Equations with Variables on Both Sides

Show your work for each problem.

1.
Solve:
1. a)
2. b)
3. c)
4. d)
Answer:
→ → → . Check: and \checkmark
2.
Solve for :
Answer: 4
→ → →
3.
Solve:
Answer: 4
1. Subtract from both sides. What do you get on the left side? 3x + 3
2. Now subtract from both sides. What is the result? 3x = 12
3. Divide both sides by . What is ? 4
4.
Which equation has as its solution?
1. a)
2. b)
3. c)
4. d)
Answer:
Check: and \checkmark. The other equations give , , and respectively.
5.
Solve:
Answer: 5
→ → →
6.
Solve:
Answer: 4
1. Subtract from both sides. What do you get? 7 = 6x - 17
2. Add to both sides. What is the result? 24 = 6x
3. Divide both sides by . What is ? 4
7.
Solve:
1. a)
2. b)
3. c)
4. d)
Answer:
→ → →
8.
Solve:
Answer: 7
→ → → →
9.
Solve:
Answer: 4
1. Distribute the on the left. What is the equation now? 5x + 5 = 3x + 13
2. Subtract from both sides. What do you get? 2x + 5 = 13
3. Subtract from both sides. What is the result? 2x = 8
4. Divide both sides by . What is ? 4
10.
Solve:
Answer: 6
→ → →
11.
What type of equation is ?
1. a)Identity (infinitely many solutions)
2. b)Contradiction (no solution)
3. c)One solution:
4. d)One solution:
Answer: Identity (infinitely many solutions)
→ → . This is always true, so any value of works.
12.
Solve:
Answer: infinite
1. Distribute on the left: 6x - 3
2. Distribute on the right: 6x + 8
3. Simplify the right side: 6x - 3
4. What do you get when you compare both sides? (type 'identity' or 'contradiction') identity
13.
What is the solution to ?
1. a)
2. b)
3. c)No solution (contradiction)
4. d)All real numbers (identity)
Answer: No solution (contradiction)
→ . This is false, so there is no value of that makes the equation true.
14.
Solve:
Answer: all
Left: . Right: . So → , which is false. No solution exists.
15.
Solve:
Answer: infinite
1. Simplify the left side: 5x - 7
2. Simplify the right side: 5x - 5
3. Compare: . Subtract from both sides. What do you get? -7 = -5
4. Is true or false? false
16.
Solve:
Answer: 0
Left: . Right: . Both sides are equal for all , so this is an identity with infinitely many solutions.
17.
Solve:
Answer: infinite
1. Distribute on the left: 2x + 3
2. Simplify the right side: 2x + 3
3. Compare both sides. What type of equation is this? identity