Question

3.06: multi-step equations
complete the worksheet on lined notebook paper.
solve each equation.
you must show inverse operations on both sides of the equation.
you must maintain the equation on all steps.
no ambiguous fractions.
the only time you can have a decimal solution is when the original equation had a decimal.
show all work! no work = no credit

1. 5x + 8 = 23 2. 6a - 11 = -13 3. 3a + 2 + 4a = -23
2. 12n - 7 - 3n = 11 5. x + 6 + 2x = 15 6. 40 = 15 - 9x - 2
3. 3x - 1 - 7x = -33 8. 2(2x - 9) = 38 9. -2(v - 6) = -8
4. 6(w + 8) = 54 11. 6(x - 1) - x = -21 12. 2(z + 5) + 4 = -12
5. 4(c - 3) - 6 = 10 14. -13 = 4(5x - 1) - 11x 15. 4 - 2(3x + 1) = 32

Explanation:

Step1: Solve \(5x + 8 = 23\)

Subtract 8 from both sides: \(5x + 8 - 8 = 23 - 8\) → \(5x = 15\)
Divide both sides by 5: \(\frac{5x}{5} = \frac{15}{5}\) → \(x = 3\)

Step2: Solve \(6a - 11 = -13\)

Add 11 to both sides: \(6a - 11 + 11 = -13 + 11\) → \(6a = -2\)
Divide by 6: \(a = \frac{-2}{6} = -\frac{1}{3}\)

Step3: Solve \(3a + 2 + 4a = -23\)

Combine like terms: \(7a + 2 = -23\)
Subtract 2: \(7a = -25\)
Divide: \(a = -\frac{25}{7}\)

Step4: Solve \(12n - 7 - 3n = 11\)

Combine like terms: \(9n - 7 = 11\)
Add 7: \(9n = 18\)
Divide: \(n = 2\)

Step5: Solve \(x + 6 + 2x = 15\)

Combine like terms: \(3x + 6 = 15\)
Subtract 6: \(3x = 9\)
Divide: \(x = 3\)

Step6: Solve \(40 = 15 - 9x - 2\)

Simplify RHS: \(40 = 13 - 9x\)
Subtract 13: \(27 = -9x\)
Divide: \(x = -3\)

Step7: Solve \(3x - 1 - 7x = -33\)

Combine like terms: \(-4x - 1 = -33\)
Add 1: \(-4x = -32\)
Divide: \(x = 8\)

Step8: Solve \(2(2x - 9) = 38\)

Divide by 2: \(2x - 9 = 19\)
Add 9: \(2x = 28\)
Divide: \(x = 14\)

Step9: Solve \(-2(v - 6) = -8\)

Divide by -2: \(v - 6 = 4\)
Add 6: \(v = 10\)

Step10: Solve \(6(w + 8) = 54\)

Divide by 6: \(w + 8 = 9\)
Subtract 8: \(w = 1\)

Step11: Solve \(6(x - 1) - x = -21\)

Distribute: \(6x - 6 - x = -21\)
Combine like terms: \(5x - 6 = -21\)
Add 6: \(5x = -15\)
Divide: \(x = -3\)

Step12: Solve \(2(z + 5) + 4 = -12\)

Distribute: \(2z + 10 + 4 = -12\)
Combine like terms: \(2z + 14 = -12\)
Subtract 14: \(2z = -26\)
Divide: \(z = -13\)

Step13: Solve \(4(c - 3) - 6 = 10\)

Distribute: \(4c - 12 - 6 = 10\)
Combine like terms: \(4c - 18 = 10\)
Add 18: \(4c = 28\)
Divide: \(c = 7\)

Step14: Solve \(-13 = 4(5x - 1) - 11x\)

Distribute: \(-13 = 20x - 4 - 11x\)
Combine like terms: \(-13 = 9x - 4\)
Add 4: \(-9 = 9x\)
Divide: \(x = -1\)

Step15: Solve \(4 - 2(3x + 1) = 32\)

Distribute: \(4 - 6x - 2 = 32\)
Combine like terms: \(2 - 6x = 32\)
Subtract 2: \(-6x = 30\)
Divide: \(x = -5\)

Answer:

1. \(x = 3\)
2. \(a = -\frac{1}{3}\)
3. \(a = -\frac{25}{7}\)
4. \(n = 2\)
5. \(x = 3\)
6. \(x = -3\)
7. \(x = 8\)
8. \(x = 14\)
9. \(v = 10\)
10. \(w = 1\)
11. \(x = -3\)
12. \(z = -13\)
13. \(c = 7\)
14. \(x = -1\)
15. \(x = -5\)