System of Linear Equations: Using Graphs

System of Linear Equations: Using Graphs

How to solve the system of linear equations problems by using their graphs: number of solutions, examples, and their solutions.

Number of Solutions

One Solution: Consistent and Independent. Infinitely Many Solutions: Consistent and Dependent. No Solution: Inconsistent

Solving the system of linear equations means
finding (x, y)
that satisfy both of the linear equations.

So it can have three types of solutions.

The first type is having [one solution].

This is the case
when two linear equations are intersecting at a point:
the intersecting point is the solution.

This happens
when the linear equations' slopes are different.

If the system has one solution,
then it's called
the system is [consistent] and [independent].

[Consistent] means
the system has at least one solution.

[Independent] means
the graphs of the linear equations are different.
(The definition of independent
is way more complicated.
But let's think the term independent like this.)

The second type is having [infinitely many solutions].

This is the case
when the graphs of two linear equations are the same.

Every point on the line is the solution of the system.

In this case, it's called
the system is [consistent] and [dependent].

[Consistent] means
the system has at least one solution.

[Dependent] means
the graphs of the linear equations are the same.

The last type is having [no solution].

This is the case
when the graphs of two linear equations
don't have any intersecting points.

This happens when the linear equations are parallel.

Parallel Lines

In this case, it's called
the system is [inconsistent].

[Inconsistent] means the system has no solution.

Example 1: One Solution

Solve the system of linear equations by using their graphs. x - y = 4, 2x + y = 5

To graph the linear equations,
change the linear equations
into slope-intercept form.

First change x - y = 4.

Move x to the right side.

And multiply (-) on both sides.

Then y = x - 4.

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

Move 2x to the right side.

Then y = -2x + 5.

So the system of linear equations is

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

The slopes of the linear equations are different.

So this system has one solution.

Let's check this
by graphing the linear equations.

Graph y = x - 4.

Slope-intercept form - Example 2

Start from the y-intercept: -4.

The slope is 1.
So move 1 unit to the right
and move 1 unit upward.
Let's call this point the 'endpoint'.

Draw a line that passes through
the y-intercept and the endpoint.

Graph y = -2x + 5.

Start from the y-intercept: 5.

The slope is -2.
So move 1 unit to the right
and move 2 units downward.
Let's call this point the 'endpoint'.

Draw a line that passes through
the y-intercept and the endpoint.

Find the intersecting point.

The intersecting point is (3, -1).
So (3, -1) is the answer.

You can also write the answer as
x = 3, y = -1.

Example 2: Infinitely Many Solutions

Solve the system of linear equations by using their graphs. x - y = 4, 2x - 2y = 8

Change x - y = 4
into slope-intercept form.

Move x to the right side.

And multiply (-) on both sides.

Then y = x - 4.

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

Move 2x to the right side.

Divide both sides by -2.

Then y = x - 4.

So the system of linear equations is

y = x - 4
y = x - 4.

2x - 2y = 8 turns out to be y = x - 4.

So this system only has one linear equation:
y = x - 4.

So this system has infinitely many solutions.

Let's check this
by graphing the linear equations.

Graph y = x - 4.

Slope-intercept form - Example 2

Start from the y-intercept: -4.

The slope is 1.
So move 1 unit to the right
and move 1 unit upward.
Let's call this point the 'endpoint'.

Draw a line that passes through
the y-intercept and the endpoint.

Every point on y = x - 4
is the solution of the system.

So this system has infinitely many solutions.

Example 3: No Solution

Solve the system of linear equations by using their graphs. x - y = 4, x - y = -3

Change x - y = 4
into slope-intercept form.

Move x to the right side.

And multiply (-) on both sides.

Then y = x - 4.

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

Move x to the right side.

And multiply (-) on both sides.

Then y = x + 3.

So the system of linear equations is

y = x - 4
y = x + 3.

These two linear equations
have the same slope (1)
and have different y-intercepts (-4, +3).

So these two linear equations are parallel lines.

So this system has no solution.

Let's check this
by graphing the linear equations.

Graph y = x - 4.

Slope-intercept form - Example 2

Start from the y-intercept: -4.

The slope is 1.
So move 1 unit to the right
and move 1 unit upward.
Let's call this point the 'endpoint'.

Draw a line that passes through
the y-intercept and the endpoint.

Graph y = x - 3.

Start from the y-intercept: 3.

The slope is 1.
So move 1 unit to the right
and move 1 unit upward.
Let's call this point the 'endpoint'.

Draw a line that passes through
the y-intercept and the endpoint.

As you can see,
the linear equations are parallel lines.

So there's no intersecting point.

So this system has no solution.