Substitution Method

Show your work for each problem.

1.
In the system , what expression should you substitute for in the second equation?
1. a)
2. b)
3. c)
4. d)
Answer:
The first equation tells us , so we substitute for in the second equation.
2.
If and , what is ?
Answer: 7
Substituting into gives , so .
3.
Which variable is easiest to isolate in the equation ?
1. a)Both are equally easy
2. b) (coefficient is 3)
3. c) (coefficient is 1)
4. d)Neither can be isolated
Answer: (coefficient is 1)
has coefficient 1, so isolating gives directly. Isolating would require dividing by 3.
4.
Solve the system:
Answer: (4, 6)
1. Substitute for in the second equation. What equation do you get? x + x + 2 = 10
2. Simplify and solve for . What is ? 4
3. Find using . What is ? 6
5.
Solve for :
Answer: 3
Substituting gives , so and .
6.
Solve the system:
Answer: (5, 3)
1. From equation 1, isolate . What is equal to? 8 - x
2. Substitute into equation 2: . Simplify the left side. 3x - 8
3. Solve for . What is ? 5
4. Find using . What is ? 3
7.
What error was made? "Solve into : , so , "
1. a)The +1 wasn't subtracted (should be , so )
2. b)Wrong substitution
3. c)Should have solved for first
4. d)No error - the solution is correct
Answer: The +1 wasn't subtracted (should be , so )
The equation was incorrectly simplified to . Correct: , so .
8.
Solve for :
Answer: 2
Substituting: , so , , .
9.
Solve the system:
Answer: (4, 3)
1. From equation 2, isolate . What is equal to? y + 1
2. Substitute into equation 1: . Simplify. 3y + 2 = 11
3. Solve for . What is ? 3
4. Find using . What is ? 4
10.
Solve for :
Answer: 2
Substituting: , so , , .
11.
When you substitute and get , what does this mean about the system?
1. a)No solution (parallel lines)
2. b)Exactly one solution
3. c)Infinitely many solutions (same line)
4. d)The calculation has an error
Answer: Infinitely many solutions (same line)
When substitution gives , the equations represent the same line, so there are infinitely many solutions (any point on the line).
12.
Solve the system:
Answer: (14/5, 9/5)
1. From equation 2, isolate . What is ? y + 1
2. Substitute into equation 1: . Simplify the left side. 5y + 3
3. Solve . What is ? 9/5
4. Find using . What is ? 14/5
13.
If , what is the value of ?
Answer: 4
Substituting: , so , , . Then . So .
14.
Solve the system:
Answer: (1, -1)
1. From equation 2, isolate . What is equal to? -1 - 2y
2. Substitute into equation 1: . Distribute and simplify. -2 - 7y = 5
3. Solve for . What is ? -1
4. Find using . What is ? 1
15.
A store sold 60 items for a total of 800 euros. T-shirts cost 10 euros and jeans cost 20 euros. Which system represents this situation?
1. a)
2. b)
3. c)
4. d)
Answer:
(total items) and (total revenue, price times quantity for each).
16.
Solve: A parking lot has cars and motorcycles. There are 50 vehicles and 160 wheels total. Cars have 4 wheels, motorcycles have 2. How many of each?
Answer: (30, 20)
1. Let = cars, = motorcycles. Write the equation for total vehicles. c + m = 50
2. Write the equation for total wheels. 4c + 2m = 160
3. From equation 1, isolate . What is ? 50 - c
4. Substitute and solve for . How many cars? 30
5. How many motorcycles? 20