System of Linear Equations: Using Graph

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

Example 1

Example

Solution

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.

Let's 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.

Example 2

Example

Solution

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.

Let's 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 graph of 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.

Example 3

Example

Solution

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.

Let's 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.