Linear Equations with Brackets - Grade 8 - Medium Solve each equation or word problem. Give each answer as a whole number. 1. Solve 3(x + 4) = 27. 2. Solve 2(3x - 5) = 20. 3. Solve 5(2x - 1) = 3x + 16. 4. Solve 4(x + 3) = 2(x + 9). 5. Solve 7 - 2(x - 1) = 13. 6. Solve 3(2x + 1) - 4(x - 2) = 17. 7. Solve 3(x - 2)/4 = 6. 8. A rectangle has a width of x cm and a length of (x + 6) cm. Its perimeter is 52 cm. Find the width of the rectangle. 9. Sara thinks of a number, adds 7, and multiplies the result by 3. She gets the same answer as when she multiplies her number by 5 and then subtracts 3. Find her number. 10. A cinema charges $x for an adult ticket and $(x - 2) for a child ticket. Three adult tickets and two child tickets cost $41 altogether. Find the price of an adult ticket in $. Worked Solutions 1. 3(x + 4) = 27 Expand the bracket: 3x + 12 = 27 Subtract 12 from both sides: 3x = 15 Divide both sides by 3: x = 5 Answer: x = 5 2. 2(3x - 5) = 20 Expand the bracket: 6x - 10 = 20 Add 10 to both sides: 6x = 30 Divide both sides by 6: x = 5 Answer: x = 5 3. 5(2x - 1) = 3x + 16 Expand the bracket: 10x - 5 = 3x + 16 Subtract 3x from both sides: 7x - 5 = 16 Add 5 to both sides: 7x = 21 Divide both sides by 7: x = 3 Answer: x = 3 4. 4(x + 3) = 2(x + 9) Expand both brackets: 4x + 12 = 2x + 18 Subtract 2x and 12 from both sides: 2x = 6 Divide both sides by 2: x = 3 Answer: x = 3 5. 7 - 2(x - 1) = 13 Expand the bracket (multiply -2 by each term): 7 - 2x + 2 = 13 Collect the numbers: 9 - 2x = 13 Subtract 9 from both sides: -2x = 4 Divide both sides by -2: x = -2 Answer: x = -2 6. 3(2x + 1) - 4(x - 2) = 17 Expand both brackets (multiply -4 by each term): 6x + 3 - 4x + 8 = 17 Collect like terms: 2x + 11 = 17 Subtract 11 from both sides: 2x = 6 Divide both sides by 2: x = 3 Answer: x = 3 7. 3(x - 2)/4 = 6 Multiply both sides by 4: 3(x - 2) = 24 Expand the bracket: 3x - 6 = 24 Add 6 to both sides: 3x = 30 Divide both sides by 3: x = 10 Answer: x = 10 8. Perimeter = 2 x (width + length) Write the equation: 2(x + x + 6) = 52, so 2(2x + 6) = 52 Expand the bracket: 4x + 12 = 52 Subtract 12 from both sides: 4x = 40 Divide both sides by 4: x = 10 Answer: the width is 10 cm 9. Let the number be n. Write the equation: 3(n + 7) = 5n - 3 Expand the bracket: 3n + 21 = 5n - 3 Subtract 3n and add 3 to both sides: 24 = 2n Divide both sides by 2: n = 12 Answer: the number is 12 10. Total cost = 3 adult tickets + 2 child tickets Write the equation: 3x + 2(x - 2) = 41 Expand the bracket: 3x + 2x - 4 = 41, so 5x - 4 = 41 Add 4 to both sides: 5x = 45 Divide both sides by 5: x = 9 Answer: an adult ticket costs $9 Teacher note: Students often multiply only the first term inside a bracket or lose a negative sign when a minus sits in front of a bracket. They also make errors when moving terms across the equals sign, so each answer should be checked by substituting it back into the original equation.