Download Lesson 23: Using the Elimination Method to Solve Systems of

Lesson 23
Hart Interactive – Algebra 1
M1
ALGEBRA I
Lesson 23: Using the Elimination Method to Solve Systems of
Equations
Exploratory Exercise
1. A. Are the following statements true?
(A) 2 + 7 = 9
(B) 5 – 8 = – 3
B. What if we multiply equation (A) by 3, will it still be true? Why? We’ll call this new equation (C).
(C) __________________
C. What if we add equations (A) and (B)?
(A) + (B) __________________
D. How about if we add equations (B) and (C)?
(B) + (C) __________________
E. What about subtracting equations (A) and (B)?
(A) – (B) __________________
F. What can we do to true equations to make new, but also true equations?
In the last lesson, we looked systems of equations and found their solutions by either graphing or with
algebra using substitution. In this lesson, we’ll use the ideas from the Exploratory Exercise to eliminate one
variable.
Lesson 23:
Using the Elimination Method to Solve Systems of Equations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG I-M1-TE-1.3.0-07.2015
S.193
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 23
Hart Interactive – Algebra 1
M1
ALGEBRA I
2. A. Here is a system of two linear equations. Verify that the solution to this system is (3,4).
Equation A1: 𝑦𝑦 = 𝑥𝑥 + 1
Equation A2: 𝑦𝑦 = −2𝑥𝑥 + 10
B. Instead of using the substitution method, let’s subtract Equation A2 from Equation A1.
C. Solve for x and then determine the value of y. Your answer should be (3, 4).
3. Solve this system of linear equations algebraically using the Elimination Method.
ORIGINAL SYSTEM
2𝑥𝑥 + 𝑦𝑦 = 6
NEW SYSTEM
Multiply by 3
𝑥𝑥 − 3𝑦𝑦 = −11
Keep the same.
SOLUTION
4. Solve this system of linear equations algebraically using the Elimination Method.
ORIGINAL SYSTEM
2𝑥𝑥 + 3𝑦𝑦 = 7
𝑥𝑥 − 𝑦𝑦 = 1
NEW SYSTEM
SOLUTION
Keep the same.
Multiply by __ or__
5. Solve this system of linear equations algebraically using the Elimination Method.
ORIGINAL SYSTEM
2𝑥𝑥 + 3𝑦𝑦 = 6
NEW SYSTEM
SOLUTION
𝑥𝑥 − 𝑦𝑦 = 3
Lesson 23:
Using the Elimination Method to Solve Systems of Equations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG I-M1-TE-1.3.0-07.2015
S.194
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 23
Hart Interactive – Algebra 1
M1
ALGEBRA I
Lesson Summary
There are three ways to solve a system of equations.
Graphing
Substitution
Elimination
Graph the lines and find the
Rearrange one equation to
Add or subtract the
point of ______________.
get one of the variables
equations to eliminate one
_________, then substitute
of the variables. You may
into the other equation.
need to ____________ one
or both of the equations by
a number before adding or
subtracting.
x − y = −1 → y = x + 1
2x + y = 4
2x + x + 1 = 4
3x
+1 = 4
3x
=3
x
=1
y = x +1 =1+1 =2
solution: (1,2)
x − y = −1
2x + y = 4
3x + 0 = 3
x
=1
y = x +1 =1+1 =2
solution: (1,2)
Homework Problem Set
1.
Try to answer the following without solving for 𝑥𝑥 and 𝑦𝑦 first.
If 3𝑥𝑥 + 2𝑦𝑦 = 6 and 𝑥𝑥 + 𝑦𝑦 = 4, then
a.
2.
2𝑥𝑥 + 𝑦𝑦 = ?
b.
4𝑥𝑥 + 3𝑦𝑦 = ?
You always get the same solution no matter which two of the four equations you choose from Problem 1
to form a system of two linear equations. Explain why this is true.
Lesson 23:
Using the Elimination Method to Solve Systems of Equations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG I-M1-TE-1.3.0-07.2015
S.195
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 23
Hart Interactive – Algebra 1
M1
ALGEBRA I
1
3.
𝑦𝑦 = 4 𝑥𝑥
Solve the system of equations �
by graphing. Then, create a new system of equations that
𝑦𝑦 = −𝑥𝑥 + 5
has the same solution. Show either algebraically or graphically that the systems have the same solution.
4. Without solving the systems, explain why the following systems must have the same solution.
System (i): 4𝑥𝑥 − 5𝑦𝑦 = 13
3𝑥𝑥 + 6𝑦𝑦 = 11
System (ii):
8𝑥𝑥 − 10𝑦𝑦 = 26
𝑥𝑥 − 11𝑦𝑦 = 2
Solve each system of equations by writing a new system that eliminates one of the variables.
5. 2𝑥𝑥 + 𝑦𝑦 = 25
4𝑥𝑥 + 3𝑦𝑦 = 9
6. 3𝑥𝑥 + 2𝑦𝑦 = 4
4𝑥𝑥 + 7𝑦𝑦 = 1
Lesson 23:
Using the Elimination Method to Solve Systems of Equations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG I-M1-TE-1.3.0-07.2015
S.196
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.