^{6-4a-5} = y^{1-4a}. Setting this equal to the target expression y−7y^{-7} gives the equation 1−4a=−71-4a = -7. Solving for aa gives −4a=−8-4a = -8, which simplifies to 22.

Adım Adım Çözüm

1
Apply the power of a power rule, (xm)n=xmn(x^m)^n = x^{mn}, to the exponential terms in the numerator.
(y3)2=y6(y^3)^2 = y^6 and (y−4)a=y−4a(y^{-4})^a = y^{-4a}
This simplifies nested exponent terms into single base terms.
2
Apply the product rule of exponents, xm⋅xn=xm+nx^m \cdot x^n = x^{m+n}, to combine the numerator terms.
y6⋅y−4a=y6−4ay^6 \cdot y^{-4a} = y^{6-4a}
This simplifies the numerator to a single power of yy.
3
Apply the quotient rule of exponents, xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}, to divide by the denominator.
y6−4ay5=y(6−4a)−5=y1−4a\frac{y^{6-4a}}{y^5} = y^{(6-4a) - 5} = y^{1-4a}
This simplifies the entire rational expression into a single exponential expression.
4
Equate the simplified exponent to the exponent of the equivalent expression and solve the linear equation for aa.
1−4a=−7  ⟹  −4a=−8  ⟹  a=21-4a = -7 \implies -4a = -8 \implies a = 2
Since the bases are equal and non-zero, their exponents must be equal for the expressions to be equivalent.

Anahtar Kavram

Properties of Exponents in Algebraic Expressions
Tahmini Süre:1m 30s
Soru 171Soru →

If the quadratic expression 6x2−7x−56x^2 - 7x - 5 is factored completely into the product of two linear binomials of the form (ax+b)(cx+d)(ax + b)(cx + d), where aa, bb, cc, and dd are integers such that a>c>0a > c > 0, what is the value of the expression ad−bcad - bc?

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Cevap: 13

Cevap

The value of the expression ad−bcad - bc is 13.

Adım Adım Çözüm

1
Factor the quadratic expression 6x2−7x−56x^2 - 7x - 5.
(3x−5)(2x+1)(3x - 5)(2x + 1)
Find two numbers that multiply to 6×(−5)=−306 \times (-5) = -30 and add to −7-7, which are −10-10 and 33. Rewrite the middle term and factor by grouping: 6x2−10x+3x−5=2x(3x−5)+1(3x−5)=(3x−5)(2x+1)6x^2 - 10x + 3x - 5 = 2x(3x - 5) + 1(3x - 5) = (3x - 5)(2x + 1).
2
Determine the values of the coefficients aa, bb, cc, and dd.
a=3a = 3, b=−5b = -5, c=2c = 2, and d=1d = 1
The expression is factored into the form (ax+b)(cx+d)(ax + b)(cx + d) where a>c>0a > c > 0. Comparing the factors (3x−5)(3x - 5) and (2x+1)(2x + 1), we see the coefficients of xx are 33 and 22. Since 3>2>03 > 2 > 0, we have a=3a = 3 and c=2c = 2. This leaves b=−5b = -5 and d=1d = 1.
3
Calculate the value of ad−bcad - bc.
13
Substitute a=3a = 3, b=−5b = -5, c=2c = 2, and d=1d = 1 into the expression: ad−bc=(3)(1)−(−5)(2)=3+10=13ad - bc = (3)(1) - (-5)(2) = 3 + 10 = 13.

Anahtar Kavram

Factoring quadratic polynomials of the form ax2+bx+cax^2 + bx + c with a>1a > 1.
Soru 172Soru →
If xx and yy are positive real numbers such that
(x3y−2)k⋅(x−1y4)3=x6y6(x^3 y^{-2})^k \cdot (x^{-1} y^4)^3 = x^6 y^6
what is the value of the exponent kk?
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Cevap: 3

Cevap

The value of the exponent kk is 3.
Applying the exponent rules simplifies the left side of the equation to x3k−3y12−2kx^{3k-3}y^{12-2k}. Equating the exponent of xx to the right side gives 3k−3=63k - 3 = 6, which yields k=3k = 3. This value is confirmed by equating the exponent of yy, since 12−2(3)=612 - 2(3) = 6.

Adım Adım Çözüm

1
Apply the power of a power rule (am)n=amn(a^m)^n = a^{mn} to expand the terms in the expression.
(x3y−2)k=x3ky−2k(x^3 y^{-2})^k = x^{3k} y^{-2k} and (x−1y4)3=x−3y12(x^{-1} y^4)^3 = x^{-3} y^{12}
To remove the outer parentheses by multiplying the internal exponents of each variable by the outer exponent.
2
Multiply the terms together by applying the product rule for exponents, am⋅an=am+na^m \cdot a^n = a^{m+n}.
(x3ky−2k)(x−3y12)=x3k−3y12−2k(x^{3k} y^{-2k})(x^{-3} y^{12}) = x^{3k-3} y^{12-2k}
To combine the like bases of xx and yy into a single simplified expression.
3
Set the exponents of like bases equal to the exponents on the right-hand side of the equation, x6y6x^6 y^6.
3k−3=63k - 3 = 6 and 12−2k=612 - 2k = 6
Since the bases are equal and non-zero, their respective exponents must also be equal.
4
Solve the linear equation 3k−3=63k - 3 = 6 for kk.
3k=9  ⟹  k=33k = 9 \implies k = 3
To determine the numerical value of the variable kk.
5
Verify the solution by solving the second linear equation, 12−2k=612 - 2k = 6.
−2k=−6  ⟹  k=3-2k = -6 \implies k = 3
To ensure consistency across both variable exponents in the expression.

Anahtar Kavram

Properties of Exponents in Algebraic Expressions
Soru 173Soru →

A manufacturing plant operates two machines, Machine X and Machine Y, to produce identical components. Machine Y produces 12 fewer components per hour than Machine X. On a certain day, Machine X operated for 5 hours and Machine Y operated for 7 hours. If the two machines produced a combined total of 636 components, how many components did Machine X produce on that day?

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Cevap: 300

Cevap

300
To find the number of components Machine X produced, let xx represent Machine X's hourly production rate. This makes Machine Y's hourly rate x−12x - 12. Since Machine X operated for 5 hours and Machine Y operated for 7 hours to produce a total of 636 components, we can write the equation: 5x+7(x−12)=6365x + 7(x - 12) = 636. Distributing the 7 yields 5x+7x−84=6365x + 7x - 84 = 636, which simplifies to 12x−84=63612x - 84 = 636. Adding 84 to both sides gives 12x=72012x = 720, and dividing by 12 yields x=60x = 60 components per hour for Machine X. To find the total produced by Machine X, we multiply its hourly rate by the 5 hours it operated: 5×60=3005 \times 60 = 300. This matches the correct option.

Adım Adım Çözüm

1
Define variables for the hourly rates of both machines.
Let xx be the number of components Machine X produces per hour. Since Machine Y produces 12 fewer components per hour, Machine Y's hourly rate is x−12x - 12.
Establishing algebraic expressions for each machine's rate allows us to set up a linear equation.
2
Set up an equation representing the total combined production.
The total production is the sum of the components produced by Machine X in 5 hours and Machine Y in 7 hours: 5x+7(x−12)=6365x + 7(x - 12) = 636.
The sum of the products of each machine's rate and its operating time equals the total combined production of 636 components.
3
Solve the equation for the hourly rate of Machine X, xx.
Distribute the 7: 5x+7x−84=6365x + 7x - 84 = 636. Combine like terms: 12x−84=63612x - 84 = 636. Add 84 to both sides: 12x=72012x = 720. Divide by 12: x=60x = 60.
Solving for xx gives the hourly rate of Machine X.
4
Calculate the total components produced by Machine X.
Multiply the hourly rate of Machine X by its hours of operation: 5 hours×60 components/hour=3005 \text{ hours} \times 60 \text{ components/hour} = 300.
The question asks for the total components produced by Machine X, not its hourly rate.

Anahtar Kavram

Setting up and solving single-variable linear equations from verbal descriptions

Alternatif Yöntem

Instead of defining the variable as Machine X's rate, we could define yy as Machine Y's hourly rate. Then Machine X's rate is y+12y + 12. The total equation becomes 5(y+12)+7y=636  ⟹  5y+60+7y=636  ⟹  12y=576  ⟹  y=485(y + 12) + 7y = 636 \implies 5y + 60 + 7y = 636 \implies 12y = 576 \implies y = 48. Since Machine X's hourly rate is y+12=60y + 12 = 60, it produced 5×60=3005 \times 60 = 300 components.
Tahmini Süre:1m 30s
Soru 174Soru →

When 55 is subtracted from 22 times a number xx, the result is at least 99. Which of the following inequalities represents all possible values of xx?

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Cevap: x≥7x \geq 7

Cevap

The inequality x≥7x \geq 7 represents all possible values of xx.
The verbal statement translates to the inequality 2x−5≥92x - 5 \geq 9. Adding 55 to both sides gives 2x≥142x \geq 14, and dividing both sides by the positive number 22 results in x≥7x \geq 7. Since we divide by a positive number, the direction of the inequality does not change.

Adım Adım Çözüm

1
Translate the verbal phrase into an algebraic inequality.
2x−5≥92x - 5 \geq 9
'5 subtracted from 2 times a number xx' translates to 2x−52x - 5, and 'at least 9' means greater than or equal to 9.
2
Add 5 to both sides of the inequality to isolate the variable term.
2x≥142x \geq 14
Adding 5 to both sides maintains the inequality and simplifies the left side.
3
Divide both sides by 2.
x≥7x \geq 7
Dividing by a positive number does not change the direction of the inequality sign.

Anahtar Kavram

Solving linear inequalities by translating verbal statements into algebraic forms and applying inverse operations.
Soru 175Soru →

What is the smallest integer value of xx that satisfies the inequality 11−3x<211 - 3x < 2?

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Cevap: 4

Cevap

The smallest integer value of xx that satisfies the inequality is 4.
Solving the inequality 11−3x<211 - 3x < 2 leads to −3x<−9-3x < -9. Dividing by −3-3 and reversing the inequality sign gives x>3x > 3. The smallest integer that is strictly greater than 3 is 4.

Adım Adım Çözüm

1
Isolate the variable term on one side of the inequality by subtracting 11 from both sides.
−3x<−9-3x < -9
Subtracting 11 from both sides of the inequality keeps the relationship balanced.
2
Divide both sides of the inequality by the coefficient of xx, which is −3-3.
x>3x > 3
Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.
3
Determine the smallest integer that is strictly greater than 3.
4
Since the inequality is strict (x>3x > 3), 3 is not included in the solution set. The smallest integer greater than 3 is 4.

Anahtar Kavram

Solving linear inequalities and reversing the inequality sign when dividing by a negative number.
Soru 176Soru →

A storage tank contains 1818 gallons of water and is being filled at a constant rate of 23\frac{2}{3} gallons per minute. A second storage tank contains 3333 gallons of water and is being drained at a constant rate of 56\frac{5}{6} gallons per minute. If both tanks begin their processes at the same time, they will contain the same amount of water after mm minutes. What is the value of 2m+32m + 3?

Cevabı ve açıklamayı göster

Cevap: 23

Cevap

23
The correct answer is 23. By representing the volume of the first tank as 18+23m18 + \frac{2}{3}m and the second tank as 33−56m33 - \frac{5}{6}m, setting them equal gives 18+23m=33−56m18 + \frac{2}{3}m = 33 - \frac{5}{6}m. Solving this equation by finding a common denominator for the fraction coefficients yields 96m=15\frac{9}{6}m = 15, which simplifies to 32m=15\frac{3}{2}m = 15, and thus m=10m = 10. Substituting m=10m = 10 into the expression 2m+32m + 3 results in 2(10)+3=232(10) + 3 = 23.

Adım Adım Çözüm

1
Set up the equation representing the water volume in both tanks over time.
18+23m=33−56m18 + \frac{2}{3}m = 33 - \frac{5}{6}m
The first tank starts with 1818 gallons and increases by 23\frac{2}{3} gallons per minute, while the second starts with 3333 gallons and decreases by 56\frac{5}{6} gallons per minute. We set their volumes equal to find when they contain the same amount.
2
Isolate the variable terms on one side and the constant terms on the other side.
23m+56m=33−18\frac{2}{3}m + \frac{5}{6}m = 33 - 18
Adding 56m\frac{5}{6}m to both sides and subtracting 1818 from both sides groups like terms together.
3
Find a common denominator to add the fraction coefficients.
46m+56m=15⇒96m=15⇒32m=15\frac{4}{6}m + \frac{5}{6}m = 15 \Rightarrow \frac{9}{6}m = 15 \Rightarrow \frac{3}{2}m = 15
A common denominator of 66 is used to add the fractions 23\frac{2}{3} and 56\frac{5}{6}.
4
Solve for mm by multiplying both sides by the reciprocal of the coefficient.
m=15×23=10m = 15 \times \frac{2}{3} = 10
Multiplying by 23\frac{2}{3} isolates mm on the left side.
5
Evaluate the expression 2m+32m + 3 using the value of mm.
2(10)+3=20+3=232(10) + 3 = 20 + 3 = 23
Substitute m=10m = 10 into the expression and follow the correct order of operations (multiply first, then add).

Anahtar Kavram

Solving linear equations with fractional coefficients and translating word problems into algebraic equations.
Soru 177Soru →

A triangle has a base of 2x+42x + 4 inches and a height of x−3x - 3 inches. What is the coefficient of xx when the expression representing the area of the triangle, in square inches, is written in standard form?

Cevabı ve açıklamayı göster

Cevap: -1

Cevap

The coefficient of xx is −1-1.
The area of a triangle is calculated using Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}. Substituting the given expressions, we get Area=12(2x+4)(x−3)\text{Area} = \frac{1}{2}(2x + 4)(x - 3). First, we can multiply the 12\frac{1}{2} by (2x+4)(2x + 4), which simplifies to x+2x + 2. Next, we expand (x+2)(x−3)(x + 2)(x - 3) using FOIL to get x2−3x+2x−6x^2 - 3x + 2x - 6. Combining like terms yields x2−x−6x^2 - x - 6. The coefficient of the xx term in this simplified expression is −1-1.

Adım Adım Çözüm

1
State the formula for the area of a triangle.
Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
To establish the mathematical relationship.
2
Substitute the given values into the formula.
Area=12(2x+4)(x−3)\text{Area} = \frac{1}{2}(2x + 4)(x - 3)
To express the area in terms of the variable xx.
3
Distribute the fraction to the first binomial.
x+2x + 2
Distributing 12\frac{1}{2} to (2x+4)(2x + 4) simplifies the expression before multiplying.
4
Expand the product of the binomials.
x2−3x+2x−6x^2 - 3x + 2x - 6
Using the FOIL method to multiply (x+2)(x + 2) and (x−3)(x - 3).
5
Combine the linear terms.
x2−x−6x^2 - x - 6
To simplify the polynomial and write it in standard form.
6
Identify the coefficient of xx.
−1-1
The coefficient of the xx term in x2−x−6x^2 - x - 6 is −1-1.

Anahtar Kavram

Multiplying binomials and applying formulas in geometric contexts
Tahmini Süre:1m 0s
Soru 178Soru →

If the expression (x2y3)4(x−1y2)3\frac{(x^2 y^3)^4}{(x^{-1} y^2)^3} is written in the equivalent form xaybx^a y^b, what is the value of a−ba - b?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5
Simplifying the numerator yields x8y12x^8 y^{12} and simplifying the denominator yields x−3y6x^{-3} y^6. Dividing these expressions by subtracting the exponents of like bases results in x8−(−3)y12−6=x11y6x^{8 - (-3)} y^{12 - 6} = x^{11} y^6. Comparing this to the expression xaybx^a y^b shows that a=11a = 11 and b=6b = 6. The value of a−ba - b is 11−6=511 - 6 = 5.

Adım Adım Çözüm

1
Simplify the numerator of the expression.
x8y12x^8 y^{12}
Apply the power of a power and power of a product properties of exponents: (x2y3)4=x2⋅4y3⋅4(x^2 y^3)^4 = x^{2 \cdot 4} y^{3 \cdot 4}.
2
Simplify the denominator of the expression.
x−3y6x^{-3} y^6
Apply the power of a power and power of a product properties of exponents: (x−1y2)3=x−1⋅3y2⋅3(x^{-1} y^2)^3 = x^{-1 \cdot 3} y^{2 \cdot 3}.
3
Simplify the quotient by dividing the simplified numerator by the simplified denominator.
x11y6x^{11} y^6
Use the quotient property of exponents, xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}, to subtract the exponents: 8−(−3)=118 - (-3) = 11 and 12−6=612 - 6 = 6.
4
Identify the values of aa and bb and compute the difference a−ba - b.
5
By comparing x11y6x^{11} y^6 to the target form xaybx^a y^b, we find a=11a = 11 and b=6b = 6. Subtracting bb from aa yields 11−6=511 - 6 = 5.

Anahtar Kavram

Properties of exponents including power of a power, power of a product, and quotient rules.
Soru 179Soru →

For all real values of xx, which of the following inequalities represents the complete solution set to the inequality 5−2x3−3x−14≤x+22\frac{5 - 2x}{3} - \frac{3x - 1}{4} \leq \frac{x + 2}{2}?

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Cevap: x≥1123x \geq \frac{11}{23}

Cevap

The complete solution set is the set of all real numbers greater than or equal to 11/23.
The correct answer is found by clearing the denominators with the least common multiple of 12, carefully expanding the terms to get 20−8x−9x+3≤6x+1220 - 8x - 9x + 3 \leq 6x + 12, simplifying to 23−17x≤6x+1223 - 17x \leq 6x + 12, grouping terms to get −23x≤−11-23x \leq -11, and dividing by −23-23 which flips the sign to yield all real values greater than or equal to 11/23.

Adım Adım Çözüm

1
Multiply all terms of the inequality by the least common multiple of the denominators (3, 4, and 2), which is 12, to clear the fractions.
4(5−2x)−3(3x−1)≤6(x+2)4(5 - 2x) - 3(3x - 1) \leq 6(x + 2)
Multiplying by a positive number allows us to eliminate denominators without changing the direction of the inequality.
2
Distribute the coefficients on both sides of the inequality, paying close attention to the distribution of the negative sign over the second term.
20−8x−9x+3≤6x+1220 - 8x - 9x + 3 \leq 6x + 12
Distributing −3-3 to both 3x3x and −1-1 yields −9x-9x and +3+3 respectively.
3
Combine the constant terms and the variable terms on the left side of the inequality.
23−17x≤6x+1223 - 17x \leq 6x + 12
Simplifying the expressions on each side makes the inequality easier to isolate.
4
Isolate the variable terms on the left and the constant terms on the right by subtracting 6x6x and 23 from both sides.
−23x≤−11-23x \leq -11
Grouping like terms together is necessary to solve for the variable.
5
Divide both sides of the inequality by −23-23 and reverse the direction of the inequality sign.
x≥1123x \geq \frac{11}{23}
Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign from ≤\leq to ≥\geq.

Anahtar Kavram

Solving linear inequalities involving fractions and distributing negative coefficients, specifically applying the rule that multiplying or dividing by a negative number reverses the inequality direction.

Alternatif Yöntem

Instead of clearing the fractions first, write each fraction as separate terms: 53−23x−34x+14≤12x+1\frac{5}{3} - \frac{2}{3}x - \frac{3}{4}x + \frac{1}{4} \leq \frac{1}{2}x + 1. Then, collect the constant terms on one side and the variable terms on the other side using decimal or fractional conversions, and isolate the variable.
Tahmini Süre:2m 0s
Soru 180Soru →

A square playground has a side length of 2x32x^3 meters. A square sandbox with a side length of x3−2xx^3 - 2x meters is built inside the playground. Which of the following expressions represents the area, in square meters, of the playground that is NOT covered by the sandbox?

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Cevap: 3x6+4x4−4x23x^6 + 4x^4 - 4x^2

Cevap

The correct expression is 3x6+4x4−4x23x^6 + 4x^4 - 4x^2.
The expression 3x6+4x4−4x23x^6 + 4x^4 - 4x^2 is correct because the area of the playground is (2x3)2=4x6(2x^3)^2 = 4x^6 and the area of the sandbox is (x3−2x)2=x6−4x4+4x2(x^3 - 2x)^2 = x^6 - 4x^4 + 4x^2. Subtracting the sandbox area from the playground area requires distributing the negative sign, resulting in 4x6−x6−(−4x4)−4x2=3x6+4x4−4x24x^6 - x^6 - (-4x^4) - 4x^2 = 3x^6 + 4x^4 - 4x^2.

Adım Adım Çözüm

1
Calculate the area of the square playground.
Areaplayground=(2x3)2=4x6\text{Area}_{\text{playground}} = (2x^3)^2 = 4x^6
The area of a square is equal to the square of its side length, and applying the exponent rules yields (2x3)2=22⋅(x3)2=4x6(2x^3)^2 = 2^2 \cdot (x^3)^2 = 4x^6.
2
Calculate the area of the square sandbox.
Areasandbox=(x3−2x)2=x6−4x4+4x2\text{Area}_{\text{sandbox}} = (x^3 - 2x)^2 = x^6 - 4x^4 + 4x^2
Using the binomial squaring formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2, we square each term and compute the middle product, adding exponents for x3⋅x1=x4x^3 \cdot x^1 = x^4.
3
Subtract the sandbox's area from the playground's area.
4x6−(x6−4x4+4x2)=4x6−x6+4x4−4x2=3x6+4x4−4x24x^6 - (x^6 - 4x^4 + 4x^2) = 4x^6 - x^6 + 4x^4 - 4x^2 = 3x^6 + 4x^4 - 4x^2
Distribute the negative sign to all three terms inside the parentheses and combine the like terms of x6x^6.

Anahtar Kavram

Polynomial subtraction and squaring binomials with variables containing exponents.

Alternatif Yöntem

Evaluate the expression for a small integer value of xx. If x=2x = 2, the playground side length is 2(2)3=162(2)^3 = 16, giving an area of 162=25616^2 = 256. The sandbox side length is 23−2(2)=42^3 - 2(2) = 4, giving an area of 42=164^2 = 16. The remaining area is 256−16=240256 - 16 = 240. Substituting x=2x = 2 into the correct expression 3x6+4x4−4x23x^6 + 4x^4 - 4x^2 yields 3(64)+4(16)−4(4)=192+64−16=2403(64) + 4(16) - 4(4) = 192 + 64 - 16 = 240, confirming its correctness.
Tahmini Süre:2m 0s
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