Algebra 1 End-of-Course Review

Module Four: Linear Equations

This module is all about slope, writing equations of lines, and graphing!

Slope

The slope of any non-vertical line can be determined in two ways:

  • Algebraically—using the slope formula m = the quantity of y subscript 2 minus y subscript 1 all over the quantity of x subscript 2 minus x subscript 1 and any two points on the line (x1, y1) and
    (x2, y2).
  • Graphically—by counting the “rise over run” between two points on the line.
    Slope = rise with vertical change in parentheses over run with horizontal change in parentheses

Slope plays an important role when working with parallel and perpendicular lines.

Type of Lines Properties of Slopes What it Looks Like An Example
Parallel lines
(never intersect) 		Slopes are the same two lines that are always the same distance apart y equals the quantity 1 over 3, end quantity, times x plus 5 and y equals the quantity 1 over 3, end quantity, times x minus 1.
Perpendicular lines
(intersect at a 90° angle)		 Slopes are opposite reciprocals two lines that intersect to form four ninety degree angles y equals the quantity 2 over 5, end quantity, times x plus 7 and y equals the negative quantity 5 over 2, end quantity, times x plus 3.

Recall that a horizontal line has a slope equal to zero, and is written in the form y = #.

Example

Write the equation of the horizontal line that passes through the point (–1, 6).

Select to reveal Answer.

A vertical line has a slope that is undefined, and is written in the form x = #.

Example

Write the equation of the vertical line that passes through the point (–1, 6).

Select to reveal Answer.

Writing Equations of Lines

Name Equation Purpose
Slope-Intercept Form y = mx + b describes a line with slope m and y-intercept b
Point-Slope Form y – y1 = m(x – x1) describes a line with slope m passing through any point (x1, y1)
Standard Form Ax + By = C useful for identifying the x-intercept and y-intercept
A, B, and C are integers and A is positive

Let's look at an example that shows how the equation of one line can be written in all three formats!

Writing Equations of Lines from a Graph

Equations of lines (in point-slope or slope-intercept form) can also be determined by looking at the information the graph provides.

With the slope and any point on the line, the equation can be written in point-slope form.

The equation of a graphed line can only be written in slope-intercept form when the y-intercept is clearly identified.

Here are some examples of each.

Writing Equations of Parallel and Perpendicular Lines

Parallel lines have the same slope, but cross the y-axis at different points.  

Perpendicular lines have slopes that are opposite reciprocals.

Check out an example of each.

Scatter Plots and Lines of Best Fit

In some real-world settings, the data you gather doesn’t always form a straight line.  Sometimes it doesn’t even resemble a line at all.  After plotting all of the points from your set of data, you will have a scatter plot.  The scatter plot will help you to determine what type of correlation the points have, if any.

coordinate plane containing several random points rising from the left to the right showing positive correlation
Positive Correlation
coordinate plane containing several random points falling from the left to the right showing negative correlation
Negative Correlation
coordinate plane containing several random points showing no correlation
No Correlation

If your data has either a positive correlation or a negative correlation, you can draw a line of best fit, and write an equation to represent the data.  This equation can be used to interpret and make predictions about the data.

Check out this example of using a line of best fit to write an equation and make a prediction.

Two values within the equation of the line of best fit provide useful information about the data. These values are the y-intercept and the slope.

The y-intercept represents the value of y when x is zero. In real-life problems, this is usually a one-time start-up fee. The slope, also called the rate of change, represents the relationship between two variables. Slope, or rate of change, is the change in y over the change in x.This can be described as the change in y for every one unit increase in x.

View an example of interpreting slope and intercepts in a line of best fit.