Graphing Systems of Linear Equations and Inequalities Study Guide
Find practice problems and solutions for these concepts at Graphing Systems of Linear Equations and Inequalities Practice Problems.
This lesson explains linear equations and systems of linear equations and inequalities and shows you how to solve them graphically.
What Is a Linear Equation?
If the graph of an equation is a straight line, the equation is a linear equation.Did you notice that the word line is part of the word linear ? That may help you remember that a linear equation will always graph into a straight line.
There are other ways to determine if an equation is linear without graphing the equation. Equations of this type can be put into the form Ax + By = C, where A and B are not both equal to zero. This is called the standard form of a linear equation. In standard form, a linear equation can have no exponent greater than 1, cannot have any variables in the denominator if the equation contains both variables, and cannot have the product of variables in the equation.
Here are examples of equations that are linear:
2x + y = 11
5x – 2y = 16
x = 18
Although y does not appear in the last equation, note that the equation could also be written as x + 0y = 18, so it fits the form of a linear equation.
Here are examples of equations that are not linear:
x2 + y = 5 Equation contains a variable greater than 1. Equation contains the variable in the denominator. xy = 6 Equation contains the product of two variables.
What Is a System of Linear Equations?
A system is two or more equations with the same variables. If you have two different variables, you need two equations. If you have three different variables, you need three equations. There are several methods of solving systems of equations. In this lesson, you will solve systems of equations graphically. When you graphed linear equations in Lesson 8, the graph, which was a straight line, was a picture of the answers; you had an infinite number of solutions.
However, a system of linear equations has more than one equation, so its graph will be more than one line. You'll know that you've solved a system of linear equations when you determine the point(s) of intersection of the lines. Since two lines can intersect in only one point, that means the system of linear equations has one solution. What if the lines don't intersect? When the lines do not intersect, the system has no solutions.
Generally speaking, two lines can intersect in only one point, or they do not intersect at all. However, there is a third possibility: The lines could coincide, which means they are the same line. If the lines coincide, there are an infinite number of solutions, since every point on the line is a point of intersection.
The graphs of a linear system would be one of three cases, as shown on pages 85 and 86.
The lines intersect in one point. When the lines intersect in one point, you have one solution.
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