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?
Alos, Can step-by-step bot help with this question?
Sure, I can help you with this question.
Let's define the variables:
Let x be the number of toppings added to the large cheese pizza at Palanzio’s Pizzeria.
Let y be the number of toppings added to the large cheese pizza at Guido’s Pizza.
Now let's write the system of equations:
The cost of the large cheese pizza at Palanzio’s Pizzeria is $6.80 plus $0.90 for each topping. So the equation for the cost at Palanzio’s Pizzeria can be written as:
Cost at Palanzio's Pizzeria = $6.80 + $0.90(x)
The cost of the large cheese pizza at Guido’s Pizza is $7.30 plus $0.65 for each topping. So the equation for the cost at Guido’s Pizza can be written as:
Cost at Guido's Pizza = $7.30 + $0.65(y)
We want to find the number of toppings that makes the cost equal at both restaurants. So we can set the two equations equal to each other and solve for the variable x:
$6.80 + $0.90(x) = $7.30 + $0.65(y)
Now we can solve for x. Here are the steps:
1. First, subtract $6.80 from both sides of the equation:
$0.90(x) = $7.30 + $0.65(y) - $6.80
$0.90(x) = $0.50 + $0.65(y)
2. Next, divide both sides of the equation by $0.90 to isolate x:
x = ($0.50 + $0.65(y)) / $0.90
x ≈ 0.56 + 0.72(y)
So the solution for x is x ≈ 0.56 + 0.72(y).
Now, the question asks for the number of toppings needed to make the pizzas cost the same. Since y is not given, we cannot find the exact number of toppings. However, we have expressed x in terms of y, so you can choose any value for y and find the corresponding value for x using the equation x ≈ 0.56 + 0.72(y).
I hope this helps! Let me know if you have any further questions.
Yes, Step-by-Step Bot can help with this question. Let's start by defining our variables:
Let x be the number of toppings added to the large cheese pizza from Palanzio’s Pizzeria.
Let y be the number of toppings added to the large cheese pizza from Guido’s Pizza.
Now, we can create the system of equations based on the given information:
For Palanzio’s Pizzeria:
Cost = $6.80 + $0.90 * x (since each topping costs $0.90)
For Guido’s Pizza:
Cost = $7.30 + $0.65 * y (since each topping costs $0.65)
Since we want the pizzas to cost the same, we can set up the equation:
$6.80 + $0.90 * x = $7.30 + $0.65 * y
To solve this system of equations, we can use substitution or elimination method. In this case, let's use the elimination method:
Step 1: Multiply the second equation by 10 to eliminate the decimals:
$6.80 + $0.90 * x = $7.30 + $0.65 * y
Multiply both sides of the equation by 10:
68 + 9x = 73 + 6.5y
Step 2: Rearrange the equation and make the variables aligned:
9x - 6.5y = 73 - 68
Step 3: Simplify the equation:
9x - 6.5y = 5
Now, we can solve this equation for one variable and substitute the resulting value back into either of the original equations to solve for the other variable. However, in this case, we can notice that the coefficients of x and y (9 and -6.5) have a common factor of 0.5. To simplify the equation, let's divide all coefficients by 0.5:
18x - 13y = 10
Now, let's solve this simplified equation using the elimination method:
Step 1: Multiply the first equation by 13 and the second equation by 18 to create opposite coefficients for y:
13(18x - 13y) = 13(10)
18(9x - 6.5y) = 18(5)
Step 2: Distribute:
234x - 169y = 130
162x - 117y = 90
Step 3: Subtract the second equation from the first equation to eliminate y:
(234x - 169y) - (162x - 117y) = 130 - 90
Step 4: Simplify and solve for x:
234x - 169y - 162x + 117y = 40
72x - 52y = 40
72x = 52y + 40
x = (52y + 40) / 72
x = (13y + 10) / 18
Now, we have an expression for x in terms of y.
We can substitute this value of x into either of the original equations to solve for y. Let's substitute it into the equation for Palanzio's Pizzeria:
$6.80 + $0.90 * (13y + 10) / 18 = $7.30 + $0.65 * y
Simplify the equation and solve for y. Once we find the value of y, we can substitute it back into x to find the number of toppings needed for the pizzas to cost the same.
To solve this word problem, let's define some variables:
Let's say:
x = the number of toppings added to a large cheese pizza at Palanzio's Pizzeria
y = the number of toppings added to a large cheese pizza at Guido's Pizza
Now let's write the system of equations:
Equation 1: Cost of pizza at Palanzio's Pizzeria
The cost of a large pizza at Palanzio's Pizzeria is $6.80 plus $0.90 for each topping:
Cost of large pizza at Palanzio's Pizzeria = 6.80 + 0.90x
Equation 2: Cost of pizza at Guido's Pizza
The cost of a large cheese pizza at Guido's Pizza is $7.30 plus $0.65 for each topping:
Cost of large pizza at Guido's Pizza = 7.30 + 0.65y
Now let's set up an equation to find when the pizzas cost the same:
Equation 3: Cost of pizza at Palanzio's Pizzeria = Cost of pizza at Guido's Pizza
6.80 + 0.90x = 7.30 + 0.65y
Now, we have a system of equations:
Equation 1: Cost of large pizza at Palanzio's Pizzeria = 6.80 + 0.90x
Equation 2: Cost of large pizza at Guido's Pizza = 7.30 + 0.65y
Equation 3: Cost of pizza at Palanzio's Pizzeria = Cost of pizza at Guido's Pizza (6.80 + 0.90x = 7.30 + 0.65y)
To solve for the number of toppings needed to make the pizzas cost the same, we need to find the values of x and y that satisfy the system of equations.
One way to solve this system is by using the method of substitution or elimination. However, since the equations involve decimals, it might be easier to solve graphically.
Step-by-step, this is how you can solve the system of equations graphically:
1. Plot the graphs of the equations.
- For Equation 1: Cost of large pizza at Palanzio's Pizzeria, plot the points (0, 6.80) and (1, 7.70) since for every one topping added, the cost increases by $0.90.
- For Equation 2: Cost of large pizza at Guido's Pizza, plot the points (0, 7.30) and (1, 7.95) since for every one topping added, the cost increases by $0.65.
2. Now, identify the point where the two graphs intersect. That point represents the number of toppings needed to make the pizzas cost the same.
3. Read the x-coordinate of the intersection point, which will give you the number of toppings needed for the pizzas to cost the same at Palanzio's Pizzeria.
4. If you want to find the number of toppings needed for the pizzas to cost the same at Guido's Pizza, read the y-coordinate of the intersection point.
And that's it! You have now solved for the number of toppings needed to make the pizzas cost the same at Palanzio's Pizzeria and Guido's Pizza using a graphical method.