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? ESSENTIAL QUESTION
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How can you use systems of equations to solve real-world problems?
Solving Systems of Linear Equations 8
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MODULE
The distance contestants in a race travel over time can be modeled by a system of equations. Solving such a system can tell you when one contestant will overtake another who has a head start, as in a boating race or marathon.
LESSON 8.1
Solving Systems of Linear Equations by Graphing
8.EE.8, 8.EE.8a,
8.EE.8c
LESSON 8.2
Solving Systems by Substitution
8.EE.8b, 8.EE.8c
LESSON 8.3
Solving Systems by Elimination
8.EE.8b, 8.EE.8c
LESSON 8.4
Solving Systems by Elimination with Multiplication
8.EE.8b, 8.EE.8c
LESSON 8.5
Solving Special Systems
8.EE.8b, 8.EE.8c
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Complete these exercises to review skills you will need for this module.
Simplify Algebraic ExpressionsEXAMPLE Simplify 5 - 4y + 2x - 6 + y.
-4y + y + 2x - 6 + 5 -3y + 2x - 1
Simplify.
1. 14x - 4x + 21
2. -y - 4x + 4y
3. 5.5a - 1 + 21b + 3a
4. 2y - 3x + 6x - y
Graph Linear EquationsEXAMPLE Graph y = - 1 _ 3 x + 2.
Step 1: Make a table of values.
x y =- 1 _ 3 x + 2 (x, y)
0 y = - 1 _ 3 (0) + 2 = 2 (0, 2)
3 y = - 1 _ 3 (3) + 2 = 1 (3, 1)
Step 2: Plot the points.Step 3: Connect the points with a line.
Graph each equation.
5. y = 4x - 1 6. y = 1 _ 2 x + 1 7. y = -x
Group like terms.Combine like terms.
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Reading Start-Up
Active ReadingFour-Corner Fold Before beginning the module, create a four-corner fold to help you organize what you learn about solving systems of equations. Use the categories Solving by Graphing, Solving by Substitution, Solving by Elimination, and Solving by Multiplication. As you study this module,note similarities and differences among the four methods. You can use your four-corner fold later to study for tests and complete assignments.
VocabularyReview Words linear equation (ecuacin
lineal) ordered pair
(par ordenado) slope (pendiente) slope-intercept
form (forma pendiente interseccin)
x-axis (eje x) x-intercept (interseccin
con el eje x) y-axis (eje y) y-intercept (interseccin
con el eje y)
Preview Words solution of a system of
equations (solucin de un sistema de ecuaciones)
system of equations (sistema de ecuaciones)
Visualize VocabularyUse the words to complete the graphic.
Understand VocabularyComplete the sentences using the preview words.
1. A is any ordered pair
that satisfies all the equations in a system.
2. A set of two or more equations that contain two or more variables is
called a .
m (x, y)
(- b __ m , 0) (0, b)
y = mx + b
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Use the elimination method.A. -x = -1 + y x + y = 4
y = y + 3
This is never true, so the system has no solution. The graphs never intersect.
Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module.
What It Means to YouYou will understand that the points of intersection of two or more graphs represent the solution to a system of linear equations.
Use the substitution method.B. 2y + x = 1 y - 2 = x
2y + (y - 2) = 1 3y - 2 = 1 y = 1
x = y - 2 x = 1 - 2 x = -1
Only one solution: x = -1, y = 1.The graphs intersect at the point (-1, 1).
Use the multiplication method.C. 3y - 6x = 3 y - 2x = 1
3y - 6x = 3 3y - 6x = 3
0 = 0
This is always true. So the system has infinitely many solutions.The graphs are the same line.
Solving Systems of Linear EquationsGETTING READY FOR
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
Key Vocabularysolution of a system of
equations (solucin de un sistema de ecuaciones) A set of values that make all equations in a system true.
system of equations (sistema de ecuaciones) A set of two or more equations that contain two or more variables.
EXAMPLE 8.EE.8a, 8.EE.8b
8.EE.8a
8.EE.8b
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ESSENTIAL QUESTIONHow can you solve a system of equations by graphing?
Investigating Systems of EquationsYou have learned several ways to graph a linear equation in slope-intercept form. For example, you can use the slope and y-intercept or you can find two points that satisfy the equation and connect them with a line.
Graph the pair of equations together:
Explain how to tell whether (2, -1) is a solution of the equation y = 3x - 2 without using the graph.
Explain how to tell whether (2, -1) is a solution of the equation y = -2x + 3 without using the graph.
Use the graph to explain whether (2, -1) is a solution of each equation.
Determine if the point of intersection is a solution of both equations.
Point of intersection: ,
y = 3x - 2
= 3 - 2
1 =
y = -2x + 3
= -2 + 3
1 =
The point of intersection is / is not the solution of both equations.
A { y = 3x - 2 y = -2x + 3 .B
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EXPLORE ACTIVITY
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L E S S O N
8.1Solving Systems of Linear Equations by Graphing
8.EE.8a
8.EE.8a
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Also 8.EE.8, 8.EE.8c
Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
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My Notes
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Solving Systems GraphicallyAn ordered pair (x, y) is a solution of an equation in two variables if substituting the x- and y-values into the equation results in a true statement. A system of equations is a set of equations that have the same variables. An ordered pair is a solution of a system of equations if it is a solution of every equation in the set.
Since the graph of an equation represents all ordered pairs that are solutions of the equation, if a point lies on the graphs of two equations, the point is a solution of both equations and is, therefore, a solution of the system.
Solve each system by graphing.
{ y = -x + 4 y = 3x Start by graphing each
equation.
Find the point of intersection of the two lines. It appears to be (1, 3). Substitute to check if it is a solution of both equations.
y = -x + 4 y = 3x
3 ?= -(1) + 4 3 ?= 3(1)
3 = 3 3 = 3
The solution of the system is (1, 3).
{ y = 3x - 3 y = x - 3 Start by graphing each
equation.
Find the point of intersection of the two lines. It appears to be (0, -3). Substitute to check if it is a solution of both equations.
y = 3x - 3 y = x - 3
-3 ?= 3(0) - 3 -3 ?= 0 - 3
-3 = -3 -3 = -3
The solution of the system is (0, -3).
EXAMPLE 1
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8.EE.8
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