Systems of Linear Equations Alıştırma Soruları — SAT — Sayfa 4

Soru 61Soru →

A tour guide offers two types of guided tours: a walking tour and a trolley tour. On Saturday, the guide led $3$ walking tours and $2$ trolley tours, which lasted a total of $8$ hours. On Sunday, the guide led $2$ walking tours and $4$ trolley tours, which lasted a total of $12$ hours. If each walking tour has a duration of $w$ hours and each trolley tour has a duration of $t$ hours, what is the total duration, in hours, of $1$ walking tour and $1$ trolley tour?

2.0

3.5

4.0

4.25

Cevabı ve açıklamayı göster

Cevap: 3.5

Cevap

The correct answer is 3.5.

The correct answer is

3.5

. The scenario can be represented by the system of linear equations

3w + 2t = 8

and

2w + 4t = 12

. Multiplying the first equation by

2

gives

6w + 4t = 16

. Subtracting the second equation from this yields

4w = 4

, which simplifies to

w = 1

. Substituting

w = 1

into the first equation yields

3(1) + 2t = 8

, which simplifies to

2t = 5

or

t = 2.5

. The sum of the durations of one walking tour and one trolley tour is

w + t = 1 + 2.5 = 3.5

hours.

Adım Adım Çözüm

1

Represent the given scenarios with a system of linear equations using the variables

w

(duration of a walking tour) and

t

(duration of a trolley tour).

The system of equations is:
1)

3w + 2t = 8

2)

2w + 4t = 12

This translates the word problem into mathematical equations.

2

Solve the system of equations for

w

and

t

. For example, multiply the first equation by

2

to align the

t

coefficients, and then subtract the second equation from it.

Multiplying the first equation by

2

gives

6w + 4t = 16

.
Subtracting the second equation (

2w + 4t = 12

) from this yields

4w = 4

, which simplifies to

w = 1

.

This eliminates the variable

t

to find the value of

w

.

3

Substitute

w = 1

back into one of the original equations to solve for

t

.

Using the first equation:

3(1) + 2t = 8 \implies 3 + 2t = 8 \implies 2t = 5 \implies t = 2.5

.

This determines the value of the second variable,

t

.

4

Calculate the sum of

w

and

t

to find the total duration of

1

walking tour and

1

trolley tour.

w + t = 1 + 2.5 = 3.5

.

This directly answers the question asked in the problem stem.

Anahtar Kavram

Solving a system of linear equations to find a linear combination of variables

Tahmini Süre:1m 30s

Soru 62Soru →

Consider the system of linear equations below:

\begin{aligned} 3x + 4y &= 8 \\ ax + 8y &= 15 \end{aligned}

If the system of equations has no solution, and $a$ is a constant, what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The correct answer is 6.

For a system of linear equations to have no solution, the lines representing the equations must be parallel, which requires them to have the same slope but different y-intercepts. Writing

3x + 4y = 8

in slope-intercept form gives

y = -\frac{3}{4}x + 2

, so its slope is

-\frac{3}{4}

. Writing

ax + 8y = 15

in slope-intercept form gives

y = -\frac{a}{8}x + \frac{15}{8}

, so its slope is

-\frac{a}{8}

. Equating the slopes gives

-\frac{3}{4} = -\frac{a}{8}

. Solving for

a

gives

a = 6

. Since the y-intercepts (

2

and

\frac{15}{8}

) are different, the lines are parallel and have no intersection points.

Adım Adım Çözüm

1

Convert the first equation

3x + 4y = 8

to slope-intercept form.

y = -\frac{3}{4}x + 2

To find the slope of the first line, which is

-\frac{3}{4}

.

2

Convert the second equation

ax + 8y = 15

to slope-intercept form.

y = -\frac{a}{8}x + \frac{15}{8}

To find the slope of the second line in terms of

a

, which is

-\frac{a}{8}

.

3

Equate the slopes of the two lines.

-\frac{3}{4} = -\frac{a}{8}

Parallel lines have the same slope, and a system with parallel lines has no solution.

4

Solve the equation for

a

.

a = 6

Isolating the variable

a

yields the value of the constant.

Anahtar Kavram

A system of linear equations has no solution if the lines represented by the equations are parallel, meaning they have the same slope but different y-intercepts.

Soru 63Soru →

The table below shows several values of $x$ and the corresponding values for the linear functions $f$ and $g$ .

$x$	$f(x)$	$g(x)$
1	7	17
2	9	14
4	13	8

If $(x, y)$ is the solution to the system of equations formed by $y = f(x)$ and $y = g(x)$ , what is the value of $2x + y$ ?

5

17

15

25

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

The correct value of the expression is 17.

The correct answer is the value obtained by finding the linear equations for both functions, solving the resulting system of equations, and substituting the coordinates into the expression. The equation for the first function is

y = 2x + 5

and the equation for the second function is

y = -3x + 20

. Equating the two expressions yields

2x + 5 = -3x + 20

, which simplifies to

5x = 15

or

x = 3

. Substituting

x = 3

back into either equation yields

y = 11

. Evaluating the expression

2x + y

with these values gives

2(3) + 11 = 17

.

Adım Adım Çözüm

1

Determine the linear equation for

f(x)

using the points in the table.

The slope of

f(x)

is

\frac{9 - 7}{2 - 1} = 2

. Using the point-slope form with

(1, 7)

, the equation is

y - 7 = 2(x - 1)

, which simplifies to

y = 2x + 5

.

Finding the equation of the first line is necessary to set up the system of linear equations.

2

Determine the linear equation for

g(x)

using the points in the table.

The slope of

g(x)

is

\frac{14 - 17}{2 - 1} = -3

. Using the point-slope form with

(1, 17)

, the equation is

y - 17 = -3(x - 1)

, which simplifies to

y = -3x + 20

.

Finding the equation of the second line completes the system of equations.

3

Find the intersection point of

y = f(x)

and

y = g(x)

by setting the two equations equal to each other.

2x + 5 = -3x + 20 \Rightarrow 5x = 15 \Rightarrow x = 3

. Substituting

x = 3

into

y = 2x + 5

gives

y = 2(3) + 5 = 11

.

Solving the system of equations yields the values of

x

and

y

at the intersection point.

4

Calculate the value of

2x + y

using the solution

(3, 11)

.

2(3) + 11 = 6 + 11 = 17

.

This evaluates the specific linear combination requested in the question.

Anahtar Kavram

Solving systems of linear equations by translating tabular data into linear equations and finding their point of intersection.

Soru 64Soru →

At a local farmer's market, a vendor sells two types of fruit baskets: a standard basket and a deluxe basket. The standard basket contains $3$ apples and $2$ oranges. The deluxe basket contains $5$ apples and $4$ oranges. If a customer bought a combination of these baskets containing a total of $31$ apples and $22$ oranges, what is the total number of baskets the customer bought?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The total number of baskets the customer bought is 9.

The correct answer is obtained by setting up a system of equations where

s

is the number of standard baskets and

d

is the number of deluxe baskets. The equations

3s + 5d = 31

and

2s + 4d = 22

represent the totals of apples and oranges respectively. Dividing the second equation by 2 gives

s + 2d = 11

, which can be rearranged to

s = 11 - 2d

. Substituting this expression into the first equation yields

3(11 - 2d) + 5d = 31

, which simplifies to

33 - d = 31

, meaning

d = 2

. Substituting

d = 2

back gives

s = 7

. The total number of baskets is the sum of both types,

7 + 2 = 9

.

Adım Adım Çözüm

1

Define variables and set up the system of linear equations.

Let

s

be the number of standard baskets and

d

be the number of deluxe baskets. The equations are:

3s + 5d = 31

(for apples) and

2s + 4d = 22

(for oranges).

To represent the relationships between the number of baskets and the total quantities of fruits mathematically.

2

Solve the system of equations using substitution.

From the second equation, dividing by 2 yields

s + 2d = 11

, so

s = 11 - 2d

. Substituting this into the first equation gives

3(11 - 2d) + 5d = 31

, which simplifies to

33 - d = 31

, leading to

d = 2

.

To find the number of deluxe baskets purchased.

3

Calculate the number of standard baskets and sum the two counts to find the total number of baskets.

Substitute

d = 2

into

s = 11 - 2d

to get

s = 7

. The total number of baskets is

s + d = 7 + 2 = 9

.

To answer the specific question asking for the total number of baskets bought.

Anahtar Kavram

Systems of Linear Equations

Soru 65Soru →

In a certain video game, players earn points by completing daily quests and weekly challenges. Last month, a player completed a total of $24$ activities, which consisted of daily quests and weekly challenges, and earned a total of $190$ points. Each daily quest completed was worth $5$ points, and each weekly challenge completed was worth $15$ points. How many weekly challenges did the player complete?

5

7

9

17

Cevabı ve açıklamayı göster

Cevap: $7$

Cevap

The player completed

7

weekly challenges.

To find the number of weekly challenges, we set up a system of linear equations. Let

d

be the number of daily quests and

w

be the number of weekly challenges. We have

d + w = 24

and

5d + 15w = 190

. Expressing

d

as

24 - w

and substituting it into the second equation gives

5(24 - w) + 15w = 190

, which simplifies to

120 - 5w + 15w = 190

. Combining like terms yields

120 + 10w = 190

, so

10w = 70

, which gives

w = 7

. This matches the correct value.

Adım Adım Çözüm

1

Represent the situation with a system of linear equations.

Let

d

represent the number of daily quests completed and

w

represent the number of weekly challenges completed. The system is:

\begin{aligned} d + w &= 24 \\ 5d + 15w &= 190 \end{aligned}

This sets up the mathematical model using variables for the two unknown quantities.

2

Express one variable in terms of the other using the first equation.

d = 24 - w

This allows for substitution into the second equation to reduce the system to a single variable.

3

Substitute the expression into the second equation and expand.

5(24 - w) + 15w = 190 \implies 120 - 5w + 15w = 190

This substitutes the representation of

d

to solve for

w

directly.

4

Simplify the equation and solve for

w

.

120 + 10w = 190 \implies 10w = 70 \implies w = 7

Combining like terms and isolating the variable yields the number of weekly challenges.

Anahtar Kavram

Solving systems of linear equations in two variables using substitution or elimination.

Alternatif Yöntem

Instead of substitution, the elimination method can be used. Multiply the first equation,

d + w = 24

, by

-5

to get

-5d - 5w = -120

. Add this equation to the second equation,

5d + 15w = 190

, to eliminate

d

, resulting in

10w = 70

, which simplifies to

w = 7

.

Tahmini Süre:1m 30s

Soru 66Soru →

A landscaping company sells two types of soil mixtures: a basic mixture and a premium mixture. Each bag of basic mixture contains $4$ pounds of compost and $8$ pounds of sand. Each bag of premium mixture contains $6$ pounds of compost and $5$ pounds of sand. A landscaper purchases a combination of bags containing a total of $46$ pounds of compost and $57$ pounds of sand. How many bags of premium mixture did the landscaper purchase?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The system of equations representing the scenario is

4b + 6p = 46

for compost and

8b + 5p = 57

for sand, where

b

represents the number of bags of basic mixture and

p

represents the number of bags of premium mixture. Multiplying the first equation by

2

gives

8b + 12p = 92

. Subtracting the second equation from this yields

(8b + 12p) - (8b + 5p) = 92 - 57

, which simplifies to

7p = 35

. Dividing by

7

gives

p = 5

. Thus, the landscaper purchased

5

bags of premium mixture.

Adım Adım Çözüm

1

Define variables and write the system of equations based on the given context.

Let

b

represent the number of basic mixture bags and

p

represent the number of premium mixture bags. The system is:

\begin{aligned} 4b + 6p &= 46 \\ 8b + 5p &= 57 \end{aligned}

Translating the word problem into a system of linear equations is necessary to solve for the unknowns.

2

Multiply the first equation by

2

to facilitate the elimination method.

8b + 12p = 92

Aligning the coefficients of

b

allows us to eliminate

b

by subtracting the two equations.

3

Subtract the second equation from the new equation.

(8b + 12p) - (8b + 5p) = 92 - 57 \implies 7p = 35

This isolates the variable

p

by eliminating the variable

b

.

4

Solve for

p

.

p = 5

Dividing both sides of the equation by

7

gives the final number of premium mixture bags.

Anahtar Kavram

Solving systems of linear equations in context

Soru 67Soru →

\begin{aligned} 3x - 2y &= 14 \\ 5x + 6y &= 42 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x - y$ ?

2

4

5

8

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

x - y

is

4

.

To find the value of

x - y

, we first solve the system of linear equations. Multiplying the first equation,

3x - 2y = 14

, by

3

gives

9x - 6y = 42

. Adding this equation to the second equation,

5x + 6y = 42

, eliminates

y

and yields

14x = 84

, which simplifies to

x = 6

. Substituting

x = 6

back into the first equation gives

3(6) - 2y = 14

, or

18 - 2y = 14

, which simplifies to

2y = 4

, so

y = 2

. Therefore, the value of

x - y

is

6 - 2 = 4

.

Adım Adım Çözüm

1

Multiply the first equation by

3

to align the coefficients of the

y

terms for elimination.

9x - 6y = 42

This makes the coefficients of

y

opposite in sign and equal in magnitude to the second equation.

2

Add the modified first equation to the second equation to eliminate

y

and solve for

x

.

(9x - 6y) + (5x + 6y) = 42 + 42 \implies 14x = 84 \implies x = 6

Adding the equations eliminates

y

, resulting in a single-variable linear equation.

3

Substitute

x = 6

back into the first equation to solve for

y

.

3(6) - 2y = 14 \implies 18 - 2y = 14 \implies -2y = -4 \implies y = 2

Substituting the known value of

x

yields the value of the other coordinate.

4

Evaluate the expression

x - y

using the values

x = 6

and

y = 2

.

x - y = 6 - 2 = 4

This computes the requested quantity.

Anahtar Kavram

Solving systems of linear equations using the elimination method and evaluating a linear combination of the variables.

Soru 68Soru →

\begin{aligned} 5x + 3y &= 22 \\ 3x + 5y &= 18 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x - y$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of the expression

x - y

is 2.

Subtracting the second equation,

3x + 5y = 18

, from the first equation,

5x + 3y = 22

, yields

(5x - 3x) + (3y - 5y) = 22 - 18

, which simplifies to

2x - 2y = 4

. Dividing both sides of this equation by 2 isolates the expression

x - y

and yields a final value of 2.

Adım Adım Çözüm

1

Subtract the second equation from the first equation.

2x - 2y = 4

Subtracting the equations aligns the coefficients of

x

and

y

to form a multiple of the target expression

x - y

.

2

Divide both sides of the equation by 2.

x - y = 2

Dividing the expression

2x - 2y

by 2 isolates the target expression

x - y

.

Anahtar Kavram

Solving systems of linear equations by linear combination and algebraic manipulation.

Soru 69Soru →

An online retailer offers two shipping options for heavy packages. The first option charges a flat fee of $15.00$ dollars plus $1.50$ dollars per pound of the package's weight. The second option charges a flat fee of $27.00$ dollars plus $0.70$ dollars per pound of the package's weight. If a package of weight $w$ pounds costs the same total amount of $c$ dollars under both options, what is the value of $w$ ?

1.5

37.5

52.5

15

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15

To find the weight of the package that yields the same cost under both shipping options, we set the cost expressions equal to each other. The first option is represented by

c = 1.50w + 15

and the second by

c = 0.70w + 27

. Setting them equal gives

1.50w + 15 = 0.70w + 27

. Subtracting

0.70w

from both sides results in

0.80w + 15 = 27

. Subtracting

15

from both sides gives

0.80w = 12

. Dividing by

0.80

yields

w = 15

. Therefore, the weight of the package is 15 pounds.

Adım Adım Çözüm

1

Set up the linear equations representing the total cost c for each shipping option based on the package's weight w.

The first option is represented by

c = 1.50w + 15

, and the second option is represented by

c = 0.70w + 27

.

This models the real-world scenario into a system of two linear equations with two variables.

2

Set the two expressions for the total cost c equal to each other to solve for the weight w.

1.50w + 15 = 0.70w + 27

Since the package costs the same under both options, the cost values are equal at this specific weight.

3

Isolate the variable w by subtracting

0.70w

and

15

from both sides of the equation.

0.80w = 12

, which simplifies to

w = 15

.

Subtracting

0.70w

from both sides isolates the variable terms on the left, and subtracting

15

isolates the constants on the right. Dividing by

0.80

solves for w.

Anahtar Kavram

Solving a system of linear equations by setting the expressions equal to find the point where two rates intersect.

Tahmini Süre:1m 30s

Soru 70Soru →

A manufacturing plant uses two types of machines, Machine X and Machine Y, to produce a certain component. Machine X produces $15$ components per hour, and Machine Y produces $25$ components per hour. On a certain day, both machines were operated for a combined total of $14$ hours, producing a total of $270$ components. For how many hours was Machine X operated?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

Machine X was operated for 8 hours.

By setting up the system of linear equations where

x

represents the hours Machine X operated and

y

represents the hours Machine Y operated, we get

x + y = 14

and

15x + 25y = 270

. Substituting

y = 14 - x

into the second equation yields

15x + 25(14 - x) = 270

. Expanding and simplifying results in

-10x + 350 = 270

, which solves to

x = 8

. Thus, Machine X operated for 8 hours.

Adım Adım Çözüm

1

Define variables for the hours each machine operated and set up the system of linear equations.

Let

x

be the hours Machine X was operated and

y

be the hours Machine Y was operated. The system is:

x + y = 14

15x + 25y = 270

This translates the verbal descriptions in the problem into algebraic expressions and equations.

2

Express

y

in terms of

x

from the first equation and substitute it into the second equation.

y = 14 - x

15x + 25(14 - x) = 270

Substitution reduces the system from two variables to a single equation in terms of

x

.

3

Solve the linear equation for

x

by distributing the constant and combining like terms.

15x + 350 - 25x = 270

-10x + 350 = 270

-10x = -80

x = 8

Isolating

x

gives the specific number of hours Machine X was operated.

Anahtar Kavram

Solving systems of linear equations in two variables using substitution or elimination.

Alternatif Yöntem

We can solve the system by multiplying the first equation by 25 to get

25x + 25y = 350

. Subtracting the second equation

15x + 25y = 270

from this yields

10x = 80

, which directly gives

x = 8

.

Tahmini Süre:1m 30s

Soru 71Soru →

Given the system of equations below:

\begin{aligned} 3x - y &= 11 \\ 2x + 3y &= 11 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x + 2y$ ?

4

6

8

9

Cevabı ve açıklamayı göster

Cevap: $6$

Cevap

The value of

x + 2y

is

6

.

To solve the system, we can isolate

y

in the first equation to get

y = 3x - 11

. Substituting this into the second equation gives

2x + 3(3x - 11) = 11

, which simplifies to

11x - 33 = 11

, or

11x = 44

, resulting in

x = 4

. Substituting

x = 4

back into the equation for

y

gives

y = 3(4) - 11 = 1

. The value of the expression

x + 2y

is therefore

4 + 2(1) = 6

.

Adım Adım Çözüm

1

Isolate

y

in the first equation.

y = 3x - 11

To prepare for substitution into the second equation.

2

Substitute the expression for

y

into the second equation and solve for

x

.

2x + 3(3x - 11) = 11 \implies 11x - 33 = 11 \implies 11x = 44 \implies x = 4

To find the value of

x

.

3

Substitute the value of

x

back into the isolated equation for

y

to solve for

y

.

y = 3(4) - 11 \implies y = 1

To find the value of

y

.

4

Evaluate the expression

x + 2y

using the values of

x

and

y

.

4 + 2(1) = 6

To find the final requested value.

Anahtar Kavram

Solving a system of linear equations using substitution or elimination, and evaluating a linear combination of the variables.

Alternatif Yöntem

Alternatively, the system can be solved using the elimination method. Multiply the first equation by

3

to align the

y

-coefficients:

9x - 3y = 33

. Add this equation to the second equation,

2x + 3y = 11

, to eliminate

y

, yielding

11x = 44

, so

x = 4

. Then substitute

x = 4

into either original equation to find

y = 1

.

Tahmini Süre:1m 30s

Soru 72Soru →

\begin{aligned} 4x - 3y &= 25 \\ -2x + 5y &= -9 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x - y$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

To solve the system of equations, we can use the elimination method. First, multiply the second equation by 2 to align the coefficients of

x

:

2(-2x + 5y) = 2(-9) \implies -4x + 10y = -18

Next, add this new equation to the first equation to eliminate

x

:

(4x - 3y) + (-4x + 10y) = 25 + (-18)

7y = 7

y = 1

Substitute

y = 1

back into the second equation to solve for

x

:

-2x + 5(1) = -9

-2x + 5 = -9

-2x = -14

x = 7

Finally, calculate the value of the requested expression

x - y

:

x - y = 7 - 1 = 6

Thus, the correct response is 6.

Adım Adım Çözüm

1

Multiply the second equation by 2 to prepare for the elimination of

x

.

-4x + 10y = -18

This creates coefficients for

x

in both equations that are additive opposites, allowing

x

to be eliminated when the equations are added.

2

Add the first equation and the modified second equation together to solve for

y

.

7y = 7

, which simplifies to

y = 1

Adding the equations eliminates the

x

terms and leaves a single-variable equation in terms of

y

.

3

Substitute

y = 1

back into one of the original equations to solve for

x

.

-2x + 5(1) = -9

, which simplifies to

-2x = -14

, yielding

x = 7

Now that the value of

y

is known, it can be substituted into either equation to find the corresponding value of

x

.

4

Calculate the value of the expression

x - y

.

7 - 1 = 6

The question asks specifically for the value of the difference

x - y

, so we subtract the value of

y

from the value of

x

.

Anahtar Kavram

Solving systems of linear equations using the elimination method and evaluating linear combinations of the solutions.

Tahmini Süre:1m 30s

Soru 73Soru →

\begin{aligned} k x - 5y &= 8 \\ 3x - 2y &= 12 \end{aligned}

In the system of equations above, $k$ is a constant. If the system has no solution, what is the value of $k$ ?

\frac{6}{5}

\frac{10}{3}

\frac{15}{2}

-\frac{15}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{15}{2}$

Cevap

\frac{15}{2}

The correct answer is the value that makes the two lines parallel. Rewriting both equations in slope-intercept form gives the slopes

\frac{k}{5}

and

\frac{3}{2}

. Equating these slopes yields

\frac{k}{5} = \frac{3}{2}

, which simplifies to

\frac{15}{2}

. Since their

y

-intercepts (

-\frac{8}{5}

and

-6

) are distinct, the lines are parallel and never intersect, meaning the system has no solution.

Adım Adım Çözüm

1

Convert the first equation to slope-intercept form (

y = mx + b

).

kx - 5y = 8 \implies -5y = -kx + 8 \implies y = \frac{k}{5}x - \frac{8}{5}

This identifies the slope (

m_1 = \frac{k}{5}

) and the

y

-intercept (

b_1 = -\frac{8}{5}

) of the first line.

2

Convert the second equation to slope-intercept form.

3x - 2y = 12 \implies -2y = -3x + 12 \implies y = \frac{3}{2}x - 6

This identifies the slope (

m_2 = \frac{3}{2}

) and the

y

-intercept (

b_2 = -6

) of the second line.

3

Equate the two slopes to find the value of

k

that makes the lines parallel.

\frac{k}{5} = \frac{3}{2} \implies k = 5 \times \frac{3}{2} = \frac{15}{2}

A system of linear equations has no solution if the lines are parallel (equal slopes) and distinct (different

y

-intercepts). Since the

y

-intercepts

-\frac{8}{5}

and

-6

are different, setting the slopes equal guarantees no solution.

Anahtar Kavram

A system of two linear equations has no solution if the lines represented by the equations have the same slope but different

y

-intercepts (parallel lines).

Tahmini Süre:1m 30s

Soru 74Soru →

A chemist mixes a $10\%$ acid solution and a $30\%$ acid solution to create $200$ milliliters of a mixture that is $18\%$ acid. How many milliliters of the $30\%$ acid solution are in the mixture?

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

80

The correct answer is

80

milliliters. By translating the problem into a system of linear equations, we let

x

be the volume of the

10\%

acid solution and

y

be the volume of the

30\%

acid solution. Since the total volume is

200

milliliters,

x + y = 200

. The amount of acid in the solutions must sum to the amount of acid in the final mixture, so

0.10x + 0.30y = 0.18(200)

, which simplifies to

0.10x + 0.30y = 36

. Multiplying this equation by

10

gives

x + 3y = 360

. Subtracting the first equation from this yields

(x + 3y) - (x + y) = 360 - 200

, or

2y = 160

. Solving for

y

gives

80

milliliters of the

30\%

solution.

Adım Adım Çözüm

1

Define variables and set up the system of equations.

Let

x

be the number of milliliters of the

10\%

acid solution and

y

be the number of milliliters of the

30\%

acid solution. The total volume equation is

x + y = 200

. The total acid content equation is

0.10x + 0.30y = 0.18(200)

.

This sets up the system of linear equations representing the physical constraints of the mixture.

2

Simplify the acid content equation and prepare for elimination.

0.10x + 0.30y = 36

. Multiplying the entire equation by

10

yields

x + 3y = 360

.

Eliminating decimals simplifies the coefficients and makes it easier to solve the system using integer arithmetic.

3

Eliminate

x

by subtracting the total volume equation from the simplified acid content equation.

(x + 3y) - (x + y) = 360 - 200 \implies 2y = 160 \implies y = 80

.

This isolates the variable

y

, which directly represents the volume of the

30\%

acid solution requested in the problem.

Anahtar Kavram

Solving systems of linear equations in context

Soru 75Soru →

In the system of equations below, $x$ and $y$ are variables.

2x - y = 5

3x - 2y = -17

What is the value of $x$ ?

7

25

27

49

Cevabı ve açıklamayı göster

Cevap: 27

Cevap

The value of

x

is 27.

Isolating

y

from the first equation gives

y = 2x - 5

. Substituting this expression into the second equation yields

3x - 2(2x - 5) = -17

. Distributing the

-2

gives

3x - 4x + 10 = -17

, which simplifies to

-x + 10 = -17

. Subtracting

10

from both sides results in

-x = -27

, meaning

x = 27

. Alternatively, multiplying the first equation by

2

gives

4x - 2y = 10

, and subtracting the second equation from it eliminates

y

directly to give

x = 10 - (-17) = 27

.

Adım Adım Çözüm

1

Isolate the variable

y

in the first equation,

2x - y = 5

.

y = 2x - 5

This allows for substitution of the expression for

y

into the second equation.

2

Substitute the expression for

y

into the second equation,

3x - 2y = -17

.

3x - 2(2x - 5) = -17

To create a single-variable equation in terms of

x

.

3

Distribute the coefficient

-2

to the terms inside the parentheses and simplify the equation.

3x - 4x + 10 = -17

, which simplifies to

-x + 10 = -17

Applying the distributive property correctly, noting that

-2 \times -5 = 10

.

4

Subtract

10

from both sides of the equation to isolate the term with

x

.

-x = -27

To isolate the variable term on one side of the equation.

5

Multiply both sides of the equation by

-1

to solve for

x

.

x = 27

To find the positive value of the variable

x

.

Anahtar Kavram

Solving systems of linear equations using the substitution method.

Tahmini Süre:1m 30s

Systems of Linear Equations

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