Solving linear equations with brackets just got easier

In this guide, we continue solving linear equations by expanding brackets and simplifying expressions, picking up from question 6e. The focus is on distributing terms correctly, simplifying, and isolating variables step by step. Let’s dive into the examples.

Solving Linear Equations with BRACKETS Just Got Easier

Understanding how to expand brackets in a linear equation

Expanding brackets is a key skill in algebra. You distribute the outside number to each term inside, then simplify and solve for the variable.

Example 1: Expanding brackets in a linear equation

e) 8(4 + q) = 40

• Expand: 8 * 4 + 8 * q = 32 + 8q.
• Equation: 32 + 8q = 40.
• Solve: Subtract 32: 32 + 8q - 32 = 40 - 32, so 8q = 8. Divide by 8: 8q / 8 = 8 / 8, so q = 1.
• Answer: q = 1.

f) 7(p + 7) = 133

• Expand: 7 * p + 7 * 7 = 7p + 49.
• Equation: 7p + 49 = 133.
• Solve: Subtract 49: 7p + 49 - 49 = 133 - 49, so 7p = 84. Divide by 7: 7p / 7 = 84 / 7, so p = 12.
• Answer: p = 12.

g) 8(m + 7) = 96

• Expand: 8 * m + 8 * 7 = 8m + 56.
• Equation: 8m + 56 = 96.
• Solve: Subtract 56: 8m + 56 - 56 = 96 - 56, so 8m = 40. Divide by 8: 8m / 8 = 40 / 8, so m = 5.
• Answer: m = 5.

h) 2(b + 5) = 22

• Expand: 2 * b + 2 * 5 = 2b + 10.
• Equation: 2b + 10 = 22.
• Solve: Subtract 10: 2b + 10 - 10 = 22 - 10, so 2b = 12. Divide by 2: 2b / 2 = 12 / 2, so b = 6.
• Answer: b = 6.

Example 2: Solving a linear equation with expanded brackets

i) 5(2 + p) = 25

• Expand: 5 * 2 + 5 * p = 10 + 5p.
• Equation: 10 + 5p = 25.
• Solve: Subtract 10: 10 + 5p - 10 = 25 - 10, so 5p = 15. Divide by 5: 5p / 5 = 15 / 5, so p = 3.
• Answer: p = 3.

j) 7(p + 2) = 63

• Expand: 7 * p + 7 * 2 = 7p + 14.
• Equation: 7p + 14 = 63.
• Solve: Subtract 14: 7p + 14 - 14 = 63 - 14, so 7p = 49. Divide by 7: 7p / 7 = 49 / 7, so p = 7.
• Answer: p = 7.

k) 9(y - 6) = 27

• Expand: 9 * y - 9 * 6 = 9y - 54.
• Equation: 9y - 54 = 27.
• Solve: Add 54: 9y - 54 + 54 = 27 + 54, so 9y = 81. Divide by 9: 9y / 9 = 81 / 9, so y = 9.
• Answer: y = 9.

l) 2(r + 8) = 32

• Expand: 2 * r + 2 * 8 = 2r + 16.
• Equation: 2r + 16 = 32.
• Solve: Subtract 16: 2r + 16 - 16 = 32 - 16, so 2r = 16. Divide by 2: 2r / 2 = 16 / 2, so r = 8.
• Answer: r = 8.

Expanding and solving more linear equations

Question 7: Solve by expanding brackets first.

a) 6(3 + 2d) = 54

• Expand: 6 * 3 + 6 * 2d = 18 + 12d.
• Equation: 18 + 12d = 54.
• Solve: Subtract 18: 18 + 12d - 18 = 54 - 18, so 12d = 36. Divide by 12: 12d / 12 = 36 / 12, so d = 3.
• Answer: d = 3.

b) 8(h - 7) = 56

• Expand: 8 * h - 8 * 7 = 8h - 56.
• Equation: 8h - 56 = 56.
• Solve: Add 56: 8h - 56 + 56 = 56 + 56, so 8h = 112. Divide by 8: 8h / 8 = 112 / 8, so h = 14.
• Answer: h = 14.

Handling complex linear equations

c) 3(2x - 4) = 18

• Expand: 3 * 2x - 3 * 4 = 6x - 12.
• Equation: 6x - 12 = 18.
• Solve: Add 12: 6x - 12 + 12 = 18 + 12, so 6x = 30. Divide by 6: 6x / 6 = 30 / 6, so x = 5.
• Answer: x = 5.

d) 3(3 + 6e) = 27

• Expand: 3 * 3 + 3 * 6e = 9 + 18e.
• Equation: 9 + 18e = 27.
• Solve: Subtract 9: 9 + 18e - 9 = 27 - 9, so 18e = 18. Divide by 18: 18e / 18 = 18 / 18, so e = 1.
• Answer: e = 1.

More practice problems with linear equations

e) 4(3a + 8) = 44

• Expand: 4 * 3a + 4 * 8 = 12a + 32.
• Equation: 12a + 32 = 44.
• Solve: Subtract 32: 12a + 32 - 32 = 44 - 32, so 12a = 12. Divide by 12: 12a / 12 = 12 / 12, so a = 1.
• Answer: a = 1.

f) 6(5r - 10) = 30

• Expand: 6 * 5r - 6 * 10 = 30r - 60.
• Equation: 30r - 60 = 30.
• Solve: Add 60: 30r - 60 + 60 = 30 + 60, so 30r = 90. Divide by 30: 30r / 30 = 90 / 30, so r = 3.
• Answer: r = 3.

g) 5(9u - 7) = 10

• Expand: 5 * 9u - 5 * 7 = 45u - 35.
• Equation: 45u - 35 = 10.
• Solve: Add 35: 45u - 35 + 35 = 10 + 35, so 45u = 45. Divide by 45: 45u / 45 = 45 / 45, so u = 1.
• Answer: u = 1.

Conclusion

Expanding brackets and simplifying expressions are foundational steps in solving linear equations. By distributing correctly and isolating the variable, you can handle any equation with brackets. Practice these problems to build confidence. Thanks for following along—see you next time!