154. Solving Equations by Expanding Brackets

Learning Intentions

• To understand that the distributive law can be used to expand brackets within equations
• solve equations by expanding brackets

Pre-requisite Summary

• The distributive law allows a factor outside brackets to multiply every term inside the brackets
• Expanding brackets creates an equivalent expression
• An equation is a mathematical statement that two expressions are equal
• Equivalent equations have the same solution
• Like terms can be combined after expanding
• The same term can be added to or subtracted from both sides of an equation to keep the equation balanced
• The same non-zero number can multiply or divide both sides of an equation to produce an equivalent equation
• A solution should be checked by substitution into the original equation

Worked Examples

Worked Example 1

Solve $2(x+3)=14$.

Worked Example 2

Solve $3(a-4)=9$.

Worked Example 3

Solve $2(x+5)=x+9$.

Worked Example 4

Solve $4(y-1)=2y+6$.

Worked Example 5

Solve $3(2m+1)=15$.

Worked Example 6

Solve $5(p-2)=2p+9$.

Problems

Problem 1

Solve $3(x+2)=18$.

Problem 2

Solve $2(a-5)=8$.

Problem 3

Solve $4(x+1)=x+10$.

Problem 4

Solve $3(y-2)=y+8$.

Problem 5

Solve $2(3m+4)=20$.

Problem 6

Solve $4(p-3)=2p+6$.

Exercises

Understanding and Fluency

1. Expand each side of the equation.
   a) $2(x+4)=12$
   b) $3(a-1)=15$
   c) $4(m+2)=20$

2. Expand each side of the equation.
   a) $2(x+3)=x+7$
   b) $5(y-2)=y+8$
   c) $3(p+4)=2p+9$

3. Solve each equation.
   a) $2(x+6)=16$
   b) $3(a-2)=12$
   c) $5(k+1)=25$

4. Solve each equation.
   a) $4(y-3)=8$
   b) $2(m+7)=18$
   c) $6(p-1)=24$

5. Solve each equation.
   a) $2(x+4)=x+11$
   b) $3(a-1)=a+7$
   c) $4(t+2)=2t+14$

6. Solve each equation.
   a) $5(y-2)=3y+6$
   b) $3(m+5)=2m+11$
   c) $2(p-4)=p-7$

7. Solve each equation.
   a) $2(2x+3)=18$
   b) $3(2a-1)=15$
   c) $4(3m+2)=32$

8. Solve each equation.
   a) $3(2y+1)=y+13$
   b) $2(4p-3)=p+12$
   c) $5(2k-1)=3k+16$

Reasoning

9. Explain why expanding both sides of an equation produces an equivalent equation.

10. A student writes $3(x+4)=12$ as $3x+4=12$. Explain the error.

11. Noah says the first step in solving $2(x+5)=x+11$ is to divide both sides by $2$. Explain why expanding first is more helpful.

12. Explain why solving $4(y-1)=2y+6$ requires both expanding and then solving an equation with pronumerals on both sides.

Problem-solving

13. A gym charges $2 per class plus a $3 joining fee. One person pays $17 in total. Write and solve an equation to find the number of classes attended.

14. A rectangle has width $x+2$ cm and fixed length $3$ cm. Its area is $24{\text{ cm}}^2$. Write and solve an equation to find $x$.

15. A builder makes $4$ identical shelves, each of length $p-1$ metres. The total length of timber used is $20$ metres. Write and solve an equation to find $p$.

16. A school hires buses. Each bus holds $2n+5$ students. If $3$ buses carry $51$ students altogether, write and solve an equation to find $n$.

Potential Misunderstandings

• Thinking the distributive law only applies in expressions and not in equations
• Multiplying the outside factor by only the first term inside the brackets
• Forgetting to multiply both the number and the pronumeral terms inside the brackets
• Writing $2(x+3)$ as $2x+3$ instead of $2x+6$
• Expanding correctly but then making errors when combining like terms
• Thinking that expanding changes the solution of the equation rather than creating an equivalent equation
• Trying to solve the equation before removing the brackets when expansion would simplify the structure
• Making sign errors when expanding brackets with subtraction, such as $3(a-4)$
• Forgetting that after expanding, the equation may still need terms collected on one side
• Failing to check the solution in the original equation