8th Grade Algebra Review - Graphing Linear Equations  

Linear equations can be graphed on the coordinate plane.

The slope of the graph of a linear equation is defined as the change in y-coordinates of two points on the graph divided by the change in x-coordinates of the same two points.

slope =    x₂ – x₁

          y₂  - y₁

The graph of a linear equation can have a positive slope, a negative slope, a zero slope or an undefined slope.

Slope can be used to describe the rate of change. For example, the rate of change of the position of a car traveling along a road is the distance traveled in miles divided by the time to travel that distance. This rate of change is also called velocity or speed of the car in miles per hour.

Direct variation is described by a linear equation in the form of y = kx, where k ≠ 0.

k is the constant of variation.

The graphs of all linear equations in the form of y = kx pass through the origin. 

In this graph, the slope = 1, y = 1x. Remember, the slope can be determined from any two points on the graph. One of the points on all graphs of a direct variation is the origin (0,0), so we only need one other point to determine the slope. Therefore, for these equations, the slope is y/x for any point on the graph. When x = 3, y = 3. The slope = 3/3 = 1.

Slope - Intercept form of a linear equation - y = mx + b,

where m = the slope, and b = the y-intercept of the graph of the linear equation.

In the graph above, the slope is 1, and the y-intercept is 3.

The equation for this graph is:  y = 1x + 3

Using the slope intercept form of a linear equation, y = mx + b, you can write an equation for the line if you know:

a) the slope and the coordinates for one point:

y = mx + b,   given the slope m = - 3, and the coordinates for the point are (4, 7)

y = - 3 x + b

7 = - 3(4) + b

b = 7 + 12 = 19

y = -3x + 19

b) the coordinates for any two points: (-2, -4), (8, 12)

y = mx + b = (y₂ - y₁ )  x  + b    = (12 - (-4))  x + b  = 16 x  + b  = 1.6x + b

                      (x₂ - x₁ )                  8 - (-2)                     10

 

y = 1.6x + b, for x = 8, and y = 12

12 = 1.6(8) + b

b = 12 - 12.8 = - 0.8

 

the equation is: y = 1.6x - 0.8

c) the slope and the y-intercept:

y = mx + b, given m = 4.5 and b = 7

The equation is: y = 4.5 x + 7

The point-slope form of a linear equation:

A line passing through the point ( 3, 2 ), and having a slope of 4 has the equation:

y - 2 = 4 (x - 3)

Parallel lines have the equal slopes.

The graphs of the following equations are parallel lines:

y = 4x - 7

y = 4x + 12

They both have a slope of 4.

The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the other line.

For the line whose equation is y = 3x - 7, the slope of the line that is perpendicular to this line is - 1/3. The y intercept can be anything.

The equation for the perpendicular line is:

y = - 1/3 x - b, where b is the y intercept.

Summary of forms of linear equations:

standard form: 4x + 2y = 8

slope-intercept form: y = - 2x + 4

point-slope form: y - 2 = -2 (x - 1)

horizontal line: y + 3 = 0

vertical line: x - 4 = 0

Scatter Plots