Solve a System of Equations by Graphing

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Question 1 of 7

1. Question
Find the solution to the system of equations by graphing.

$x - 3 = 0$ $x-3=0$

$x + y = 1$ $x+y=1$
- $x =$ $x=$ (3) $y =$ $y=$ (-2)
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First graph the equations.

Graph: $x - 3 = 0$ $x-3=0$

Make $x$ $x$ the subject to get the equation of the line.

$x - 3 + 3$ $x-3 \ color(crimson)(+3)$ $=$ $=$ $0 + 3$ $0 \ color(crimson)(+3)$ Add $3$ $3$ to both sides

$x \ cancel(-3) cancel(+3)$ $=$ $0 \ color(crimson)(+3)$

$x$ $=$ $3$

Graph: $x+y=1$

First, make $y$ the subject.

$x+y$ $=$ $1$

$y$ $=$ $-x+1$ Make $y$ the subject

Working out the y-values for each x-value in the table of values.

$x$ $0$ $1$ $2$ $3$

$y$ $1$ $0$ $-1$ $-2$

Plot the points and draw the graph.

The graphs intersect at the point $(3,-2)$

$:. color(darkviolet)(x=3, \ y=-2)$

$x=3, y=-2$
Question 2 of 7

2. Question
Find the solution to the system of equations by graphing.

$2x+y=5$

$5x-3y=7$
- $x=$ (2) $y=$ (1)
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First graph the equations.

Graph: $2x+y=5$

Make $y$ the subject to get the equation of the line.

$2x+y \ color(crimson)(-2x)$ $=$ $5 \ color(crimson)(-2x)$ Add $-2x$ to both sides

$y \ cancel(+2x) cancel(-2x)$ $=$ $5 \ color(crimson)(-2x)$

$y$ $=$ $-2x+5$

Working out the y-values for each x-value in the table of values.

$x$ $0$ $1$ $2$ $3$

$y$ $5$ $3$ $1$ $-1$

Plot the points and draw the graph.

Graph: $5x-3y=7$

First, make $y$ the subject.

$5x-3y$ $=$ $7$ Make $y$ the subject

$-3y$ $=$ $-5x+7$ Add $-5x$ to both sides

$y$ $=$ $5/3x-7/3$ Divide both sides by $-3$

Working out the y-values for each x-value in the table of values.

$x$ $0$ $1$ $2$ $3$

$y$ $-7/3$ $-2/3$ $1$ $8/3$

Plot the points and draw the graph.

The graphs intersect at the point $(2,1)$

$:. color(darkviolet)(x=2, \ y=1)$

$x=2, y=1$

Question 3 of 7

Find the solution to the system of equations by graphing.

$4x-3y=1$

$3x+2y=5$

$x=$ (1) $y=$ (1)

Question 4 of 7

Find the solution to the system of equations by graphing.

$x+3y=15$

$y-x=1$

$x=$ (3) $y=$ (4)

Question 5 of 7

5. Question
Find the solution to the system of equations by graphing.

$2x+y=2$

$x+y=2$
- $x=$ (0) $y=$ (2)
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First graph the equations.

Graph: $2x+y=2$

Make $y$ the subject to get the equation of the line.

$2x+y \ color(crimson)(-2x)$ $=$ $2 \ color(crimson)(-2x)$ Add $-2x$ to both sides

$y \ cancel(+2x) cancel(-2x)$ $=$ $2\ color(crimson)(-2x)$

$y$ $=$ $-2x+2$

Working out the y-values for each x-value in the table of values.

$x$ $0$ $1$ $2$ $3$

$y$ $2$ $0$ $-2$ $-4$

Plot the points and draw the graph.

Graph: $x+y=2$

First, make $y$ the subject.

$x+y$ $=$ $2$

$y$ $=$ $-x+2$ Make $y$ the subject

Working out the y-values for each x-value in the table of values.

$x$ $0$ $1$ $2$ $3$

$y$ $2$ $1$ $0$ $-1$

Plot the points and draw the graph.

The graphs intersect at the point $(0,2)$

$:. color(darkviolet)(x=0, \ y=2)$

$x=0, y=2$
Question 6 of 7

6. Question
Find the solution to the system of equations by graphing.

$y-x=4$

$x-y=1$
- $x=$ (no solution) $y=$ (no solution)
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First graph the equations.

Graph: $y-x=4$

Make $y$ the subject to get the equation of the line.

$y-x \ color(crimson)(+x)$ $=$ $4 \ color(crimson)(+x)$ Add $x$ to both sides

$y \ cancel(-x) cancel(+x)$ $=$ $4 \ color(crimson)(+x)$

$y$ $=$ $x+4$

Working out the y-values for each x-value in the table of values.

$x$ $-2$ $0$ $2$ $4$

$y$ $2$ $4$ $6$ $8$

Plot the points and draw the graph.

Graph: $x+y=1$

First, make $y$ the subject.

$x-y$ $=$ $1$

$y$ $=$ $x-1$ Make $y$ the subject

Working out the y-values for each x-value in the table of values.

$x$ $-2$ $0$ $2$ $4$

$y$ $-3$ $-1$ $1$ $3$

Plot the points and draw the graph.

The graphs are parallel the lines do not intersect.

There is no solution.

There is no solution.

Question 7 of 7

Find the solution to the system of equations by graphing.

$-6x+2y=8$

$y=3x+4$

$x=$ (infinite solutions) $y=$ (infinite solutions)

$4x-3y \ color(crimson)(-4x)$	$=$	$1 \ color(crimson)(-4x)$	Add $-4x$ to both sides
$-3y \ cancel(+4x) cancel(-4x)$	$=$	$1 \ color(crimson)(-4x)$
$\ color(crimson)(-3)y \ color(crimson)(\div(-3))$	$=$	$(-4x+1 )\ color(crimson)(\div(-3))$	Divide both sides by $-3$
$y$	$=$	$4/3x-1/3$

$x+3y \ color(crimson)(-x)$	$=$	$15 \ color(crimson)(-x)$	Add $-x$ to both sides
$3y \ cancel(+x) cancel(-x)$	$=$	$15 \ color(crimson)(-x)$
$\ color(crimson)(3)y\ color(crimson)(\div(3))$	$=$	$(-x+15) \ color(crimson)(\div(3))$	Divide both sides by $3$
$y$	$=$	$-1/3x+5$

$-6x+2y \ color(crimson)(+6x)$	$=$	$8 \ color(crimson)(+6x)$	Add $6x$ to both sides
$2y \ cancel(-6x) cancel(+6x)$	$=$	$8 \ color(crimson)(+6x)$
$\ color(crimson)(2)y\ color(crimson)(\div(2))$	$=$	$(6x+8) \ color(crimson)(\div(2))$	Divide both sides by $2$
$y$	$=$	$3x+4$

Solve a System of Equations by Graphing

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2. Question

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5. Question

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7. Question

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Quizzes

$x - 3 + 3$ $x-3 \ color(crimson)(+3)$	$=$ $=$	$0 + 3$ $0 \ color(crimson)(+3)$	Add $3$ $3$ to both sides
$x \ cancel(-3) cancel(+3)$	$=$	$0 \ color(crimson)(+3)$
$x$	$=$	$3$

$2x+y \ color(crimson)(-2x)$	$=$	$5 \ color(crimson)(-2x)$	Add $-2x$ to both sides
$y \ cancel(+2x) cancel(-2x)$	$=$	$5 \ color(crimson)(-2x)$
$y$	$=$	$-2x+5$

$5x-3y$	$=$	$7$	Make $y$ the subject
$-3y$	$=$	$-5x+7$	Add $-5x$ to both sides
$y$	$=$	$5/3x-7/3$	Divide both sides by $-3$

$3x+2y$	$=$	$5$	Make $y$ the subject
$2y$	$=$	$-3x+5$	Add $-3x$ to both sides
$y$	$=$	$-3/2x+5/2$	Divide both sides by $2$

$2x+y \ color(crimson)(-2x)$	$=$	$2 \ color(crimson)(-2x)$	Add $-2x$ to both sides
$y \ cancel(+2x) cancel(-2x)$	$=$	$2\ color(crimson)(-2x)$
$y$	$=$	$-2x+2$

$y-x \ color(crimson)(+x)$	$=$	$4 \ color(crimson)(+x)$	Add $x$ to both sides
$y \ cancel(-x) cancel(+x)$	$=$	$4 \ color(crimson)(+x)$
$y$	$=$	$x+4$

$x$	$0$	$1$	$2$	$3$
$y$	$1$	$0$	$-1$	$-2$

$x$	$0$	$1$	$2$	$3$
$y$	$-7/3$	$-2/3$	$1$	$8/3$

$x$	$0$	$1$	$2$	$3$
$y$	$-1/3$	$1$	$7/3$	$11/3$

$x$	$0$	$1$	$2$	$3$
$y$	$5/2$	$1$	$-1/2$	$-2$