Equations with Variables on Both Sides

Show your work for each problem.

1.
Solve: $4x + 5 = 2x + 13$
1. a) $x = 9$
2. b) $x = 4$
3. c) $x = 3$
4. d) $x = 6$
Answer: $x = 4$
$4x + 5 = 2x + 13$ → $4x - 2x = 13 - 5$ → $2x = 8$ → $x = 4$ . Check: $4(4) + 5 = 21$ and $2(4) + 13 = 21$ \checkmark
2.
Solve for $x$ : $6x - 2 = 3x + 10$
Answer: 4
$6x - 2 = 3x + 10$ → $6x - 3x = 10 + 2$ → $3x = 12$ → $x = 4$
3.
Solve: $5x + 3 = 2x + 15$
Answer: 4
1. Subtract $2x$ from both sides. What do you get on the left side? 3x + 3
2. Now subtract $3$ from both sides. What is the result? 3x = 12
3. Divide both sides by $3$ . What is $x$ ? 4
4.
Which equation has $x = 5$ as its solution?
1. a) $2x + 8 = 4x - 6$
2. b) $4x + 1 = 2x + 9$
3. c) $3x + 2 = x + 12$
4. d) $5x - 3 = 2x + 9$
Answer: $3x + 2 = x + 12$
Check: $3(5) + 2 = 17$ and $5 + 12 = 17$ \checkmark. The other equations give $x = 7$ , $x = 4$ , and $x = 4$ respectively.
5.
Solve: $8x - 4 = 5x + 11$
Answer: 5
$8x - 4 = 5x + 11$ → $3x - 4 = 11$ → $3x = 15$ → $x = 5$
6.
Solve: $3x + 7 = 9x - 17$
Answer: 4
1. Subtract $3x$ from both sides. What do you get? 7 = 6x - 17
2. Add $17$ to both sides. What is the result? 24 = 6x
3. Divide both sides by $6$ . What is $x$ ? 4
7.
Solve: $2(x + 3) = x + 10$
1. a) $x = 8$
2. b) $x = 4$
3. c) $x = 2$
4. d) $x = 6$
Answer: $x = 4$
$2(x + 3) = x + 10$ → $2x + 6 = x + 10$ → $x + 6 = 10$ → $x = 4$
8.
Solve: $4(x - 2) = 2x + 6$
Answer: 7
$4(x - 2) = 2x + 6$ → $4x - 8 = 2x + 6$ → $2x - 8 = 6$ → $2x = 14$ → $x = 7$
9.
Solve: $5(x + 1) = 3x + 13$
Answer: 4
1. Remove the parentheses on the left: $5(x + 1) = 5x + \square$ . What number goes in the box? 5
2. Subtract $3x$ from both sides. What do you get? 2x + 5 = 13
3. Subtract $5$ from both sides. What is the result? 2x = 8
4. Divide both sides by $2$ . What is $x$ ? 4
10.
Solve: $7x - 5 = 4x + 13$
Answer: 6
$7x - 5 = 4x + 13$ → $3x - 5 = 13$ → $3x = 18$ → $x = 6$
11.
What type of equation is $2x + 4 = 2(x + 2)$ ?
1. a)Contradiction (no solution)
2. b)One solution: $x = 0$
3. c)Identity (infinitely many solutions)
4. d)One solution: $x = 2$
Answer: Identity (infinitely many solutions)
$2x + 4 = 2(x + 2)$ → $2x + 4 = 2x + 4$ → $0 = 0$ . This is always true, so any value of $x$ works.
12.
Solve: $3(2x - 1) = 2(3x + 4) - 11$
Answer: Infinitely many solutions (identity)
1. Remove the parentheses on the left: $3(2x - 1) = 6x + \square$ . What number goes in the box? -3
2. Remove the parentheses on the right: $2(3x + 4) = 6x + \square$ . What number goes in the box? 8
3. Combine the numbers on the right: $6x + 8 - 11 = 6x + \square$ . What number goes in the box? -3
4. Compare both sides. What type of equation is this? identity (infinitely many solutions)
13.
What is the solution to $x + 5 = x - 3$ ?
1. a) $x = -8$
2. b)No solution (contradiction)
3. c)All real numbers (identity)
4. d) $x = 8$
Answer: No solution (contradiction)
$x + 5 = x - 3$ → $5 = -3$ . This is false, so there is no value of $x$ that makes the equation true.
14.
Solve: $2(3x + 4) - 5 = 4(x - 1) + x + 1$
Answer: -6
Left: $6x + 8 - 5 = 6x + 3$ . Right: $4x - 4 + x + 1 = 5x - 3$ . So $6x + 3 = 5x - 3$ → $x + 3 = -3$ → $x = -6$ . Check: $2(-14) - 5 = -33$ and $4(-7) - 6 + 1 = -33$ \checkmark
15.
Solve: $5(x - 2) + 3 = 2(x + 4) + 3x - 13$
Answer: No solution (contradiction)
1. Remove the parentheses on the left and combine the numbers: $5(x - 2) + 3 = 5x + \square$ . What number goes in the box? -7
2. Remove the parentheses on the right and combine like terms: $2(x + 4) + 3x - 13 = 5x + \square$ . What number goes in the box? -5
3. Compare: $5x - 7 = 5x - 5$ . Subtract $5x$ from both sides: $-7 = \square$ . What number goes in the box? -5
4. Is $-7 = -5$ true or false? false
16.
Solve: $4(2x - 3) = 3x + 2(x + 6)$
Answer: 8
$8x - 12 = 5x + 12$ → $3x - 12 = 12$ → $3x = 24$ → $x = 8$ . Check: $4(13) = 52$ and $24 + 2(14) = 52$ \checkmark
17.
Solve: $\frac{1}{2}(4x + 6) = x + 3 + x$
Answer: Infinitely many solutions (identity)
1. Remove the parentheses on the left: $\frac{1}{2}(4x + 6) = 2x + \square$ . What number goes in the box? 3
2. Combine the $x$ terms on the right: $x + 3 + x = \square x + 3$ . What number goes in the box? 2
3. Compare both sides. What type of equation is this? identity (infinitely many solutions)