LESSON 10 | SESSION 4 6. Mateo is making a school spirit flag. He has $\frac{1}{3}$ as many yards of red fabric as blue fabric. He buys $2 \frac{2}{3}$ yards more red fabric. Now he has equal amounts of red and blue fabric. Use $x$ to represent the amount of blue fabric Mateo has. Which equations could you use to find the amount of red fabric Mateo has? Select all the correct answers. A $x=\frac{1}{3} x+2 \frac{2}{3}$ B $\frac{1}{3} x=x+2 \frac{2}{3}$ c $x=\frac{1}{3} x-2 \frac{2}{3}$ D $x-2 \frac{2}{3}=\frac{1}{3} x$ E $x+2 \frac{2}{3}=\frac{1}{3} x+2 \frac{2}{3}$ $F \quad x=\frac{1}{3}\left(x+2 \frac{2}{3}\right)$ (7) Greg draws three lines on the board. Lines $m$ and $n$ are parallel. Greg says that the measure of $\angle 2$ is $\frac{2}{3}$ the measure of $\angle 1$. What is the measure of $\angle 2$ ?
Solution
Step 1 ：The problem states that Mateo initially has \(\frac{1}{3}\) as many yards of red fabric as blue fabric. After buying \(2 \frac{2}{3}\) yards more red fabric, he has equal amounts of red and blue fabric. We can represent the amount of blue fabric Mateo has as \(x\).
Step 2 ：The initial amount of red fabric Mateo has is \(\frac{1}{3}x\). After buying more fabric, the total amount of red fabric he has is \(\frac{1}{3}x + 2 \frac{2}{3}\). Since this is equal to the amount of blue fabric he has, we can write the equation as \(x = \frac{1}{3}x + 2 \frac{2}{3}\).
Step 3 ：So, the correct equation to represent the problem is option A: \(x = \frac{1}{3}x + 2 \frac{2}{3}\).
Step 4 ：\(x = x\)
Step 5 ：Final Answer: \(\boxed{x = \frac{1}{3}x + 2 \frac{2}{3}}\)
