Examkin

Algebra

282 questions

All types282Multiple choice147True / False7Numeric33Matching33Fill in the blank39Short answer23
Question 81Question →

Determine the value of hh that satisfies the equation below:

5−2(3h−4)=6h−4(h+4)−3 5 - 2(3h - 4) = 6h - 4(h + 4) - 3
Show answer & explanation

Answer: 4

Answer

The correct value is 4.
The equation is solved by first distributing the coefficients across the parentheses, which yields 5−6h+8=6h−4h−16−35 - 6h + 8 = 6h - 4h - 16 - 3. Combining like terms on both sides simplifies the equation to 13−6h=2h−1913 - 6h = 2h - 19. Adding 6h6h and 1919 to both sides isolates the variable, resulting in 32=8h32 = 8h. Dividing by 88 gives the final answer of 44.

Step-by-Step Solution

1
Apply the distributive property.
5−6h+8=6h−4h−16−35 - 6h + 8 = 6h - 4h - 16 - 3
Multiply the terms inside the parentheses by their outside coefficients, carefully distributing the negative signs.
2
Combine like terms on the left side.
13−6h=6h−4h−16−313 - 6h = 6h - 4h - 16 - 3
Combine the constant terms 55 and 88.
3
Combine like terms on the right side.
13−6h=2h−1913 - 6h = 2h - 19
Combine the variable terms (6h−4h=2h6h - 4h = 2h) and the constant terms (−16−3=−19-16 - 3 = -19).
4
Isolate the variable hh.
13+19=2h+6h13 + 19 = 2h + 6h
Add 6h6h to both sides and add 1919 to both sides to group the variables and constants.
5
Solve for the final value.
32=8h→h=432 = 8h \rightarrow h = 4
Simplify the addition and divide by 88 to find hh.

Key Concept

Solving One-Variable Linear Equations
Estimated Time:1m 30s
Question 82Question →

Expand the given product of a binomial and a trinomial, combine like terms, and determine the values of the missing coefficients.

Fill in the blanks below

When the expression (3x−5)(2x2+4x−7)(3x - 5)(2x^2 + 4x - 7) is fully expanded and simplified, the resulting polynomial in standard form is 6x3+6x^3 + x2+x^2 + x+35x + 35.
Show answer & explanation

Answer

The coefficient for the x^2 term is 2, and the coefficient for the x term is -41.
By distributing each term of the binomial (3x−5)(3x - 5) to every term of the trinomial (2x2+4x−7)(2x^2 + 4x - 7), we get six terms. Grouping and combining the like terms yields an x2x^2 coefficient of 12−10=212 - 10 = 2 and an xx coefficient of −21−20=−41-21 - 20 = -41.

Step-by-Step Solution

1
Distribute the first term of the binomial, 3x3x, to each term in the trinomial.
3x(2x2)+3x(4x)+3x(−7)=6x3+12x2−21x3x(2x^2) + 3x(4x) + 3x(-7) = 6x^3 + 12x^2 - 21x
To apply the distributive property for the first part of the polynomial multiplication.
2
Distribute the second term of the binomial, −5-5, to each term in the trinomial.
−5(2x2)−5(4x)−5(−7)=−10x2−20x+35-5(2x^2) - 5(4x) - 5(-7) = -10x^2 - 20x + 35
To apply the distributive property for the second part, being careful to carry the negative sign.
3
Combine the results and group the like terms.
6x3+(12x2−10x2)+(−21x−20x)+356x^3 + (12x^2 - 10x^2) + (-21x - 20x) + 35
To arrange the expression so that coefficients of terms with the same exponent can be simplified.
4
Simplify the grouped terms.
6x3+2x2−41x+356x^3 + 2x^2 - 41x + 35
By adding the coefficients: 12−10=212 - 10 = 2 and −21−20=−41-21 - 20 = -41.

Key Concept

Distributive Property with Polynomials

Alternative Method

You can use the 'box method' (a grid) by creating a 2x3 table with 3x3x and −5-5 on the rows, and 2x22x^2, 4x4x, and −7-7 on the columns. Multiply the row and column headers to fill each cell, then add all the cells together along the diagonals to combine like terms.
Estimated Time:1m 30s
Question 83Question →

Evaluate the algebraic expression 3a2b−4bca−c\frac{3a^2b - 4bc}{a - c} given that a=−2a = -2, b=3b = 3, and c=−5c = -5. What is the final numerical value?

Show answer & explanation

Answer: 32

Answer

The numerical value of the expression is 32.
After properly substituting the given values, evaluating the exponents, and following the order of operations, the numerator calculates to 96 and the denominator simplifies to 3. The final division of 96 by 3 yields 32.

Step-by-Step Solution

1
Substitute the assigned values of aa, bb, and cc into the given algebraic expression.
3(−2)2(3)−4(3)(−5)−2−(−5)\frac{3(-2)^2(3) - 4(3)(-5)}{-2 - (-5)}
Direct variable substitution is required to compute the numerical value.
2
Evaluate the exponent applied to the negative base.
(−2)2=4(-2)^2 = 4
According to the order of operations (PEMDAS), exponents must be resolved prior to multiplication.
3
Carry out the multiplication operations in the numerator.
3(4)(3)=363(4)(3) = 36 and 4(3)(−5)=−604(3)(-5) = -60
Multiplication is processed from left to right within the grouping of the numerator.
4
Complete the subtraction operation in the numerator.
36−(−60)=36+60=9636 - (-60) = 36 + 60 = 96
Subtracting a negative quantity is algebraically identical to adding its positive counterpart.
5
Evaluate the subtraction operation in the denominator.
−2−(−5)=−2+5=3-2 - (-5) = -2 + 5 = 3
The entire denominator acts as a single grouping and must be fully simplified before division.
6
Divide the evaluated numerator by the evaluated denominator.
963=32\frac{96}{3} = 32
This final arithmetic step produces the precise evaluation of the fractional expression.

Key Concept

Evaluating complex rational algebraic expressions using substitution and the order of operations with signed integers.
Question 84Question →

Which of the following represents the fully simplified form of the expression (3x−4)2−x(2x+5)(3x - 4)^2 - x(2x + 5)?

Show answer & explanation

Answer: 7x2−29x+167x^2 - 29x + 16

Answer

The correct simplified expression is 7x2−29x+167x^2 - 29x + 16.
Expanding the squared binomial yields 9x2−24x+169x^2 - 24x + 16. Distributing the −x-x yields −2x2−5x-2x^2 - 5x. Combining the like terms gives (9x2−2x2)+(−24x−5x)+16(9x^2 - 2x^2) + (-24x - 5x) + 16, which simplifies perfectly to 7x2−29x+167x^2 - 29x + 16.

Step-by-Step Solution

1
Expand the squared binomial (3x−4)2(3x - 4)^2.
(3x)2−2(3x)(4)+42=9x2−24x+16(3x)^2 - 2(3x)(4) + 4^2 = 9x^2 - 24x + 16
Using the perfect square formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 is required to properly multiply the binomial by itself.
2
Distribute the term −x-x across the binomial (2x+5)(2x + 5).
−x(2x)−x(5)=−2x2−5x-x(2x) - x(5) = -2x^2 - 5x
The distributive property must be applied, ensuring the negative sign is carried to both terms inside the parentheses.
3
Combine the results from the previous steps and combine like terms.
(9x2−24x+16)+(−2x2−5x)=(9x2−2x2)+(−24x−5x)+16=7x2−29x+16(9x^2 - 24x + 16) + (-2x^2 - 5x) = (9x^2 - 2x^2) + (-24x - 5x) + 16 = 7x^2 - 29x + 16
Like terms (those with the same variable raised to the same power) must be grouped and added together to find the fully simplified polynomial form.

Key Concept

Expanding perfect square binomials and distributing monomial terms correctly.
Question 85Question →

What is the exact numerical value of the variable pp that balances the given algebraic equation?

Fill in the blanks below

−2(3p−5)+4p=3(p+5)−20-2(3p - 5) + 4p = 3(p + 5) - 20

Value:
Show answer & explanation

Answer

3
The correct answer is found by first distributing the outer values into the parentheses, taking special care to distribute the negative sign accurately so that −2(−5)=10-2(-5) = 10. After expanding, the equation becomes −6p+10+4p=3p+15−20-6p + 10 + 4p = 3p + 15 - 20. Combining the like terms on each side reduces the equation to −2p+10=3p−5-2p + 10 = 3p - 5. Adding 2p2p to both sides and adding 55 to both sides isolates the terms, resulting in 15=5p15 = 5p. Finally, dividing both sides by 5 yields a value of 3.

Step-by-Step Solution

1
Distribute the constants into the parentheses on both sides of the equation.
−6p+10+4p=3p+15−20-6p + 10 + 4p = 3p + 15 - 20
Expanding the grouped terms is necessary to access and combine all the variable and constant terms.
2
Combine the like terms on each side of the equation separately.
−2p+10=3p−5-2p + 10 = 3p - 5
Simplifying each side independently reduces the complexity before moving variables across the equals sign.
3
Isolate the variable by moving all pp terms to one side and all constants to the opposite side.
10+5=3p+2p10 + 5 = 3p + 2p, which simplifies to 15=5p15 = 5p
Adding 2p2p and adding 55 to both sides gathers the unknown variable exclusively to the right and constants to the left.
4
Divide by the coefficient of the variable to solve for pp.
p=3p = 3
Dividing both sides by 5 leaves the variable pp completely isolated, providing the final solution.

Key Concept

Applying the distributive property and balancing operations to solve multi-step linear equations.
Question 86Question →

Consider the following algebraic equation:

5(2m−1)−3m=2(m+10) 5(2m - 1) - 3m = 2(m + 10)

What is the value of the variable mm?

Show answer & explanation

Answer: 5

Answer

5
By correctly applying the distributive property on both sides of the equation, combining the resulting like terms, and applying algebraic operations to isolate the variable, the equation simplifies to 5m=255m = 25, which yields exactly 55.

Step-by-Step Solution

1
Distribute the constants outside the parentheses into the terms inside.
10m−5−3m=2m+2010m - 5 - 3m = 2m + 20
This removes the parentheses and allows us to combine like terms.
2
Combine the like variable terms on the left side (10m10m and −3m-3m).
7m−5=2m+207m - 5 = 2m + 20
Simplifying each side independently makes the equation easier to solve.
3
Subtract 2m2m from both sides and add 55 to both sides.
5m=255m = 25
This groups all variable terms on the left and all constant terms on the right.
4
Divide both sides by 55.
m=5m = 5
This isolates the variable mm to give the final solution.

Key Concept

Solving one-variable linear equations requiring distribution and combining like terms
Estimated Time:1m 0s
Question 87Question →

Solve the following linear equation for kk:

3−2(4k−5)=−5(k+1) 3 - 2(4k - 5) = -5(k + 1)

What is the exact value of kk?

Show answer & explanation

Answer: 66

Answer

The correct value is 66.
The equation is correctly solved by first using the distributive property on both sides to get 3−8k+10=−5k−53 - 8k + 10 = -5k - 5. Combining the constants on the left side yields 13−8k=−5k−513 - 8k = -5k - 5. Adding 8k8k to both sides gives 13=3k−513 = 3k - 5. Finally, adding 55 to both sides results in 18=3k18 = 3k, and dividing by 33 gives the solution k=6k = 6.

Step-by-Step Solution

1
Distribute the coefficients into the parentheses on both sides of the equation.
3−8k+10=−5k−53 - 8k + 10 = -5k - 5
To eliminate the parentheses and expose all terms for simplification.
2
Combine the constant terms on the left side of the equation.
13−8k=−5k−513 - 8k = -5k - 5
To simplify the left side.
3
Add 8k8k to both sides of the equation.
13=3k−513 = 3k - 5
To gather all terms with the variable kk on the right side.
4
Add 55 to both sides of the equation.
18=3k18 = 3k
To isolate the variable term.
5
Divide both sides by 33.
k=6k = 6
To solve for kk.

Key Concept

Solving one-variable linear equations by applying the distributive property and combining like terms.
Question 88Question →

Multiply the binomial (3x−4)(3x - 4) by the trinomial (5x2−2x−7)(5x^2 - 2x - 7). After fully expanding the product and combining all like terms, what is the numerical coefficient of the xx term?

Show answer & explanation

Answer: -13

Answer

The coefficient of the x term is -13.
The coefficient of the xx term is found by locating all the xx terms in the expanded expression (−21x-21x and 8x8x) and adding their coefficients together (−21+8=−13-21 + 8 = -13).

Step-by-Step Solution

1
Distribute the term 3x3x to every term in the trinomial (5x2−2x−7)(5x^2 - 2x - 7).
3x(5x2)+3x(−2x)+3x(−7)=15x3−6x2−21x3x(5x^2) + 3x(-2x) + 3x(-7) = 15x^3 - 6x^2 - 21x
The distributive property requires multiplying each term of the first polynomial by each term of the second polynomial.
2
Distribute the term −4-4 to every term in the trinomial (5x2−2x−7)(5x^2 - 2x - 7).
−4(5x2)−4(−2x)−4(−7)=−20x2+8x+28-4(5x^2) - 4(-2x) - 4(-7) = -20x^2 + 8x + 28
This completes the distribution for the second term of the binomial, taking care to properly multiply the negative signs.
3
Write out the fully expanded polynomial before combining like terms.
15x3−6x2−21x−20x2+8x+2815x^3 - 6x^2 - 21x - 20x^2 + 8x + 28
Listing all terms together makes it easier to identify which terms share the same variables and exponents.
4
Identify and combine the like terms that contain exactly the variable xx to the first power.
−21x+8x=−13x-21x + 8x = -13x
The question specifically asks for the coefficient of the xx term, so we combine −21x-21x and 8x8x.

Key Concept

Multiplying polynomials and combining like terms
Question 89Question →

An architect is designing a rectangular layout where the area is represented by the algebraic expression 4y2+12y−404y^2 + 12y - 40. To determine the possible dimensions of the layout in terms of yy, the architect needs to rewrite this expression in its completely factored form. Which of the following represents the completely factored expression?

Show answer & explanation

Answer: 4(y+5)(y−2)4(y + 5)(y - 2)

Answer

The completely factored expression is 4(y+5)(y−2)4(y + 5)(y - 2).
The correct answer completely factors out the GCF of 4 from all terms and correctly factors the remaining trinomial y2+3y−10y^2 + 3y - 10 into (y+5)(y−2)(y + 5)(y - 2).

Step-by-Step Solution

1
Identify and extract the greatest common factor (GCF) from all terms in the expression 4y2+12y−404y^2 + 12y - 40.
4(y2+3y−10)4(y^2 + 3y - 10)
Factoring out the GCF simplifies the polynomial and makes the remaining trinomial easier to factor.
2
Find two integers that multiply to the constant term (-10) and add to the coefficient of the middle term (3).
The correct integers are 5 and -2.
These integers are required to construct the binomial factors of the simplified trinomial y2+3y−10y^2 + 3y - 10.
3
Write the trinomial as a product of two binomials using the integers found, and combine them with the GCF.
4(y+5)(y−2)4(y + 5)(y - 2)
This represents the original expression fully broken down into its prime polynomial factors.

Key Concept

Factoring Polynomials by Extracting the GCF and Grouping
Question 90Question →

Which of the following represents the completely factored form of the quadratic expression 3x2−3x−183x^2 - 3x - 18?

Show answer & explanation

Answer: 3(x−3)(x+2)3(x - 3)(x + 2)

Answer

The completely factored form is 3(x−3)(x+2)3(x - 3)(x + 2).
The correct answer properly extracts the GCF of 3 first, leaving the trinomial x2−x−6x^2 - x - 6. This trinomial factors into (x−3)(x+2)(x - 3)(x + 2) because −3-3 and 22 multiply to −6-6 and add up to −1-1.

Step-by-Step Solution

1
Identify and factor out the Greatest Common Factor (GCF) from all terms in the expression.
3(x2−x−6)3(x^2 - x - 6)
The coefficients 3,−3,3, -3, and −18-18 are all divisible by 33.
2
Find two numbers that multiply to the constant term (−6-6) and add to the coefficient of the middle term (−1-1).
The numbers are −3-3 and 22.
−3×2=−6-3 \times 2 = -6 and −3+2=−1-3 + 2 = -1.
3
Rewrite the quadratic trinomial as the product of two binomials using these numbers.
3(x−3)(x+2)3(x - 3)(x + 2)
This represents the final, completely factored form of the original expression.

Key Concept

Factoring out a Greatest Common Factor (GCF) followed by factoring a quadratic trinomial.
Question 91Question →

Pair the given polynomial expressions on the left with their fully factored equivalents on the right.

Click a left item, then click its matching right item

Items

4x2−254x^2 - 25
x2−8x+15x^2 - 8x + 15
3x2+12x−153x^2 + 12x - 15
2x3−8x2x^3 - 8x

Matches

Show answer & explanation

Answer

The correct pairings are: 4x2−254x^2 - 25 with (2x−5)(2x+5)(2x - 5)(2x + 5); x2−8x+15x^2 - 8x + 15 with (x−3)(x−5)(x - 3)(x - 5); 3x2+12x−153x^2 + 12x - 15 with 3(x+5)(x−1)3(x + 5)(x - 1); and 2x3−8x2x^3 - 8x with 2x(x−2)(x+2)2x(x - 2)(x + 2).
Each polynomial expression was broken down into its fundamental algebraic components by identifying patterns such as the difference of squares, standard trinomial factoring, and extracting the greatest common factor.

Step-by-Step Solution

1
Analyze the first expression 4x2−254x^2 - 25.
It factors to (2x−5)(2x+5)(2x - 5)(2x + 5).
It follows the difference of squares pattern a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b) with a=2xa = 2x and b=5b = 5.
2
Analyze the second expression x2−8x+15x^2 - 8x + 15.
It factors to (x−3)(x−5)(x - 3)(x - 5).
The constant term is positive and the middle coefficient is negative, so both factors must contain negative constants that multiply to 1515 and add to −8-8.
3
Analyze the third expression 3x2+12x−153x^2 + 12x - 15.
It factors to 3(x+5)(x−1)3(x + 5)(x - 1).
Factoring out the GCF of 33 gives 3(x2+4x−5)3(x^2 + 4x - 5). The resulting quadratic factors into binomials with constants 55 and −1-1.
4
Analyze the fourth expression 2x3−8x2x^3 - 8x.
It factors to 2x(x−2)(x+2)2x(x - 2)(x + 2).
Factoring out the GCF of 2x2x gives 2x(x2−4)2x(x^2 - 4). The expression x2−4x^2 - 4 is a difference of squares that further factors into (x−2)(x+2)(x - 2)(x + 2).

Key Concept

Factoring Polynomials
Estimated Time:2m 0s
Question 92Question →

If (x,y)(x, y) is the solution to the system of equations below, what is the value of the expression x+3yx + 3y?

3x−2y=143x - 2y = 14
x=y+5x = y + 5
Show answer & explanation

Answer: 1

Answer

The value of the expression is 1.
The system can be efficiently solved using substitution because the second equation already isolates xx. Substituting x=y+5x = y + 5 into the first equation yields 3(y+5)−2y=143(y + 5) - 2y = 14. Distributing the 33 gives 3y+15−2y=143y + 15 - 2y = 14, which simplifies to y=−1y = -1. Substituting this back into the second equation gives x=−1+5=4x = -1 + 5 = 4. Finally, the question asks for the value of x+3yx + 3y, which evaluates to 4+3(−1)=14 + 3(-1) = 1.

Step-by-Step Solution

1
Substitute the expression for xx from the second equation into the first equation.
3(y+5)−2y=143(y + 5) - 2y = 14
This isolates a single variable, allowing us to solve for yy.
2
Distribute the 3 and solve the resulting equation for yy.
3y+15−2y=14⇒y+15=14⇒y=−13y + 15 - 2y = 14 \Rightarrow y + 15 = 14 \Rightarrow y = -1
Expanding the terms allows combining like terms to find the value of yy.
3
Substitute y=−1y = -1 back into the second equation to determine xx.
x=−1+5⇒x=4x = -1 + 5 \Rightarrow x = 4
Both values are needed for the final expression.
4
Evaluate the target expression x+3yx + 3y using the values found for xx and yy.
4+3(−1)=4−3=14 + 3(-1) = 4 - 3 = 1
The question specifically asks for the evaluated value of this expression, not just the ordered pair.

Key Concept

Solving a system of linear equations using the substitution method and evaluating an algebraic expression.
Question 93Question →

Given the system of linear equations below, what is the value of yy?

5x+2y=125x + 2y = 12
3x−4y=23x - 4y = 2
Show answer & explanation

Answer: 1

Answer

The correct value for y is 1.
By multiplying the first equation by 2, we get 10x + 4y = 24. Adding this to the second equation (3x - 4y = 2) eliminates y, resulting in 13x = 26, which simplifies to x = 2. Substituting x = 2 back into the first equation gives 5(2) + 2y = 12, or 10 + 2y = 12. Subtracting 10 from both sides leaves 2y = 2, yielding the final correct answer of y = 1.

Step-by-Step Solution

1
Multiply the first equation by 2 to align the coefficients of y for elimination.
10x+4y=2410x + 4y = 24
Creating a +4y term in the first equation allows y to be eliminated when added to the -4y term in the second equation.
2
Add the modified first equation to the second equation.
(10x+3x)+(4y−4y)=24+2⇒13x=26(10x + 3x) + (4y - 4y) = 24 + 2 \Rightarrow 13x = 26
This eliminates the variable y, leaving a single-variable equation that can be easily solved for x.
3
Divide both sides by 13 to solve for x.
x=2x = 2
Isolating x provides the first part of the coordinate solution.
4
Substitute x = 2 back into the first original equation to solve for y.
5(2)+2y=12⇒10+2y=12⇒2y=2⇒y=15(2) + 2y = 12 \Rightarrow 10 + 2y = 12 \Rightarrow 2y = 2 \Rightarrow y = 1
Since the question asks specifically for the value of y, this final substitution step is required.

Key Concept

Solving Systems of Linear Equations using the Elimination Method
Question 94Question →

A logistics team is packing a shipment containing standard crates and heavy crates. A load of 44 standard crates and 33 heavy crates weighs a total of 170170 pounds. Another load consisting of 22 standard crates and 55 heavy crates weighs a total of 190190 pounds. Based on this information, what is the weight, in pounds, of a single heavy crate?

Show answer & explanation

Answer: 30

Answer

30
The correct answer is found by translating the word problem into the system of equations 4x+3y=1704x + 3y = 170 and 2x+5y=1902x + 5y = 190. By multiplying the second equation by −2-2, the xx terms are eliminated when the equations are added, yielding −7y=−210-7y = -210, which simplifies to y=30y = 30.

Step-by-Step Solution

1
Define variables for the unknown quantities.
Let xx be the weight of a standard crate and yy be the weight of a heavy crate.
To translate the word problem into a mathematical system of equations.
2
Set up a system of linear equations based on the given information.
Equation 1: 4x+3y=1704x + 3y = 170
Equation 2: 2x+5y=1902x + 5y = 190
To mathematically represent the total weights of the two different loads.
3
Use the elimination method to cancel out the xx variable. Multiply Equation 2 by −2-2.
−2(2x+5y)=−2(190)⇒−4x−10y=−380-2(2x + 5y) = -2(190) \Rightarrow -4x - 10y = -380
Creating a −4x-4x term allows the xx variables to be eliminated when added to the 4x4x in Equation 1.
4
Add the new Equation 2 to Equation 1.
(4x+3y)+(−4x−10y)=170−380⇒−7y=−210(4x + 3y) + (-4x - 10y) = 170 - 380 \Rightarrow -7y = -210
This leaves a single-variable equation that can be directly solved.
5
Solve for yy.
y=−210−7=30y = \frac{-210}{-7} = 30
To isolate yy, which represents the target value: the weight of the heavy crate.

Key Concept

Solving Systems of Linear Equations
Question 95Question →

The two linear equations below form a system.

2x+3y=122x + 3y = 12
2x−y=42x - y = 4

Based on the solution to this system, what is the value of the expression x+2yx + 2y?

Show answer & explanation

Answer: 7

Answer

The correct answer is 7.
The correct answer is found by first solving the system of equations. Subtracting the second equation from the first eliminates the x variable, giving 3y - (-y) = 12 - 4, which simplifies to 4y = 8 and y = 2. Substituting y = 2 into the second equation yields 2x - 2 = 4, meaning 2x = 6 and x = 3. Finally, evaluating the expression x + 2y with these values gives 3 + 2(2) = 3 + 4 = 7.

Step-by-Step Solution

1
Eliminate the x variable by subtracting the second equation from the first.
(2x−2x)+(3y−(−y))=12−4⇒4y=8(2x - 2x) + (3y - (-y)) = 12 - 4 \Rightarrow 4y = 8
Because the coefficient of x is identical in both equations, subtracting them cancels out the x terms entirely, leaving an equation with only y.
2
Solve for y.
y=2y = 2
Dividing both sides of the equation by 4 isolates the y variable.
3
Substitute the value of y back into the second equation to solve for x.
2x−(2)=4⇒2x=6⇒x=32x - (2) = 4 \Rightarrow 2x = 6 \Rightarrow x = 3
Using the known value of y allows us to find the corresponding x value for the system's ordered pair.
4
Evaluate the target expression x+2yx + 2y using the found values.
3+2(2)=3+4=73 + 2(2) = 3 + 4 = 7
This answers the specific question asked in the stem by applying the solution to the given expression.

Key Concept

Solving Systems of Linear Equations by Elimination
Question 96Question →

A motor pool contains a total of 24 vehicles, consisting exclusively of transport trucks and utility vehicles. The number of transport trucks is 6 less than twice the number of utility vehicles. How many utility vehicles are in the motor pool?

Show answer & explanation

Answer: 10; ten; 10 utility vehicles

Answer

10
By defining tt as transport trucks and uu as utility vehicles, we can create the system t+u=24t + u = 24 and t=2u−6t = 2u - 6. Substituting the value of tt into the first equation yields (2u−6)+u=24(2u - 6) + u = 24. Combining the terms gives 3u−6=243u - 6 = 24. Adding 6 to both sides results in 3u=303u = 30, and dividing by 3 shows there are 1010 utility vehicles.

Step-by-Step Solution

1
Define variables for the unknown quantities.
Let tt be the number of transport trucks and uu be the number of utility vehicles.
This allows us to translate the word problem into a system of algebraic equations.
2
Set up the system of equations based on the problem statement.
Equation 1: t+u=24t + u = 24 (total vehicles).
Equation 2: t=2u−6t = 2u - 6 (transport trucks are 6 less than twice the utility vehicles).
These equations accurately represent the given conditions in mathematical terms.
3
Substitute the expression for tt from Equation 2 into Equation 1.
(2u−6)+u=24(2u - 6) + u = 24
Substitution eliminates the variable tt, leaving a single-variable equation that can be solved directly for uu.
4
Combine like terms and solve for uu.
3u−6=24⇒3u=30⇒u=103u - 6 = 24 \Rightarrow 3u = 30 \Rightarrow u = 10
Isolating uu gives the exact number of utility vehicles requested by the problem.

Key Concept

Solving Systems of Linear Equations Using Substitution

Alternative Method

This can also be solved by elimination if we rearrange the second equation to −t+2u=6-t + 2u = 6. Adding this to t+u=24t + u = 24 yields 3u=303u = 30, which gives u=10u = 10.
Estimated Time:1m 0s
Question 97Question →

Calculate the solution (x,y)(x, y) for the system of linear equations shown below.

3x−2y=73x - 2y = 7
2x+4y=102x + 4y = 10

What is the value of the expression x−yx - y?

Show answer & explanation

Answer: 2

Answer

The correct value of the expression is 2.
The system can be efficiently solved using either substitution or elimination. By dividing the second equation by 2, we get x+2y=5x + 2y = 5, which means x=5−2yx = 5 - 2y. Substituting this into the first equation gives 3(5−2y)−2y=73(5 - 2y) - 2y = 7. Distributing the 3 results in 15−6y−2y=715 - 6y - 2y = 7, which simplifies to 15−8y=715 - 8y = 7. Subtracting 15 from both sides gives −8y=−8-8y = -8, so y=1y = 1. Substituting y=1y = 1 back into x=5−2yx = 5 - 2y yields x=5−2(1)=3x = 5 - 2(1) = 3. Finally, the requested expression is x−yx - y, which evaluates to 3−1=23 - 1 = 2.

Step-by-Step Solution

1
Simplify the second equation by dividing all terms by 2.
x+2y=5x + 2y = 5
Dividing the entire equation by a common factor makes the coefficients smaller and allows us to easily isolate xx for the substitution method.
2
Isolate xx in the simplified equation.
x=5−2yx = 5 - 2y
Expressing xx in terms of yy prepares us to substitute this expression into the first equation.
3
Substitute the expression for xx into the first equation and solve for yy.
3(5−2y)−2y=7⇒15−6y−2y=7⇒15−8y=7⇒−8y=−8⇒y=13(5 - 2y) - 2y = 7 \Rightarrow 15 - 6y - 2y = 7 \Rightarrow 15 - 8y = 7 \Rightarrow -8y = -8 \Rightarrow y = 1
This step eliminates xx entirely, creating a single-variable linear equation that can be solved to find the exact value of yy.
4
Substitute the value of yy back into the isolated xx equation to find xx.
x=5−2(1)⇒x=5−2⇒x=3x = 5 - 2(1) \Rightarrow x = 5 - 2 \Rightarrow x = 3
Now that we have the numerical value for yy, we can plug it back in to find the corresponding value for xx.
5
Evaluate the final expression x−yx - y using the calculated values.
3−1=23 - 1 = 2
The question asks specifically for the value of x−yx - y, not just the ordered pair.

Key Concept

To solve a system of linear equations, you can use substitution by isolating a variable in one equation and substituting it into the other, or use elimination by combining the equations to cancel out a variable.
Question 98Question →

If 5m+12=2m+275m + 12 = 2m + 27, what is the value of mm?

Show answer & explanation

Answer: 5

Answer

The correct value of the variable is 5.
By using inverse operations to collect the variable terms on one side and the constant terms on the other, the equation simplifies to 3m=153m = 15, which directly yields a result of 55.

Step-by-Step Solution

1
Move all terms containing the variable to one side of the equation.
3m+12=273m + 12 = 27
Subtracting 2m2m from both sides eliminates the variable from the right side and groups it on the left.
2
Move all constant terms to the opposite side of the equation.
3m=153m = 15
Subtracting 1212 from both sides isolates the term containing the variable.
3
Solve for the variable.
m=5m = 5
Dividing both sides by the coefficient 33 provides the final value of mm.

Key Concept

Solving Linear Equations with Variables on Both Sides
Estimated Time:45s
Question 99Question →

Which of the following represents the simplified form of the expression 4b5⋅2b−24b^5 \cdot 2b^{-2}?

Show answer & explanation

Answer: 8b38b^3

Answer

The simplified expression is 8b38b^3.
The correct expression is found by multiplying the numerical coefficients (4×2=84 \times 2 = 8) and adding the exponents for the common base bb (5+(−2)=35 + (-2) = 3), resulting in 8b38b^3.

Step-by-Step Solution

1
Multiply the numerical coefficients.
4×2=84 \times 2 = 8
When multiplying algebraic terms, their numerical coefficients are multiplied together.
2
Apply the product rule of exponents to the variable terms.
b5⋅b−2=b5+(−2)=b3b^5 \cdot b^{-2} = b^{5 + (-2)} = b^3
When multiplying exponential expressions with the same base, add their exponents.
3
Combine the coefficient and the variable term.
8b38b^3
This is the final simplified form of the expression.

Key Concept

When multiplying two exponential terms with the same base, multiply their coefficients and add their exponents.
Question 100Question →

Which simplified algebraic expression corresponds to each unsimplified expression? Match each item on the left with its correctly simplified equivalent on the right.

Click a left item, then click its matching right item

Items

4x+3x−54x + 3x - 5
2(3x+4)2(3x + 4)
9x−4x+29x - 4x + 2
3(x−2)+53(x - 2) + 5

Matches

Show answer & explanation

Answer

The correct pairings are: 4x+3x−54x + 3x - 5 matches with 7x−57x - 5; 2(3x+4)2(3x + 4) matches with 6x+86x + 8; 9x−4x+29x - 4x + 2 matches with 5x+25x + 2; and 3(x−2)+53(x - 2) + 5 matches with 3x−13x - 1.
Each expression is correctly matched to its simplified equivalent by strictly applying the distributive property to eliminate any grouping symbols, followed by combining like terms (terms with identical variables) and grouping constants.

Step-by-Step Solution

1
Simplify the first expression, 4x+3x−54x + 3x - 5.
7x−57x - 5
Combining the like variable terms (4x4x and 3x3x) gives 7x7x. The constant −5-5 remains unchanged.
2
Simplify the second expression, 2(3x+4)2(3x + 4).
6x+86x + 8
Applying the distributive property multiplies each term inside the parentheses by 22.
3
Simplify the third expression, 9x−4x+29x - 4x + 2.
5x+25x + 2
Combining the like variable terms (9x9x and −4x-4x) gives 5x5x. The constant 22 remains unchanged.
4
Simplify the fourth expression, 3(x−2)+53(x - 2) + 5.
3x−13x - 1
Distributing the 33 yields 3x−63x - 6. Combining the constant terms (−6-6 and 55) results in −1-1.

Key Concept

Simplifying Algebraic Expressions by Combining Like Terms and Distributing
PreviousPage 5 / 15Next
Algebra Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 5 | Examkin