# Linear Equation (Two Variables)

See how to write a linear equation (two variables)

and graph a linear inequality (two variables).

18 examples and their solutions.

- (x
_{1}, y_{1}), (x_{2}, y_{2})

m = y_{2}- y_{1}x_{2}- x_{1} - x-intercept

→ y = 0 - y-intercept

→ x = 0 - Slope: m, y-intercept: b

y = mx + b - Slope: m, (x
_{1}, y_{1})

y = m(x - x_{1}) + y_{1} - (x
_{1}, y_{1}), (x_{2}, y_{2})

y = y_{2}- y_{1}x_{2}- x_{1}(x - x_{1}) + y_{1} - x-intercept: a, y-intercept: b

xa + yb = 1 - Parallel lines

→ Same slopes - Perpendicular lines

→ m_{1}⋅m_{2}= -1 - Linear Inequality (Two Variables)

## Slope between Two Points

### Formula

m = y

_{2}- y

_{1}x

_{2}- x

_{1}

### Example

Slope between (1, 1), (3, 5)

Solution (1, 1), (3, 5)

m = 5 - 13 - 1

= 42

= 2

Graph

If a line goes ↗,

then the slope is (+).

m = 5 - 13 - 1

= 42

= 2

Graph

If a line goes ↗,

then the slope is (+).

### Example

Slope between (-4, 1), (1, -2)

Solution (-4, 1), (1, -2)

m = -2 - 11 - (-4)

= -31 + 4

= - 35

Graph

If a line goes ↘,

then the slope is (-).

m = -2 - 11 - (-4)

= -31 + 4

= - 35

Graph

If a line goes ↘,

then the slope is (-).

### Example

Slope between (-1, 2), (5, 2)

Solution (-1, 2), (5, 2)

m = 2 - 25 - (-1)

= 05 + 1

= 0

Graph

If a line is →,

then the slope is 0.

m = 2 - 25 - (-1)

= 05 + 1

= 0

Graph

If a line is →,

then the slope is 0.

### Example

Slope between (1, 2), (1, 3)

Solution (1, 2), (1, -3)

m = -3 - 21 - 1

= -50

→ No slope

Graph

If a line is ↕,

then the line has no slope.

m = -3 - 21 - 1

= -50

→ No slope

Graph

If a line is ↕,

then the line has no slope.

## x-Intercept

### Definition

of the graph and the x-axis.

### Example

x-intercept of 3x + 4y = 12

Solution 3x + 4y = 12

3x + 4⋅0 = 12- [1]

3x = 12

x = 4

[1] Put 0 in the y.

3x + 4⋅0 = 12- [1]

3x = 12

x = 4

[1] Put 0 in the y.

## y-Intercept

### Definition

of the graph and the y-axis.

### Example

y-intercept of 3x + 4y = 12

Solution 3x + 4y = 12

3⋅0 + 4y = 12- [1]

4y = 12

y = 3

[1] Put 0 in the x.

3⋅0 + 4y = 12- [1]

4y = 12

y = 3

[1] Put 0 in the x.

## Slope-Intercept Form

### Formula

y = mx + b

### Example

Slope: 2, y-intercept: 1

Linear equation?

Solution Linear equation?

m = 2

y-intercept: 1

y = 2x + 1

y-intercept: 1

y = 2x + 1

### Example

Graph y = 2x + 1.

Solution y = 2x + 1

Point the y-intercept + 1.

y = 2x + 1

The slope is 2.

So move [→ ×1] and [↑ ×2].

(2 = 2/1)

Mark this endpoint.

Draw a line that passes through

the y-intercept and the endpoint.

This is the graph of y = 2x + 1.

Close

Point the y-intercept + 1.

y = 2x + 1

The slope is 2.

So move [→ ×1] and [↑ ×2].

(2 = 2/1)

Mark this endpoint.

Draw a line that passes through

the y-intercept and the endpoint.

This is the graph of y = 2x + 1.

Close

### Example

5x + 3y = 6

→ slope-intercept form

Solution → slope-intercept form

5x + 3y = 6

3y = -5x + 6

y = -53x + 2

3y = -5x + 6

y = -53x + 2

## Point-Slope Form

### Formula

y = m(x - x

_{1}) + y

_{1}

### Example

Slope: 3

Point on the line: (1, 2)

Linear equation?

Solution Point on the line: (1, 2)

Linear equation?

m = 3

Point on the line: (1, 2)

y = 3(x - 1) + 2

= 3x - 3 + 2

y = 3x - 1- [1]

[1] Write the answer in [y = ...] form.

Point on the line: (1, 2)

y = 3(x - 1) + 2

= 3x - 3 + 2

y = 3x - 1- [1]

[1] Write the answer in [y = ...] form.

## Two-Point Form

### Formula

y = y

_{2}- y

_{1}x

_{2}- x

_{1}(x - x

_{1}) + y

_{1}

_{2}- y

_{1}x

_{2}- x

_{1}(x - x

_{2}) + y

_{2}

is also true.

### Example

Points on the line: (2, 1), (5, 4)

Linear equation?

Solution Linear equation?

(2, 1), (5, 4)

y = 4 - 15 - 2(x - 2) + 1

= 33⋅(x - 2) + 1

= 1⋅(x - 2) + 1

= x - 2 + 1

y = x - 1

y = 4 - 15 - 2(x - 2) + 1

= 33⋅(x - 2) + 1

= 1⋅(x - 2) + 1

= x - 2 + 1

y = x - 1

## Intercept Form

### Formula

xa + yb = 1

### Example

x-intercept: -3, y-intercept: 2

Linear equation?

Solution Linear equation?

x-intercept: -3

y-intercept: 2

x-3 + y2 = 1

y2 = x3 + 1

y = 23x + 2

y-intercept: 2

x-3 + y2 = 1

y2 = x3 + 1

y = 23x + 2

## Linear Equations of Parallel Lines

### Definition

that have the same slope

and have the different y-intercepts.

The red triangle symbols mean

these two lines are parallel.

### Example

Parallel to y = 3x + 4,

Passes through (1, -2)

Linear equation?

Solution Passes through (1, -2)

Linear equation?

y = 3x + 4

→ m = 3- [1]

(1, -2)

y = 3(x - 1) - 2- [2]

= 3x - 3 - 2

y = 3x - 5

[1] The line is parallel to y = 3x + 4.

So the slope of the line is 3.

[2] m = 3, Point on the line: (1, -2)

→ m = 3- [1]

(1, -2)

y = 3(x - 1) - 2- [2]

= 3x - 3 - 2

y = 3x - 5

[1] The line is parallel to y = 3x + 4.

So the slope of the line is 3.

[2] m = 3, Point on the line: (1, -2)

## Linear Equations of Perpendicular Lines

### Formula

m

_{1}⋅m

_{2}= -1

that form a right angle:

90 degrees.

The red symbol is the right angle symbol.

If the slopes of two perpendicular lines are

m

_{1}, m

_{2},

then m

_{1}⋅m

_{2}= -1.

### Example

Perpendicular to y = 3x + 4,

Passes through (3, 1)

Linear equation?

Solution Passes through (3, 1)

Linear equation?

y = 3x + 4

3⋅m = -1- [1]

m = -13

(3, 1)

y = -13(x - 3) + 1- [2]

= -13x + 1 + 1

y = -13x + 2

[1] The line is perpendicular to y = 3x + 4.

Set the slope of the line m.

Then 3⋅m = -1.

[2] m = -1/3, Point on the line: (3, 1)

3⋅m = -1- [1]

m = -13

(3, 1)

y = -13(x - 3) + 1- [2]

= -13x + 1 + 1

y = -13x + 2

[1] The line is perpendicular to y = 3x + 4.

Set the slope of the line m.

Then 3⋅m = -1.

[2] m = -1/3, Point on the line: (3, 1)

## Linear Inequality (Two Variables)

### Example

Graph y > x + 2.

Solution 1.

y > x + 2

> does not include '='.

So use a dashed line

to draw y = x + 2.

2.

y > x + 2

y is greater than the right side.

So color the upper region of the dashed line.

Close

y > x + 2

> does not include '='.

So use a dashed line

to draw y = x + 2.

2.

y > x + 2

y is greater than the right side.

So color the upper region of the dashed line.

Close

### Example

Graph y ≤ 3x - 4.

Solution 1.

y ≤ 3x - 4

≤ does include '='.

So use a solid line

to draw y = 3x - 4.

2.

y ≤ 3x - 4

y is less than or equal to the right side.

So color the lower region of the solid line.

Close

y ≤ 3x - 4

≤ does include '='.

So use a solid line

to draw y = 3x - 4.

2.

y ≤ 3x - 4

y is less than or equal to the right side.

So color the lower region of the solid line.

Close

### Example

Graph y < 1 on a coordinate plane.

Solution 1.

y < 1

< does not include '='.

So draw a horizontal dashed line

that passes through y = 1.

2.

y < 1

y is less than the right side.

So color the lower region of the dashed line.

Close

y < 1

< does not include '='.

So draw a horizontal dashed line

that passes through y = 1.

2.

y < 1

y is less than the right side.

So color the lower region of the dashed line.

Close

### Example

Graph x ≥ -2 on a coordinate plane.

Solution 1.

x ≥ -2

≥ does include '='.

So draw a vertical solid line

that passes through x = -2.

2.

x ≥ -2

x is greater than or equal to the right side.

So color the right side of the solid line.

Close

x ≥ -2

≥ does include '='.

So draw a vertical solid line

that passes through x = -2.

2.

x ≥ -2

x is greater than or equal to the right side.

So color the right side of the solid line.

Close