Question

what is the value of x? solve the equation by using the distributive property first. there are many ways to solve this equation. try to start by distributing first. use the distributive property to rewrite the equation. \\(\frac{2}{3}(x - \frac{3}{2}) = 5\\) \\(square = 5\\)

Explanation:

Step1: Apply distributive property

The distributive property states that \(a(b - c)=ab - ac\). Here, \(a=\frac{2}{3}\), \(b = x\), and \(c=\frac{3}{2}\). So, \(\frac{2}{3}(x-\frac{3}{2})=\frac{2}{3}\times x-\frac{2}{3}\times\frac{3}{2}\).
Simplify \(\frac{2}{3}\times\frac{3}{2}\) which is \(1\). So the equation becomes \(\frac{2}{3}x - 1=5\). But first, for the first blank (rewriting using distributive property), we have \(\frac{2}{3}x-1\). Wait, the problem says "Use the distributive property to rewrite the equation" \(\frac{2}{3}(x - \frac{3}{2})=5\) as \(\square=5\). So applying distributive property:
\(\frac{2}{3}(x-\frac{3}{2})=\frac{2}{3}x-\frac{2}{3}\times\frac{3}{2}=\frac{2}{3}x - 1\)
So the first step (rewriting) is \(\frac{2}{3}x-1 = 5\)

Step2: Solve for \(x\)

Now, add \(1\) to both sides of the equation \(\frac{2}{3}x-1 = 5\)
\(\frac{2}{3}x-1 + 1=5 + 1\)
Simplify: \(\frac{2}{3}x=6\)
Multiply both sides by \(\frac{3}{2}\) to isolate \(x\):
\(x=6\times\frac{3}{2}\)
\(x = 9\)

Answer:

First, rewriting the equation using distributive property: \(\boldsymbol{\frac{2}{3}x - 1}\) (for the blank). Then, the value of \(x\) is \(\boldsymbol{9}\)