Solving systems of equations word problems worksheet For all problems, define variables, write the system of equations and solve for all variables.
-A large pizza at Palanzio’s Pizzeria costs $6.80 plus $0.90 for each topping. The cost of a large cheese pizza at Guido’s Pizza is $7.30 plus $0.65 for each topping. How many toppings need to be added to a large cheese pizza from Palanzio’s Pizzeria and Guido’s Pizza in order for the pizzas to cost the same, not including tax?
Let's define:
x = number of toppings added to the large cheese pizza at Palanzio's Pizzeria
y = number of toppings added to the large cheese pizza at Guido's Pizza
The system of equations is:
Palanzio's Pizzeria: 6.80 + 0.90x
Guido's Pizza: 7.30 + 0.65y
To find the number of toppings needed for the pizzas to cost the same, we set the two equations equal to each other:
6.80 + 0.90x = 7.30 + 0.65y
Now, we can solve for x and y.
0.90x - 0.65y = 7.30 - 6.80
0.90x - 0.65y = 0.50
To get rid of the decimals, we can multiply both sides of the equation by 100:
90x - 65y = 50
Now, this equation can be solved further depending on the specific values of x and y that are desired.
Step 1: Define variables.
Let's define the variables:
x = number of toppings for Palanzio's Pizza
y = number of toppings for Guido's Pizza
Step 2: Write the system of equations.
The cost of a large pizza at Palanzio's Pizzeria can be represented as $6.80 + $0.90x.
The cost of a large cheese pizza at Guido's Pizza can be represented as $7.30 + $0.65y.
Step 3: Set the two expressions equal to each other.
$6.80 + $0.90x = $7.30 + $0.65y
Step 4: Solve for one variable in terms of the other.
$0.90x = $7.30 - $6.80 + $0.65y
$0.90x = $0.50 + $0.65y
Divide both sides by $0.90:
x = ($0.50 + $0.65y) / $0.90
Step 5: Simplify the equation.
x = $0.5556 + $0.7222y
So the number of toppings for Palanzio's pizza in order for the pizzas to cost the same, not including tax, is $0.5556 + $0.7222y.
To solve this word problem, we need to define some variables. Let's call the number of toppings for the Palanzio's pizza "p" and the number of toppings for the Guido's pizza "g".
The cost of a large pizza at Palanzio's Pizzeria is given by the equation:
Cost of Palanzio's pizza = $6.80 + $0.90 * p
The cost of a large cheese pizza at Guido's Pizza is given by the equation:
Cost of Guido's pizza = $7.30 + $0.65 * g
We want to find the number of toppings needed for the pizzas to cost the same, so we set the two equations equal to each other and solve for the variables p and g:
$6.80 + $0.90 * p = $7.30 + $0.65 * g
Subtract $6.80 from both sides:
$0.90 * p = $7.30 + $0.65 * g - $6.80
Simplify:
$0.90 * p = $0.50 + $0.65 * g
Now, we can isolate one variable (either p or g) and substitute it into the other equation. Let's isolate p:
p = ($0.50 + $0.65 * g) / $0.90
Now we have p in terms of g. We can substitute this expression for p into the other equation and solve for g:
$0.90 * (($0.50 + $0.65 * g) / $0.90) = $7.30 + $0.65 * g
Simplify:
$0.50 + $0.65 * g = $7.30 + $0.65 * g
Subtract $0.65 * g from both sides:
$0.50 = $7.30
This equation is not possible, so there is no solution for this word problem. This means that there is no number of toppings that can be added to make the pizzas cost the same.