Question

1. the number of years that brad has been on the soccer team is 2 less than 5 times the number of years that scott has. in total, the boys have been on the soccer team for 10 years. how long has brad been on the soccer team? 4. there are 10 less trumpet players than saxophone players. the number of saxophone players is 3 times the number of trumpet players. how many trumpet players are there? 5. the length of a rectangle is 3 inches more than its width. if the perimeter is 42 inches, find the dimensions of the rectangle. 6. the length of a rectangle is 4 centimeters more than twice its width. if the perimeter of the rectangle is 86 centimeters, find the dimensions of the rectangle.

Explanation:

3.

Step1: Define variables

Let $x$ be the number of years Scott has been on the team. Then Brad has been on the team for $5x - 2$ years.

Step2: Set up equation

The total number of years they have been on the team is 10, so $x+(5x - 2)=10$.

Step3: Simplify the equation

Combine like - terms: $x + 5x-2=10$, which becomes $6x-2 = 10$.

Step4: Solve for $x$

Add 2 to both sides: $6x=10 + 2=12$. Then divide both sides by 6, so $x = 2$.

Step5: Find Brad's years

Substitute $x = 2$ into the expression for Brad's years: $5x-2=5\times2-2=8$.

Step1: Define variables

Let $t$ be the number of trumpet players and $s$ be the number of saxophone players.

Step2: Set up equations

We have $t=s - 10$ and $s = 3t$.

Step3: Substitute

Substitute $s = 3t$ into $t=s - 10$, we get $t=3t-10$.

Step4: Solve for $t$

Subtract $t$ from both sides: $0=3t - t-10$, so $2t=10$. Divide both sides by 2, $t = 5$.

Step1: Define variables

Let $w$ be the width of the rectangle. Then the length $l=w + 3$.

Step2: Recall perimeter formula

The perimeter of a rectangle $P = 2(l + w)$. Given $P = 42$, so $2((w + 3)+w)=42$.

Step3: Simplify the equation

First, simplify inside the parentheses: $2(2w + 3)=42$. Then distribute the 2: $4w+6 = 42$.

Step4: Solve for $w$

Subtract 6 from both sides: $4w=42 - 6=36$. Divide both sides by 4, $w = 9$.

Step5: Find the length

$l=w + 3=9 + 3=12$.

Answer:

8 years

4.