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Solving Systems of Linear Equations

14 questions

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Question 1Question →

Determine the exact ordered pair (x,y)(x, y) that satisfies the system of linear equations shown below.

4x−3y=254x - 3y = 25
x+2y=9x + 2y = 9

Enter your answer in the format (x,y)(x, y).

Show answer & explanation

Answer: (7, 1); (7,1); 7, 1; 7,1

Answer

The correct ordered pair is (7, 1).
The correct solution is (7, 1). To verify, substitute x = 7 and y = 1 into both original equations. For the first equation: 4(7) - 3(1) = 28 - 3 = 25. For the second equation: 7 + 2(1) = 7 + 2 = 9. Since both equations hold true simultaneously, (7, 1) is the accurate solution.

Step-by-Step Solution

1
Isolate the variable xx in the second equation.
x=9−2yx = 9 - 2y
The second equation has an xx with a coefficient of 1, making it the most straightforward variable to isolate for substitution without creating fractions.
2
Substitute the expression for xx into the first equation.
4(9−2y)−3y=254(9 - 2y) - 3y = 25
This creates a single equation containing only the variable yy, which can be solved algebraically.
3
Distribute the 4 and combine like terms.
36−8y−3y=25⇒36−11y=2536 - 8y - 3y = 25 \Rightarrow 36 - 11y = 25
Simplifying the equation prepares it for isolating the yy term.
4
Solve for yy by subtracting 36 from both sides and dividing by -11.
−11y=−11⇒y=1-11y = -11 \Rightarrow y = 1
This determines the numerical value for the y-coordinate of the solution.
5
Substitute y=1y = 1 back into the isolated xx equation from Step 1.
x=9−2(1)⇒x=9−2=7x = 9 - 2(1) \Rightarrow x = 9 - 2 = 7
This provides the corresponding x-coordinate to complete the ordered pair.

Key Concept

Solving Systems of Linear Equations by Substitution
Question 2Question →

If 3x+5y=113x + 5y = 11 and 4x+9y=34x + 9y = 3, what is the value of the expression x+2yx + 2y?

Show answer & explanation

Answer: 2

Answer

2
The correct answer is 2. To solve the system, use the elimination method by multiplying the equations to create opposite or matching coefficients. Multiplying the first equation by 4 yields 12x+20y=4412x + 20y = 44. Multiplying the second equation by 3 yields 12x+27y=912x + 27y = 9. Subtracting the first equation from the second gives 7y=−357y = -35, so y=−5y = -5. Substituting y=−5y = -5 into the first original equation gives 3x−25=113x - 25 = 11, which simplifies to 3x=363x = 36, or x=12x = 12. Finally, substituting these values into the expression x+2yx + 2y gives 12+2(−5)=12−10=212 + 2(-5) = 12 - 10 = 2.

Step-by-Step Solution

1
Multiply the first equation by 4 and the second equation by 3 to prepare for elimination.
12x+20y=4412x + 20y = 44 and 12x+27y=912x + 27y = 9
Creating matching coefficients for xx allows us to eliminate it by subtracting the equations.
2
Subtract the first new equation from the second new equation.
(12x+27y)−(12x+20y)=9−44⇒7y=−35(12x + 27y) - (12x + 20y) = 9 - 44 \Rightarrow 7y = -35
This eliminates xx and leaves a simple equation with only yy.
3
Divide by 7 to solve for yy.
y=−5y = -5
Isolating yy gives its numerical value.
4
Substitute y=−5y = -5 back into the original first equation (3x+5y=113x + 5y = 11) and solve for xx.
3x+5(−5)=11⇒3x−25=11⇒3x=36⇒x=123x + 5(-5) = 11 \Rightarrow 3x - 25 = 11 \Rightarrow 3x = 36 \Rightarrow x = 12
This finds the corresponding value for xx.
5
Substitute x=12x = 12 and y=−5y = -5 into the requested expression x+2yx + 2y.
12+2(−5)=12−10=212 + 2(-5) = 12 - 10 = 2
The question asks for the value of this specific expression, not just the ordered pair.

Key Concept

Solving systems of linear equations using the elimination method and evaluating algebraic expressions.
Estimated Time:2m 0s
Question 3Question →

Given the system of equations below, what is the value of the expression x−2yx - 2y?

5x−13=4y5x - 13 = 4y
2x+y=132x + y = 13
Show answer & explanation

Answer: -1

Answer

The correct value of the expression is -1.
Solving the system of equations by rearranging and using elimination yields the solution x = 5 and y = 3. Substituting these values into the target expression x - 2y gives 5 - 2(3), which evaluates to 5 - 6 = -1.

Step-by-Step Solution

1
Rearrange the first equation into standard form to align the variables.
5x−4y=135x - 4y = 13
Aligning the variables vertically makes it much easier to use the elimination method.
2
Multiply the entire second equation by 4.
8x+4y=528x + 4y = 52
This creates opposite coefficients (-4y and +4y) so the y variable will cancel out when the equations are added.
3
Add the rearranged first equation and the modified second equation together.
(5x−4y)+(8x+4y)=13+52⇒13x=65⇒x=5(5x - 4y) + (8x + 4y) = 13 + 52 \Rightarrow 13x = 65 \Rightarrow x = 5
Eliminating y isolates x, allowing us to find its numerical value.
4
Substitute x = 5 back into the original second equation to solve for y.
2(5)+y=13⇒10+y=13⇒y=32(5) + y = 13 \Rightarrow 10 + y = 13 \Rightarrow y = 3
This completes the solution pair (x, y) for the system of equations.
5
Substitute x = 5 and y = 3 into the requested expression x−2yx - 2y.
5−2(3)=5−6=−15 - 2(3) = 5 - 6 = -1
The question asks for the evaluated value of this specific expression, not just the solution coordinates.

Key Concept

Solving Systems of Linear Equations using elimination or substitution, followed by algebraic evaluation.
Question 4Question →

Given the system of equations below, what is the value of xx?

3x+2y=163x + 2y = 16
x−2y=0x - 2y = 0
Show answer & explanation

Answer: 4; x=4; x = 4; four

Answer

4
Because the coefficients of yy in the two equations are opposites (+2+2 and −2-2), the fastest approach is the elimination method. Adding the equations top-to-bottom yields (3x+x)+(2y−2y)=16+0(3x + x) + (2y - 2y) = 16 + 0, which simplifies directly to 4x=164x = 16. Dividing both sides by 44 gives x=4x = 4.

Step-by-Step Solution

1
Add the two equations together.
4x=164x = 16
The 2y2y and −2y-2y terms are opposites, so adding the equations eliminates the yy variable entirely.
2
Divide both sides of the resulting equation by 4.
x=4x = 4
This isolates the variable xx to find its exact numerical value.

Key Concept

Solving systems of linear equations using the elimination method.
Question 5Question →

Which ordered pair (x,y)(x, y) is the solution to the following system of linear equations?

3x+y=173x + y = 17
x−y=3x - y = 3
Show answer & explanation

Answer: (5,2)(5, 2)

Answer

The correct solution is the ordered pair (5,2)(5, 2).
The correct answer is (5, 2). By adding the two equations, the y terms cancel each other out, leaving 4x = 20. Dividing both sides by 4 gives x = 5. Substituting this value back into the second equation (5 - y = 3) allows us to solve for y, which is 2.

Step-by-Step Solution

1
Add the two equations together to eliminate the y variable.
(3x+y)+(x−y)=17+3⇒4x=20(3x + y) + (x - y) = 17 + 3 \Rightarrow 4x = 20
Since the coefficients of y are opposites (1 and -1), adding the equations cancels out the y terms.
2
Solve for x by dividing both sides of the resulting equation by 4.
x=5x = 5
Dividing by the coefficient of x isolates the variable.
3
Substitute the value of x back into the second original equation.
5−y=35 - y = 3
Substituting the known x value leaves an equation with only one variable, y.
4
Solve for y.
−y=3−5⇒−y=−2⇒y=2-y = 3 - 5 \Rightarrow -y = -2 \Rightarrow y = 2
Isolating y provides the final part of the solution.

Key Concept

Solving systems of linear equations using the elimination method
Question 6Question →

Consider the following system of linear equations:

4(x−2)+y=3x 4(x - 2) + y = 3x
2x−3(y+1)=−2 2x - 3(y + 1) = -2

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

Show answer & explanation

Answer: 2

Answer

2
The correct answer is derived by first simplifying both equations into standard form (x + y = 8 and 2x - 3y = 1). By applying the elimination method, we find the system's solution is x = 5 and y = 3. Finally, evaluating the specific requested expression gives us x - y = 5 - 3 = 2.

Step-by-Step Solution

1
Simplify the first equation.
x + y = 8
Equations must be in standard form to easily apply elimination or substitution.
2
Simplify the second equation.
2x - 3y = 1
Expanding the terms and grouping variables on one side creates the standard form.
3
Multiply the simplified first equation by 3.
3x + 3y = 24
This aligns the y-coefficients (3y and -3y) for the elimination method.
4
Add the modified first equation to the simplified second equation.
5x = 25
Adding the equations eliminates the y variable, allowing us to solve for x.
5
Divide both sides by 5.
x = 5
Isolates the variable x.
6
Substitute x = 5 into the simplified first equation (x + y = 8).
y = 3
This finds the value of the remaining variable, y.
7
Evaluate the target expression (x - y).
5 - 3 = 2
The question specifically asks for the difference between x and y, not just their individual values.

Key Concept

Solving systems of linear equations involving pre-simplification
Question 7Question →

A cafe sells muffins and scones. A customer purchases 22 muffins and 11 scone for a total of 1111 dollars. If the price of one scone is 44 dollars less than the price of 33 muffins, what is the price of one scone in dollars?

Show answer & explanation

Answer: 5

Answer

The price of one scone is 55 dollars.
By translating the word problem into a system of two equations (2m+s=112m + s = 11 and s=3m−4s = 3m - 4) and using substitution, we find that m=3m = 3 and s=5s = 5. Thus, the price of one scone is 55.

Step-by-Step Solution

1
Define variables and set up the equations based on the word problem.
Let mm = price of a muffin and ss = price of a scone. The equations are 2m+s=112m + s = 11 and s=3m−4s = 3m - 4.
Translating the text into algebraic equations allows us to solve for the unknown values.
2
Substitute the expression for ss from the second equation into the first equation.
2m+(3m−4)=112m + (3m - 4) = 11
This eliminates the variable ss, leaving an equation with only one variable, mm.
3
Solve the resulting equation for mm.
5m−4=11⇒5m=15⇒m=35m - 4 = 11 \Rightarrow 5m = 15 \Rightarrow m = 3
Isolating mm gives us the price of one muffin.
4
Substitute the value of mm back into the equation for ss.
s=3(3)−4=9−4=5s = 3(3) - 4 = 9 - 4 = 5
This calculates the price of the scone, which is the final requested value.

Key Concept

Solving systems of linear equations using the substitution method.
Question 8Question →

Find the ordered pair (x,y)(x, y) that satisfies both equations below:

y=3x−2y = 3x - 2
y=x+6y = x + 6

Which of the following is the correct solution?

Show answer & explanation

Answer: (4,10)(4, 10)

Answer

The correct solution is the ordered pair (4,10)(4, 10).
By setting the two expressions for y equal to each other (3x−2=x+63x - 2 = x + 6), we can isolate x to find x=4x = 4. Substituting 44 back into either original equation gives y=10y = 10. Therefore, the solution is (4,10)(4, 10).

Step-by-Step Solution

1
Set the right sides of both equations equal to each other.
3x−2=x+63x - 2 = x + 6
Since both expressions are equal to y, they must be equal to each other.
2
Subtract xx from both sides to group the variables.
2x−2=62x - 2 = 6
This begins the process of isolating the variable x.
3
Add 22 to both sides of the equation.
2x=82x = 8
This isolates the term with the x variable on one side and the constant on the other.
4
Divide both sides by 22.
x=4x = 4
This fully isolates x to find its numerical value.
5
Substitute x=4x = 4 into the second equation to find yy.
y=4+6=10y = 4 + 6 = 10
Substituting the known x value back into either original equation yields the corresponding y value.

Key Concept

Solving systems of linear equations using the substitution method.
Question 9Question →

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 10Question →

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 11Question →

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 12Question →

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 13Question →

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 14Question →

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.