Question

name: _____________________________. date: ________
directions: simplify each expression. show all your work for each step. no calculators allowed!

1. $-2y + \frac{1}{3}(9y - 6)$

1. $4 + 3(x + 2) + 2.5x - 1.5x$

1. $1.5(4y + 2) - 2y + 3.5$

these are tricky! take your time and show each step clearly.

1. $-x + \frac{1}{3}(6x - 9) + 2(1.5x + 3)$

1. $\frac{1}{4}(8y + 12) + 1.5(2y - 4) + y$

Explanation:

Problem 1: Simplify \(-2y + \frac{1}{3}(9y - 6)\)

Step 1: Distribute the \(\frac{1}{3}\)

We use the distributive property \(a(b + c)=ab+ac\) to expand \(\frac{1}{3}(9y - 6)\).
\(\frac{1}{3}(9y - 6)=\frac{1}{3}\times9y-\frac{1}{3}\times6 = 3y-2\)
So the expression becomes \(-2y + 3y-2\)

Step 2: Combine like terms

Combine the \(y\) terms: \(-2y+3y=( - 2 + 3)y=y\)
So the simplified expression is \(y - 2\)

Step 1: Distribute the \(3\)

Using the distributive property \(a(b + c)=ab + ac\) for \(3(x + 2)\)
\(3(x + 2)=3x+6\)
The expression becomes \(4 + 3x+6+2.5x-1.5x\)

Step 2: Combine like terms (constant terms and \(x\) terms)

First, combine the constant terms: \(4 + 6=10\)
Then, combine the \(x\) terms: \(3x+2.5x-1.5x=(3 + 2.5-1.5)x = 4x\)
So the simplified expression is \(4x + 10\)

Step 1: Distribute the \(1.5\)

Using the distributive property \(a(b + c)=ab+ac\) for \(1.5(4y + 2)\)
\(1.5(4y + 2)=1.5\times4y+1.5\times2 = 6y+3\)
The expression becomes \(6y + 3-2y+3.5\)

Step 2: Combine like terms

Combine the \(y\) terms: \(6y-2y=(6 - 2)y = 4y\)
Combine the constant terms: \(3+3.5 = 6.5\)
So the simplified expression is \(4y+6.5\)

Answer:

\(y - 2\)

Problem 2: Simplify \(4 + 3(x + 2)+2.5x-1.5x\)