1. MS Algebra A-F-IF-7 – Ch. 7.2 Solve Linear Systems by Substitution
Mr. Deyo
Solve Linear Systems by Substitution

2. By the end of the period, I will solve linear systems by substitution.
Title: 7.2 Solve Linear Systems by substitution
Date:
Learning Target
By the end of the period, I will solve linear systems by substitution.
I will demonstrate this by completing Four-Square Notes and by solving problems in a pair/group activity.

3. Home Work 1-2-3: 1) Class 4-Square Notes Put In Binder?
2) Section 7.2 Pg ) Section ______
TxtBk.Prob.#3,15,17,27,29,32,33 Notes Copied on blank sheet Solved and Put in Binder? of paper in Binder?
Table of Contents
Date Description Date Due

4. Storm Check (Think, Write, Discuss, Report) Questions on which to ponder and answer:
How are the two images similar?
How are they different?
How can these two images be related to math?
IMAGE 1
IMAGE 2

5. Daily Warm-Up Exercises
Solve the equations.
For use with pages xxx–xxx
6a – 3 + 2a = 13
4(n + 2) – n = 11

6. Daily Warm-Up Exercises
Solve the equations.
For use with pages xxx–xxx
6a – 3 + 2a = 13
4(n + 2) – n = 11
ANSWER
ANSWER 2 1
4n + 8 – n = 11
3n + 8 = 11
3n = 3
n = 1
8a – 3 = 13
8a = 16
a = 2

7. Vocabulary System of Linear Equations
Solution to a System of Equations
Solve by Graphing
Solve by Elimination
Solve by Substitution

8. Vocabulary Acquisition
Friendly Definition
Sketch
Wordwork
Sentence
DAY 2
1. Review word
Friendly Definition
Physical Representation
2. Draw a sketch
DAY 1
Use Visuals
Introduce the word
Friendly Definition
Physical Representation
Use Cognates
Write friendly definition
Word List
Vocabulary Acquisition
DAY 3 and/or DAY 4
1. Review the word
Friendly Definition
Physical Representation
2. Show how the word works
Synonyms/antonym
Word Problems
Related words/phrases
Example/non-example
DAY 5
1. Review the word
Friendly definition
Physical Representation
3. Write a sentence
at least 2 rich words (1 action)
correct spelling
correct punctuation
correct subject/predicate agreement
clear and clean writing

9. System of Linear Equations
Notes:
A system of equations (linear system) has two or more linear equations with the same variables.
Equation 1: y = -x + 5
Equation 2: y = ½ x + 2
solution
A solution to the system is an ordered pair ( x, y ) that is a solution to EACH of the equations (where the lines intersect).
(2, 3)

10. Solving Systems by Substitution
Notes:
Step 1: Solve one of the equations for one of its variables that has a coefficient of 1 or -1. (Here we choose y.)
Step 2: Substitute the expression from Step 1 into the other equation and solve for the other variable. (Here the “other” variable is x).
Here we choose Equation 1 where y = (2x - 1)
Here we substitute (2x - 1) for y in Equation 2.
Equation 1: y = 2x - 1
Equation 2: 2x + y = 3
Equation 2: 2x + (2x - 1) = 3
2x + 2x -1 = 3
4x – 1 = 3
4x = 4
x = 1
Step 3: Substitute the expression from Step 2 back into Equation 1 and solve for the first variable. (Here the 1st variable is y.)
y = 2x – 1
y = 2(1) – 1
y = 2 – 1 = 1
Step 4: Check your solution
Solution:
(1, 1)
Equation 1: 1 = 2(1)- 1
Equation 2: 2(1)+(1) = 3

11. y = 3x + 2 x + 2y = 11 Solve the linear system. Problem A
Use the substitution method
Equation 1
STEP 1 Solve for y.
STEP 2 Substitute in Eq. 2
AND Solve for x.
STEP 3 Substitute in Eq. 1
AND Solve for y.
STEP 4 Check Solution
Equation 2
y = 3x + 2
x + 2y = 11

12. (1, 5) y = 3x + 2 x + 2y = 11 x + 2(3x + 2) = 11 x + 6x + 4 = 11
Problem A
Solve the linear system.
Use the substitution method
SOLUTION
Equation 1
STEP 1 Solve for y.
STEP 2 Substitute in Eq. 2
AND Solve for x.
STEP 3 Substitute in Eq. 1
AND Solve for y.
STEP 4 Check Solution
Equation 2
y = 3x + 2
x + 2y = 11
STEP 1 Solve for y.
STEP 2 Substitute 3x + 2 for y in Equation 2 and solve for x.
Equation 1 is already solved for y.
x + 2(3x + 2) = 11
x + 6x + 4 = 11
7x + 4 = 11
7x = 7
x = 1
STEP 3 Substitute 1 for x in Equation 1 to find the value of y.
y = 3x + 2
y = 3(1) + 2
y = 3 + 2
y = 5
Solution:
(1, 5)
Eq. 1: 5 = 3(1) + 2
5 = 3 + 2
5 = 5
Eq. 2: (5) = 11
= 11
11 = 11
STEP 4 Substitute 1 for x and 5 for y in each of the original equations.

13. y = 2x + 5 3x + y = 10 Solve the linear system. Problem B SOLUTION
Use the substitution method
SOLUTION
Equation 1
STEP 1 Solve for y.
STEP 2 Substitute in Eq. 2
AND Solve for x.
STEP 3 Substitute in Eq. 1
AND Solve for y.
STEP 4 Check Solution
Equation 2
y = 2x + 5
3x + y = 10

14. (1, 7) y = 2x + 5 3x + y = 10 3x + (2x + 5) = 10 3x + 2x + 5 = 10
Problem B
Solve the linear system.
Use the substitution method
SOLUTION
Equation 1
STEP 1 Solve for y.
STEP 2 Substitute in Eq. 2
AND Solve for x.
STEP 3 Substitute in Eq. 1
AND Solve for y.
STEP 4 Check Solution
Equation 2
y = 2x + 5
3x + y = 10
STEP 1 Solve for y.
STEP 2 Substitute 2x + 5 for y in Equation 2 and solve for x.
Equation 1 is already solved for y.
3x + (2x + 5) = 10
3x + 2x + 5 = 10
5x + 5 = 10
5x = 5
x = 1
STEP 3 Substitute 1 for x in Equation 1 to find the value of y.
y = 2x + 5
y = 2(1) + 5
y = 2 + 5
y = 7
Solution:
(1, 7)
Eq. 1: 7 = 2(1) + 5
7 = 2 + 5
7 = 7
Eq. 2: 3(1) + 7 = 10
= 10
10 = 10
STEP 4 Substitute 1 for x and 7 for y in each of the original equations.

15. Storm Check (Think, Write, Discuss, Report)
Explain in your own words the four steps of solving systems of equations by substitution?
The four steps of solving systems of equations by substitution are:
1) _________________________________________
2) _________________________________________
3) _________________________________________
4) _________________________________________

16. x – 2y = -6 4x + 6y = 4 Solve the linear system. Problem A Equation 1
Solve for best variable & Use the substitution method
Equation 1
STEP 1 Choose a variable
and solve.
STEP 2 Substitute in Eq. 2
AND Solve for 2nd variable.
STEP 3 Substitute in Eq. 1
AND Solve for 1st Variable.
STEP 4 Check Solution
Equation 2
x – 2y = -6
4x + 6y = 4

17. (-2, 2) x – 2y = -6 4x + 6y = 4 x = 2y – 6 4(2y – 6) + 6y = 4
Problem A
Solve the linear system.
Solve for best variable & Use the substitution method
SOLUTION
Equation 1
STEP 1 Choose a variable
and solve.
STEP 2 Substitute in Eq. 2
AND Solve for 2nd variable.
STEP 3 Substitute in Eq. 1
AND Solve for 1st Variable.
STEP 4 Check Solution
Equation 2
x – 2y = -6
4x + 6y = 4
STEP 1 Solve for x
(Coefficient of 1).
STEP 2 Substitute 2y – 6 for x in Equation 2 and solve for y.
x = 2y – 6
4(2y – 6) + 6y = 4
8y y = 4
14y – 24 = 4
14y = 28
y = 2
STEP 3 Substitute 2 for y in Equation 1 to find the value of x.
x – 2y = -6
x – 2(2) = -6
x – 4 = -6
x = -2
Solution:
(-2, 2)
Eq. 1: (-2) – 2(2) = -6
-2 – 4 = -6
-6 = -6
Eq. 2: 4(-2) + 6(2) = 4
= 4
4 = 4
STEP 4 Substitute -2 for x and 2 for y in each of the original equations.

18. 3x + y = -7 -2x + 4y = 0 Solve the linear system. Problem B Equation 1
Solve for best variable & Use the substitution method
Equation 1
STEP 1 Choose a variable
and solve.
STEP 2 Substitute in Eq. 2
AND Solve for 2nd variable.
STEP 3 Substitute in Eq. 1
AND Solve for 1st Variable.
STEP 4 Check Solution
Equation 2
3x + y = -7
-2x + 4y = 0

19. (-2, -1) 3x + y = -7 -2x + 4y = 0 y = -3x - 7 -2x + 4(-3x -7) = 0
Problem B
Solve the linear system.
Solve for best variable & Use the substitution method
SOLUTION
Equation 1
STEP 1 Choose a variable
and solve.
STEP 2 Substitute in Eq. 2
AND Solve for 2nd variable.
STEP 3 Substitute in Eq. 1
AND Solve for 1st Variable.
STEP 4 Check Solution
Equation 2
3x + y = -7
-2x + 4y = 0
STEP 1 Solve for y
(Coefficient of 1).
STEP 2 Substitute -3x – 7 for y in Equation 2 and solve for x.
y = -3x - 7
-2x + 4(-3x -7) = 0
-2x – 12x – 28 = 0
-14x – 28 = 0
-14x = 28
x = -2
STEP 3 Substitute -2 for x in Equation 1 to find the value of y.
3x + y = -7
3(-2) + y = -7
-6 + y = -7
y = -1
Solution:
(-2, -1)
Eq. 1: 3(-2)+(-1)=-7
-6 – 1 = -7
-7 = -7
Eq. 2: -2(-2) + 4(-1) = 0
4 – 4 = 0
0 = 0
STEP 4 Substitute -2 for x and -1 for y in each of the original equations.

20. Partner/Table Practice
For use with pages xxx–xxx
You burned 8 calories per minute on a treadmill and 10 calories per minute on an elliptical trainer for a total of 560 calories in 60 minutes. How many minutes did you spend on each machine?
T = Treadmill minutes
E = Elliptical minutes

21. Partner/Table Practice
ANSWER
For use with pages xxx–xxx
You burned 8 calories per minute on a treadmill and 10 calories per minute on an elliptical trainer for a total of 560 calories in 60 minutes. How many minutes did you spend on each machine?
T = Treadmill minutes
E = Elliptical minutes
Calories Burned Exercise Time
8T + 10E = T + E = 60
- E E
8(60-E) +10E = T = 60 – E (Substitution)
480 – 8E + 10E = 560
E = T = 60-40
2E = T = 20
E = 40
treadmill: 20 min,
elliptical trainer: 40 min

22. Storm Check (Think, Write, Discuss, Report)
To verify your solution to a system of linear equations, what do you have to do?
To verify my solution to a system of linear
equations, I have to _________________________
___________________________________________
___________________________________________.

23. Vocabulary System of Linear Equations
Solution to a System of Equations
Solve by Graphing
Solve by Elimination
Solve by Substitution

24. Home Work 1-2-3: 1) Class 4-Square Notes Put In Binder?
2) Section 7.2 Pg ) Section ______
TxtBk.Prob.#3,15,17,27,29,32,33 Notes Copied on blank sheet Solved and Put in Binder? of paper in Binder?
Table of Contents
Date Description Date Due

25. By the end of the period, I will solve linear systems by substitution.
Title: 7.2 Solve Linear Systems by Substitution
Date:
Learning Target
By the end of the period, I will solve linear systems by substitution.
I will demonstrate this by completing Four-Square Notes and by solving problems in a pair/group activity.

26. Solve the System of Equations:
Use the Graph and Check Method
Step 1: Graph Both Equations in the same plane
Ticket Out
Step 2: Find coordinates of the point of intersection
Step 3: Check each equation to verify solution
Solve the System of Equations:
Eq. 1: x – y = Eq. 2: 3x + y = 3
STEP 1
Graph both
equations
STEP 2
Find intersection.
STEP 3
Check solution
Eq. 1: x – y = Eq. 2: 3x + y = 3

27. -y = -x + 5 y = x - 5 y = -3x + 3 (2) - (-3) = 5 3(2) + (-3) = 3
Use the Graph and Check Method
Step 1: Graph Both Equations in the same plane
Ticket Out
Step 2: Find coordinates of the point of intersection
SOLUTION
Step 3: Check each equation to verify solution
Solve the System of Equations:
Eq. 1: x – y = Eq. 2: 3x + y = 3
STEP 1
Graph both
equations
-y = -x + 5
y = x - 5
y = mx + b
y = -3x + 3
y = mx + b
STEP 2
Find intersection.
STEP 3
Check solution
Eq. 1: x – y = Eq. 2: 3x + y = 3
(2) - (-3) = 5
2 + 3 = 5
5 = 5
(2, -3)
3(2) + (-3) = 3
6 – 3 = 3
3 = 3