Graphing Systems of Linear Equations and Inequalities Study Guide (page 2)
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.
The lines do not intersect. When the lines do not intersect, you have no solutions (Ø).
The lines coincide. When the lines coincide, you have an infinite number of solutions.
Graphing a System of Linear Equations
To graph a system of linear equations, use what you already know about graphing linear equations. To graph a system of linear equations, you will use the slope-intercept form of graphing. The first step is to transform the equations into slope-intercept form, or y = mx + b. Then use the slope and y-intercept to graph the line. Once you have both lines graphed, determine your solutions.
x – y = 6
2x + y = 3
|Transform the first equation into y = mx + b.||x – y = 6|
|Subtract x from both sides of the equation.||x – x – y = 6 – x|
|Simplify.||–y = 6 – x|
|Use the commutative property.||–y = –x + 6|
|Multiply both sides of the equation by –1.||–1 · –y = –1(–x + 6)|
|Simplify both sides.||y = x – 6|
|Transform the second equation into y = mx + b.||2x + y = 3|
|Subtract 2x from both sides of the equation.||2x – 2x + y = 3 – 2x|
|Simplify.||y = 3 – 2x|
|Use the commutative property.||y = –2x + 3|
The slope of the first equation is 1, and the y-intercept is –6. The slope of the second equation is –2, and the y-intercept is 3. In the first equation, the line cuts the y-axis at –6. From that point, go up 1 and right 1. Draw a line through your beginning point and the endpoint—this line can extend as long as you want in both directions since it is endless. In the second equation, the line cuts the y-axis at 3. From that point, go down 2 and to the right 1. Draw a line through your beginning point and the endpoint, extending it as long as you want. The point of intersection of the two lines is (3,–3), so there is one solution.
x + 2y = –4
2x + 4y = –8
|Transform the first equation into y = mx + b.||x + 2y = –4|
|Subtract x from both sides of the equation.||x – x + 2y = –4 – x|
|Simplify.||2y = –4 – x|
|Use the commutative property.||2y = –x – 4|
|Divide both sides of the equation by 2.|
|Simplify both sides of the equation.|
|Transform the second equation into y = mx + b.||2x + 4y = –8|
|Subtract 2x from both sides of the equation.||2x – 2x + 4y = –8 – 2x|
|Simplify.||4y = –8 – 2x|
|Use the commutative property.||4y = –2x – 8|
|Divide both sides of the equation by 4.|
|Simplify both sides of the equation.|
The first and second equations transformed into the same equation, . If you have the same equation, it will graph into the same line. This is a case where the two lines coincide. Graph the line. Start with the y-intercept, which is –2. From that point, go down 1 and to the right 2. Since the lines coincide, there are an infinite number of solutions.
Here's a time saver! You can determine the nature of the solutions of a system without actually graphing them. Once you have the equations transformed into y = mx + b, compare the slopes and y-intercepts of each equation. If the slopes and y-intercepts are the same, you will have the same line. If the slopes are the same, but the y-intercepts are different, then the lines will be parallel. If the slopes are different, then the lines will intersect.
Case 1: The slopes and y-intercepts are the same, so the lines coincide. There will be an infinite number of solutions.
y = 2x + 3 y = 2x + 3
Case 2: The slopes are the same, but the y-intercepts are different. The lines will be parallel, so there is no solution.
y = 2x + 4 y = 2x – 5
Case 3: The slopes are different. The lines will intersect, so there will be one solution.
y = 3x + 2 y = 2x + 3
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