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Graphing Linear Equations

What is a linear equation

A linear equation involves one or two variables, typically denoted as x and y. In a linear equation, these variables are only raised to the first power, which means you won't find variables in the denominator of a fraction, inside a radical, or raised to any power greater than 1.

These are some equations and graphs of linear equations. Their graphs form straight lines:

y = 1

a graph with a straight line with a slope of zero and a y-intercept of 1.

 

−3x − 5y =4

a graph with a straight line with a negative slope and a y-intercept between zero and negative 1.

 

y is equal to seven fourths times x minus 1.

a graph with a straight line with a positive slope and a y-intercept of 1.

 

Now compare those linear graphs to the equations and graphs of non-linear equations. Their graphs form curves:

y=x²

a u shaped graph with a vertex at the origin.

 

y is equal to 1 divided by x

a graph with two curves that approach both axes without intercepting them.

 

y is equal to the square root of x, or y is equal to x to the one half power.

a graph of a curve that has an endpoint at the origin and extends into quadrant 1.

 

Recall that the y is called the dependent variable. We call this set the range.

The x is called the independent variable becuase the value of y is dependent upon the value of x. We call this set the domain.

We will continue reviewing graphing linear equations. You can graph linear equations using the slope-intercept form that you reviewed in the previous lesson, or you can graph linear equations by finding the x and y-intercepts.

Example #1

Watch Graph Linear Equations Using X and Y Intercepts and One More Point.

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Graphing Linear Equations

Graphing Inequalities

Remember when graphing inequalities on a number line, we marked an open or closed endpoint then drew a line in one direction. Graphing linear inequalities is done in a similar manner.

To graph y ≤ x -2, we first draw a boundary line and then shade to one side. The boundary line is the line you would graph if you changed the inequality sign to an equal sign.

The boundary line will be solid for ≥ and ≤.

The boundary line will be a dashed line for < and >.

To decide which way to shade, we choose a point that is not on the boundary line and evaluate the inequality. If it is a true statement, then the point is a solution, so we shade the side of the graph that contains the point we tested. If it is a false statement, we shade the other side.

Example #2

Watch Graph Linear Inequalities Using a Table of Values.

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Example #3

Watch Graph Linear Inequalities Using a Table of Values.

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More Linear Inequalities

Linear Inequalities

Another way to graph linear inequalities is to write in slope-intercept form, find and plot the y-intercept, and then apply the slope (rise over run) to find the next point.

Example #4

Watch Graph Linear Inequalities Using Slope and Y-Intercept.

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Example #5

Watch Graph Linear Inequalities Using Slope and Y-Intercept.

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