Elimination Method

Show your work for each problem.

1.
Which system is ready for elimination by adding (no multiplication needed)?
1. a) and
2. b) and
3. c) and
4. d) and
Answer: and
The first system has and - opposite coefficients! Adding eliminates immediately: , so .
2.
Solve by adding: and . What is ?
Answer: 7
Adding: , so . The terms cancel because .
3.
Solve: and . What is ?
Answer: 1
Adding: , so . Substitute into equation 1: , so , , .
4.
To eliminate from and , what should you multiply the second equation by?
1. a)
2. b)
3. c)
4. d)
Answer:
Multiply by : gives . Now adding eliminates : , so .
5.
Solve: and
Answer: (3, 2)
1. Multiply equation 2 by to eliminate . What do you get? -2x - 2y = -10
2. Add the equations. What is the result? y = 2
3. Substitute into equation 2. What is ? 3
6.
Solve: and . What is the value of ?
Answer: 5
Adding: , so . Substitute: , so . Therefore .
7.
What happens when you add and ?
1. a) (unique solution)
2. b) (no solution)
3. c) (unique solution)
4. d) (infinitely many solutions)
Answer: (infinitely many solutions)
Adding gives , which is always true. This means the equations represent the same line, so there are infinitely many solutions!
8.
Solve: and
Answer: (3, 4)
1. Notice the coefficients are already opposites. Add the equations. What do you get? 5x = 15
2. Solve for 3
3. Substitute into the first equation to find 4
9.
Solve: and . What is ?
Answer: 2
Subtracting: , so , .
10.
Solve: and
Answer: (2, 4)
1. To eliminate , multiply equation 1 by 3 and equation 2 by -2. What is the new equation 1? 6x + 15y = 72
2. What is the new equation 2? -6x - 4y = -28
3. Add the equations. What do you get? 11y = 44
4. Solve for 4
11.
Solve: and . What is ?
Answer: 2
Multiply equation 2 by -2: . Add to equation 1: , so , . Verify: , check ✓ and ✓.
12.
What happens when you try to solve and ?
1. a)Exactly one solution:
2. b)Infinitely many solutions (same line)
3. c)Exactly one solution:
4. d)No solution (parallel lines)
Answer: No solution (parallel lines)
Multiply equation 1 by -2: . Add: , which is false! The system has no solution - the lines are parallel (same slope, different y-intercepts).
13.
A store sold 150 items. Shirts cost 25 euros each and pants cost 40 euros each. Total sales were 4,500 euros. How many shirts were sold?
Answer: 100
1. Let = shirts and = pants. Write the equation for total items. s + p = 150
2. Write the equation for total sales. 25s + 40p = 4500
3. Multiply the first equation by -25 and add. What do you get? 15p = 750
4. Solve for , then find . 100
14.
When should you use subtraction instead of addition in elimination?
1. a)When coefficients are opposites (+3 and -3)
2. b)When one coefficient is larger than the other
3. c)When coefficients are the same (both positive or both negative)
4. d)Never - always use addition
Answer: When coefficients are the same (both positive or both negative)
When coefficients are the same (like and ), subtracting makes them cancel: . Adding would give , not elimination!
15.
Solve: and . What is ?
Answer: 3
Adding: , so , .
16.
Solve: and
Answer: (4, 3)
1. Notice that and are related. Multiply equation 1 by 2 to make the coefficients opposites. What do you get? 10x - 6y = 22
2. Now add the two equations. What is the result? 12x = 48
3. Solve for 4