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Algebra 1 : How to find the solution for a system of equations

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #2 : Inequalities

Solve for .

4x+3y=6

2x+2y=4

Possible Answers:

x=0

$x=\frac{3}{4}$

x=-2

x>0

x=2

Correct answer:

x=0

Explanation:

4x+3y=6

2x+2y=4

For the second equation, solve for in terms of .

4-2x=2y

y=2-x

Plug this value of y into the first equation.

4x+3(2-x)=6

4x + 6- 3x =6

x=0

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Example Question #3 : Systems Of Equations

x+y=32

and

$x=\frac{2}{3}y-2$

Solve for and .

Possible Answers:

$y=11\frac{2}{5},x=20\frac{3}{5}$

$y=20\frac{2}{5},x=11\frac{3}{5}$

$y=20\frac{3}{5},x=11\frac{4}{5}$

$y=19\frac{2}{5},x=12\frac{3}{5}$

None of the available answers

Correct answer:

$y=20\frac{2}{5},x=11\frac{3}{5}$

Explanation:

x+y=32 rearranges to

x=32-y

and

$x=\frac{2}{3}y-2$ , so

$32-y=\frac{2}{3}y-2$

$32-1\frac{2}{3}y=-2$

$-1\frac{2}{3}y=-34$

$-\frac{5}{3}y=-34$

$y=\frac{-34}{1}\cdot\frac{-3}{5}=\frac{102}{5}=20\frac{2}{5}$

$x=32-20\frac{2}{5}=11\frac{3}{5}$

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Example Question #4 : Systems Of Equations

Solve for (x, y) in the system of equations:

y = 3x + 4

2x + 3y = 34

Possible Answers:

(5, 19)

The system has no solution

(4, 16)

(3, 13)

(2, 10)

Correct answer:

(2, 10)

Explanation:

In the second equation, you can substitute 3x + 4 for from the first.

2x + 3y = 34

2x + 3 (3x + 4) = 34

2x + 3 (3x) + 3 (4) = 34

2x + 9x + 12 = 34

11x + 12 = 34

11x = 22

x = 2

Now, substitute 2 for in the first equation:

y = 3x + 4

y = 3 (2) + 4

y = 6 + 4

y = 10

The solution is (2, 10)

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Example Question #35 : Solving Equations

Without drawing a graph of either equation, find the point where the two lines intersect.

Line 1 : y = 3x

Line 2 : y = x - 2

Possible Answers:

(2,0)

(-1,-3)

(1,3)

(0,-2)

(0,0)

Correct answer:

(-1,-3)

Explanation:

To find the point where these two lines intersect, set the equations equal to each other, such that is substituted with the side of the second equation. Solving this new equation for will give the -coordinate of the point of intersection.

3x = x - 2

Subtract from both sides.

(3x) - x = (x - 2) - x

2x = - 2

Divide both sides by 2.

$\frac{2x}{2} = \frac{-2}{2}$

x = - 1

Now substitute into either equation to find the -coordinate of the point of intersection.

y = 3x

y= 3(-1)

y = -3

With both coordinates, we know the point of intersection is (-1,-3) . One can plug in for and for in both equations to verify that this is correct.

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Example Question #24 : How To Find The Solution For A System Of Equations

What is the sum of and for the following system of equations?

3x - 5y = 5

-2x + 5y = 0

Possible Answers:

Correct answer:

Explanation:

3x - 5y = 5

-2x + 5y = 0

Add the equations together.

3x+(-2x)=x

-5y+5y=0

5+0=5

Put the terms together to see that $x+0=5\ \textup{or}\ x=5$ .

Substitute this value into one of the original equaitons and solve for .

3x - 5y = 5

3(5) - 5y = 5

-5y=-10

y=2

Now we know that $x=5\ \textup{and}\ y=2$ , thus we can find the sum of and .

x+y=5+2=7

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Example Question #52 : Equations / Solution Sets

Two lines have equations of y=3x+4 and y=-2x+9 . At what point do these lines intersect?

Possible Answers:

(0,13)

(5,-2)

(7,1)

(1,7)

(2,5)

Correct answer:

(1,7)

Explanation:

We can solve this problem by setting up a simple system of equations. First, we want to change the equations so one variable can cancel out. Multiplying the first equation by 2 and the second equation by 3 gives us a new system of 2y = 6x + 8 and 3y=-6x+27 . These equations add up to 5y=35 or y = 7 . Plugging in 7 for in either of the original two equations shows us that is equal to 1 and the point is (1,7) .

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Example Question #22 : How To Find The Solution For A System Of Equations

Does this system of equations have one solution, no solutions, or infinite solutions?

2x - y = -3

2y - 6 = 4x

Possible Answers:

one solution: (0,0)

no solution

one solution: (1,0)

infinite solutions

Correct answer:

infinite solutions

Explanation:

This system has infinite solutions becasue the two equations are actually the exact same line. To discover this, put both equations in terms of y.

First, 2x - y = -3 . Add y to both sides:

2x = -3 + y Now add 3 to both sides:

2x + 3 = y

Now we can show that the second equation also represents the line y=2x + 3

2y-6 = 4x add 6 to both sides

2y = 4x + 6 divide both sides by 2

y=2x+3

Since both equations are the same line, literally any point on one line will also be on the other - infinite solutions.

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Example Question #23 : How To Find The Solution For A System Of Equations

Find the solution for the system of equations.

y=2x+4 and y=x-2

Possible Answers:

y= -8 and x= -2

y= -6 and x= -8

y= -2 and x= -6

y= -8 and x= -6

y= 8 and x= 6

Correct answer:

y= -8 and x= -6

Explanation:

A system of equations can be solved by subsituting one variable for another. Since we know that y=x-2 , we can subsitute this into the other equation so x-2=2x-4 . This expression can be solved to find that x=-6 . Now that we know the value of it can be subsituted into either of the original equations to find .

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Example Question #61 : Equations / Solution Sets

$\\\textup{Two years ago, Pig was three times as old as Dog. }\\\textup{Next year, Pig will be three times as old as Cat. }\\\textup{In two years, Pig will be only 2.5 times as old as Cat.}\\\textup{ How old was Dog when Cat was born?}$

Possible Answers:

3.5

1.5

$\textup{Dog was not born yet.}$

Correct answer:

Explanation:

$\\\textup{Pig, Dog and Cat are } P, D, \textup{ and } C \textup{ years old, respectively.}\\\textup{The three statements translate in the following three equations, respectively}\\P-2=3(D-2)\\P+1=3(C+1)\\P+2=2.5(C+2).\\\textup{ Subtract the first from the second equation and obtain }\\(P-2)-(P+1)=(3D-6)-(3C+3)\\-3=3(D-C)-9\\D-C=2.\\\textup{So Dog is two years older than Cat.}$

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Example Question #22 : How To Find The Solution For A System Of Equations

Determine the intersection point of the following two equations.

3x+ 2y=4

5x-2y=4

Possible Answers:

(5,1)

(0,1)

$\left(\frac{1}{2},1\right)$

$\left(1,\frac{1}{2}\right)$

(3,4)

Correct answer:

$\left(1,\frac{1}{2}\right)$

Explanation:

To find the intersection point, you must find the values of x and y that satisfy the two equations. We can use the method of adding the two equations together:

3x+ 2y=4

5x-2y=4

If we add the equations together, the y terms cancel out, so we get

8x=8

x=1

Now that we know the value of x, we can plug that into one of the equations and solve for y. Pluggin it into the first equation, we get

3(1)+2y=4

2y=1

$y=\frac{1}{2}$ .

So that point of intersection is (x,y) , or $\left(1,\frac{1}{2}\right)$ .

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Algebra 1 : How to find the solution for a system of equations

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #2 : Inequalities

Example Question #3 : Systems Of Equations

Example Question #4 : Systems Of Equations

Example Question #35 : Solving Equations

Example Question #24 : How To Find The Solution For A System Of Equations

Example Question #52 : Equations / Solution Sets

Example Question #22 : How To Find The Solution For A System Of Equations

Example Question #23 : How To Find The Solution For A System Of Equations

Example Question #61 : Equations / Solution Sets

Example Question #22 : How To Find The Solution For A System Of Equations

All Algebra 1 Resources

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