# System of Linear Equations: Using Graph

How to solve a system of linear equations by using their graphs: 3 examples and their solutions.

## Examplex - y = 4, 2x + y = 5

(x, y) = (3, -1)

So (3, -1) means

x = 3 and y = -1.

To graph the linear equations,

change the linear equations to slope-intercept form.

Start from the first equation x - y = 4.

The slope-intercept form is y = x - 4.

Change 2x + y = 5 to slope-intercept form.

Then y = -2x + 5.

So the given linear equations

in slope-intercept form are

y = x - 4 and y = -2x + 5.

Graph these equations

on the coordinate plane.

Graph y = x - 4 on the coordinate plane.

Start from the y-intercept -4.

The slope is 1.

So move 1 unit to the right

and 1 unit upward.

Mark the endpoint.

Draw a line that passes through

the y-intercept -4 and the marked endpoint.

Graph y = -2x + 5 on the coordinate plane.

Start from the y-intercept +5.

The slope is -2.

So move 1 unit to the right

and 2 units downward.

Mark the endpoint.

Draw a line that passes through

the y-intercept +5 and the marked endpoint.

Find the intersecting point of the lines.

These two lines intersect at (3, -1).

So (3, -1) is the answer.

(x, y) = (3, -1)

So (3, -1) means

x = 3 and y = -1.

## Examplex - y = 4, 2x - 2y = 8

To graph the linear equations,

change the linear equations

to slope-intercept form.

The slope-intercept form of x - y = 4 is

y = x - 4.

Change 2x - 2y = 8 to slope-intercept form.

Then y = x - 4.

So the given linear equations

in slope-intercept form

are both y = x - 4.

Graph this equation

on the coordinate plane.

Graph the first linear equation y = x - 4

on the coordinate plane.

Start from the y-intercept -4.

The slope is 1.

So move 1 unit to the right

and 1 unit upward.

Mark the endpoint.

Draw a line that passes through

the y-intercept -4 and the marked endpoint.

The second linear equation is also y = x - 4.

So the second linear equation

is the same as the first one.

Find the intersecting point of

y = x - 4 and y = x - 4.

The graphs of the lines are the same.

So every point on the line

is the intersecting point.

So there are infinitely many intersecting points.

So

infinitely many solutions

is the answer.

## Examplex - y = 4, x - y = -3

To graph the linear equations,

change the linear equations

to slope-intercept form.

Start from the first equation x - y = 4.

The slope-intercept form is y = x - 4.

Change x - y = -3 to slope-intercept form.

Then y = x + 3.

So the given linear equations

in slope-intercept form are

y = x - 4 and y = x + 3.

Graph these equations

on the coordinate plane.

Graph y = x - 4 on the coordinate plane.

Start from the y-intercept -4.

The slope is 1.

So move 1 unit to the right

and 1 unit upward.

Mark the endpoint.

Draw a line that passes through

the y-intercept -4 and the marked endpoint.

Graph y = x + 3 on the coordinate plane.

Start from the y-intercept +3.

The slope is 1.

So move 1 unit to the right

and 1 unit upward.

Mark the endpoint.

Draw a line that passes through

the y-intercept +3 and the marked endpoint.

Find the intersecting point of

y = x - 4 and y = x + 3.

Both lines have the same slope: 1.

And they have different y-intercepts.

So these two lines are parallel lines.

So there are no intersecting points.

So

no solution

is the answer.