Question

write an equation to model the situation. then solve for x.
2 twice the sum of paloma’s age and 15 is 106. how old is paloma?
3 two coaches each buy 1 soccer ball for $12 and 4 cones. each cone costs the same amount. together the coaches spend $40. how much does each cone cost?
4 explain how you determined paloma’s age in problem 2.

Explanation:

Problem 2:

Step1: Define variable

Let \( x \) be Paloma's age. The sum of Paloma's age and 15 is \( x + 15 \), and twice that sum is \( 2(x + 15) \). This equals 106, so the equation is \( 2(x + 15)=106 \).

Step2: Solve the equation

First, divide both sides by 2: \( x + 15=\frac{106}{2}=53 \). Then subtract 15 from both sides: \( x = 53 - 15 = 38 \).

Step1: Define variable

Let \( c \) be the cost of one cone. Each coach buys 1 soccer ball (\$12) and 4 cones (\( 4c \)), so one coach's total is \( 12 + 4c \). Two coaches spend \( 2(12 + 4c) \), which equals 40. So the equation is \( 2(12 + 4c)=40 \).

Step2: Solve the equation

Divide both sides by 2: \( 12 + 4c = 20 \). Subtract 12: \( 4c = 20 - 12 = 8 \). Divide by 4: \( c=\frac{8}{4}=2 \).

In problem 2, we first set up the equation \( 2(x + 15)=106 \) by translating the words "twice the sum of Paloma’s age and 15 is 106" into algebra (letting \( x \) be her age). Then we solved the equation by first dividing both sides by 2 to isolate the sum, then subtracting 15 to find \( x \).

Answer:

The equation is \( 2(x + 15)=106 \) and Paloma's age \( x = 38 \).

Problem 3: