In this chapter, you will learn:
- How to solve multi-variable equations for one of the variables.
- How to solve equations with fractional coefficients.
- How to find the point where two lines intersect.
- How to use the connections between graphs, tables, rules, and patterns to solve problems.
Linear Equations
- A linear equation is an equation that forms a line when it is graphed. This type of equation may be written in several different forms. Although these forms look different, they are equivalent; that is, they all graph the same line.
- Standard form: An equation in ax + by = c form, such as 6x + 3y = −18.
- Slope-Intercept form: An equation in y = mx + b form, such as y = 2x − 6.
- You can quickly find the growth and y-intercept of a line in y = mx + b form. For the equation y = 2x − 6, the growth is 2, while the y-intercept is (0, −6).
Equivalent Equations
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Fraction Busters |
Systems of Equations Vocabulary
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Solving Systems Using Equal Values Method
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Equal Values Method is a non-graphing method to find the point of intersection or solution to a system of equations.
Start with two equations in y = mx + b form, such as y = −2x + 5 and
y = x − 1. Take the two expressions that equal y and set them equal to each other. Then solve this new equation to find x. See the example at below.
−2x + 5 = x −1
−3x = −6
x = 2
Once you know the x-coordinate of the point of intersection, substitute your solution for x into either original equation to find y. In this example, the first equation is used.
y = −2x + 5
y = −2(2) + 5
y = 1
A good way to check your solution is to substitute your solution for x into both equations to verify that you get equal y-values.
y = x − 1
y = (2) − 1
y = 1
Write the solution as an ordered pair to represent the point on the graph where the equations intersect.(2 ,1)
Start with two equations in y = mx + b form, such as y = −2x + 5 and
y = x − 1. Take the two expressions that equal y and set them equal to each other. Then solve this new equation to find x. See the example at below.
−2x + 5 = x −1
−3x = −6
x = 2
Once you know the x-coordinate of the point of intersection, substitute your solution for x into either original equation to find y. In this example, the first equation is used.
y = −2x + 5
y = −2(2) + 5
y = 1
A good way to check your solution is to substitute your solution for x into both equations to verify that you get equal y-values.
y = x − 1
y = (2) − 1
y = 1
Write the solution as an ordered pair to represent the point on the graph where the equations intersect.(2 ,1)
Solutions to a System of Equations
- A solution to a system of equations gives a value for each variable that makes both equations true. For example, when 4 is substituted for x and 5 is substituted for y in both equations at right, both equations are true. So x = 4 and y = 5 or (4, 5) is a solution to this system of equations. When the two equations are graphed, (4, 5) is the point of intersection.
System with one solution:
intersecting lines
x − y = −1
2x − y = 3
Some systems of equations have no solutions or infinite solutions.
Notice that the Equal Values Method would yield 3 = 4 , which is never true. When the lines are graphed, they are parallel. Therefore, the system has no solution.
System with no solutions:
parallel lines
x + y = 3
x + y = 4
In the third set of equations, the second equation is just the first equation multiplied by 2. Therefore, the two lines are really the same line and have infinite solutions.
System with infinite solutions:
coinciding lines
x + y = 3