081e. Expanding Brackets and Simplifying Expressions

Learning Intentions

• expand brackets and collect like terms in algebraic expressions
• To understand that expanding brackets and collecting like terms is a useful technique when solving equations involving brackets

Pre-requisite Summary

• Know that a term is a part of an expression separated by $+$ or $-$ signs
• Know that like terms have the same pronumeral part, such as $3x$ and $-5x$
• Know how to use the distributive property, for example $a(b+c)=ab+ac$
• Know that when solving equations, the goal is to keep both sides equal
• Know basic one-step and two-step equation solving using inverse operations

Worked Examples

Worked Example 1

Expand and simplify: $3(x+4)$

Worked Example 2

Expand and simplify: $5(2a-3)+a$

Worked Example 3

Expand and simplify: $-2(y-6)+3y$

Worked Example 4

Solve: $4(x+2)=20$

Worked Example 5

Solve: $3(2x-1)=15$

Worked Example 6

Solve: $2(x+5)+3=17$

Problems

Problem 1

Expand and simplify: $4(x+3)$

Problem 2

Expand and simplify: $2(3a+4)-a$

Problem 3

Expand and simplify: $-3(y-2)+4y$

Problem 4

Solve: $5(x+1)=30$

Problem 5

Solve: $2(3x+4)=20$

Problem 6

Solve: $3(x+2)-1=14$

Exercises

Understanding and Fluency

1. Expand and simplify.
   a) $2(x+7)$
   b) $6(a+1)$
   c) $3(m+5)$

2. Expand and simplify.
   a) $4(x+2)+x$
   b) $3(y+4)+2y$
   c) $5(a+1)-2a$

3. Expand and simplify.
   a) $-2(x+3)$
   b) $-4(y-1)$
   c) $-3(a-5)$

4. Expand and simplify.
   a) $2(x+6)+3x$
   b) $5(m-2)+m$
   c) $-2(y-4)+y$

5. Solve.
   a) $3(x+1)=18$
   b) $2(x+4)=14$
   c) $6(x+2)=30$

6. Solve.
   a) $4(x-1)=20$
   b) $3(2x+1)=15$
   c) $2(x+3)+4=18$

7. Solve.
   a) $5(x+2)-3=22$
   b) $2(3x-1)=10$
   c) $4(x+1)+2=18$

8. Expand and simplify.
   a) $7(2x+1)-3x$
   b) $3(4a-2)+5a$
   c) $-2(3y-4)+y$

Reasoning

9. Noah says $3(x+4)=3x+4$. Explain why Noah is incorrect.

10. Without fully solving, decide which equation would be easier to solve first by expanding brackets, and explain why.
    a) $2(x+5)=18$
    b) $2x+5=18$

11. Mia simplified $-2(x-3)$ as $-2x-6$. Identify the error and write the correct expansion.

12. Explain why collecting like terms cannot combine $4x$ and $4x^2$.

Problem-solving

13. A rectangle has width $x+3$ cm and length $4$ cm. Write and simplify an expression for its perimeter.

14. Tickets to a school concert cost $6 each, and there is a $4 booking fee. Write and simplify an expression for the total cost of buying $x$ tickets.

15. A gardener plants $3$ rows of flowers, with $(x+2)$ flowers in each row, then adds $5$ more flowers. Write an expression and simplify it.

16. The equation $2(x+4)+1=19$ represents a number puzzle. Find the value of $x$.

Potential Misunderstandings

• A student may multiply the number outside the bracket by only the first term and forget the second term
• A student may not treat a negative sign outside the bracket as multiplying every term inside
• A student may combine unlike terms, such as adding $3x$ and $2$ to make $5x$
• A student may collect like terms before expanding brackets, even when the terms are not yet like terms
• A student may expand brackets correctly in an equation but then forget to keep both sides balanced
• A student may think expanding brackets changes the value of an expression, rather than rewriting it in an equivalent form
• A student may solve an equation involving brackets but not check the solution in the original equation

Next: 082. Using Formulas by Substitution and Solving