Linear Equations

The Parts of a Linear Equation

A linear equation is another name for a linear-function rule. In standard form, using more of algebra’s typical alphabet-soup style, a linear equation looks like this:

y = mx + b

Let’s go through what the m and the b represent. The x and y variables will still be the input and output values, respectively, allowing you to create ordered pairs to plot on a coordinate plane.

The variable b represents the y-intercept of the graph of the line. The y-intercept is the place where the line crosses the y-axis.

The variable m represents the slope of the line. Slope is the steepness of the line; the steeper the line, the greater the slope.

Lots More About Slope

Slope is either positive or negative. Slope is calculated as the amount of rise (change in height) per amount of run (horizontal change). Slope can be determined with a formula or from a graph of a line. First, we will look at how to determine the slope just by counting units from the graph of a line.

Question

What is the slope of the line below?

1. 
2. 
3. 
4. 

Reveal Answer

The correct answer is D. The slope of the line is negative, so that rules out choices A and C. The rise of the line is (-5) and the run is 2. Slope is generally left as an improper fraction, rather than as a mixed number.

Slope can also be calculated if you know any two points on the line. You can find the points yourself from a graph or you may be given the points without any graphical representation of the line. The formula follows:

Example

What is the slope of the line that passes through the points (4, -7) and (-2, 5)?

Special Slope Cases

Horizontal lines have a slope of zero (m = 0). Since this makes the “mx” term of the linear equation disappear, all horizontal lines are an equation of the form y = b.

Vertical lines have an undefined slope. The “run” on a vertical line would be zero, and when you divide by zero, you get an undefined answer. Vertical lines are equations in the form of x = n, where n is the x-intercept of the graph.

Parallel and Perpendicular Lines

Parallel lines have the same slope. Think about what parallel lines look like: they are always the same distance apart, so they must have the same amount of steepness.

Example

y = 3x + 7 and y = 3x + 12 are parallel.

The slopes of perpendicular lines are related, but it is a little more complicated. The slopes of perpendicular lines are negative reciprocals of each other. If a line has a slope of , then the line perpendicular to it has a slope of .

Example

are perpendicular.

Graphing Lines from Equations

Graphing a line from an equation is a quick task. You can graph a line without having to use several input values, now that you know about the y-intercept and slope in each equation.

Follow these steps to graph a line from its equation:

1. Mark the y-intercept on the y-axis.
2. Starting from the y-intercept, count out the slope. (Remember, the run is the denominator and the rise is the numerator. Pay attention to any negative signs!)
3. Make a mark at the point you reached after counting out the slope.
4. Connect the two points with a line.

Example

Graph

To find the intercepts of the line, you will be replacing one variable with zero.

Steps:

1. To find the x-intercept, let y = 0 and solve for x. Then you will have an ordered pair (x, 0) to graph.
2. To find the y-intercept, let x = 0 and solve for y. Then you will have an ordered pair (0, y) to graph.
3. Graph both points and connect the dots.

Question

Which graph below shows the line y = 3x + 2?

Reveal Answer

The correct answer is A. The y-intercept of the line is (0, 2). The slope is three, which means it has a rise of three and a run of one.

Writing Linear Equations from Graphs

Now that you can graph lines from equations, let’s work on going the other direction: writing an equation of a line from its graph. As you can probably imagine, this can be done in several different ways; here are two basic methods.

Depending on what the line looks like on the graph, you may be able to easily determine the y-intercept and count out the slope to another obvious point. Then all you have to do is put it into the y = mx + b form.

Therefore,

Question

What is the equation of the line shown on the graph?

1. 
2. 
3. 
4. 

Reveal Answer

The correct answer is B. This problem can be solved just by looking at the y-intercept and counting out the slope. The run is 3, and the rise is 1, so the slope is . Therefore the equation is .

A Little About Linear Inequalities

A linear inequality looks just like a linear equation, but the equal sign is replaced by an inequality sign: <, >, ≤ ≥.

To graph a linear inequality, you would use the same method you used to graph a linear equation, except for the following: if the equation uses a less than (<) or greater than (>) sign, the line will be dashed instead of solid. If the equation uses a less than or equal (≤) or greater than or equal sign (≥), the line will still be solid.

For linear inequalities, you will shade the graph either above or below the line that you have drawn. The easiest way to determine which part to shade is to choose a point either above or below the line; (0, 0) is often an easy point to choose. Plug in the values of the point and see if the inequality is true. If it is true, shade the area that contains the point. If the inequality is false, shade the area on the other side of the line.

Take a look at the following example to see how it works.

Systems of Equations

• A system of equations is a group of two or more linear equations. While a single linear equation has infinite solutions (all the points along the line), a pair of linear equations has a single solution or no solution.
• A pair of linear equations has a single solution at the point where the two lines intersect. This ordered pair is the solution to the system of equations. All pairs of nonparallel lines (that are not actually the same line in disguise) have exactly one solution.
• A system of parallel lines (lines with the same slope) has no solution because the lines will never intersect.

Example

What is the solution to the system of equations shown on the graph below?

Question

How many solutions are there to the following system of equations?

1. No solution
2. One solution
3. Two solutions
4. Infinite solutions

Reveal Answer

The correct answer is B. Even though the lines have not yet intersected, they continue off the part of the coordinate plane that we can see and will intersect at one point.

Review

• A linear equation is written as y = mx + b. The variable b represents the y-intercept of the graph of the line. The variable m represents the slope of the line.
• Slope is the steepness of a line. It is calculated as the amount of rise (change in height) per amount of run (horizontal change).
• Slope can be counted on a graph or calculated from any two points on the line.
• Horizontal lines have a slope of zero (m = 0). Vertical lines have an undefined slope.
• Parallel lines have the same slope. The slopes of perpendicular lines are negative reciprocals of each other.
• Linear inequalities look the same as linear equations, except that a region above or below the line (solid or dashed) is shaded to show the area included in the solution.
• In a system of equations, a pair of linear equations has a single solution at the point where the two lines intersect, unless the lines are parallel, in which case there is no solution.