Solving systems of equations word problems worksheet. For all problems, define variables, write the system of equations, and solve for all variables. The directions are from TAKS, so do all three (variables, equations, and solve) no matter what is asked in the problem. A large pizza at Palanzio’s Pizzeria costs
0.90 for each topping. The cost of a large cheese pizza at Guido’s Pizza is
0.65 for each topping. How many toppings need to be added to the large cheese pizza from Palanzio’s Pizzeria and Guido’s Pizza in order for the pizzas to cost the same, not including tax
The Correct Answer and Explanation is:
Problem:
A large pizza at Palanzio’s Pizzeria costs $0.90 for each topping. The cost of a large cheese pizza at Guido’s Pizza is $0.65 for each topping. How many toppings need to be added to each pizza so that they cost the same (excluding tax)?
Step 1: Define Variables
Let:
- xx = number of toppings added to each pizza
- P(x)P(x) = total cost of a Palanzio’s pizza with xx toppings
- G(x)G(x) = total cost of a Guido’s pizza with xx toppings
Assume both pizzas start at the same base cheese pizza cost. If not, we would need that base cost to solve fully. But since it’s omitted, we’ll assume we’re comparing just the cost of toppings.
Step 2: Write the System of Equations
We are told the cost per topping:
- Palanzio’s: $0.90 per topping → P(x)=0.90xP(x) = 0.90x
- Guido’s: $0.65 per topping → G(x)=0.65xG(x) = 0.65x
We want to find when the total cost is equal, so set: 0.90x=0.65×0.90x = 0.65x
Step 3: Solve the Equation
0.90x=0.65×0.90x = 0.65x
Subtract 0.65×0.65x from both sides: 0.90x−0.65x=00.25x=00.90x – 0.65x = 0 \\ 0.25x = 0
Divide both sides by 0.25: x=0x = 0
Final Answer: 0\boxed{0} toppings
Explanation
This problem compares the cost of adding toppings to pizzas from two different pizzerias: Palanzio’s and Guido’s. We are told that Palanzio’s charges $0.90 per topping and Guido’s charges $0.65 per topping. The question asks us to determine how many toppings must be added to each pizza so that the total cost of the pizzas becomes equal.
To solve this, we define the variable xx as the number of toppings added. Since we aren’t given the base price of either pizza (just the topping prices), we assume the comparison is solely based on the topping costs. Thus, the total cost from each pizzeria is a simple multiplication of the number of toppings by the price per topping.
We represent this mathematically:
- Cost from Palanzio’s: 0.90×0.90x
- Cost from Guido’s: 0.65×0.65x
To find when the pizzas cost the same, we set these expressions equal: 0.90x=0.65×0.90x = 0.65x
Solving the equation: 0.90x−0.65x=0⇒0.25x=0⇒x=00.90x – 0.65x = 0 \Rightarrow 0.25x = 0 \Rightarrow x = 0
This means only when zero toppings are added will the pizzas cost the same based on the topping prices. Any additional topping will make Palanzio’s pizza more expensive due to the higher cost per topping. Therefore, the correct answer is 0 toppings—if no toppings are added, the topping cost is zero for both pizzas, and their prices are equal.
