At a local coffee shop, one customer purchased two muffins and three cups of coffee for
$14.47. Another customer ordered 3 muffins and 5 cups of coffee for $23.30. Write and
solve a system of equations to find the price of a single muffin and a single cup of coffee.
2m + 3c = 14.47
3m + 5c = 23.30
solve by substitution or elimination
Step 1. Subtract equations
(3M + 5C=23.3) - (2M + 3C= 14.47).
You get 1M + 2C = 8.83
Step 2. Do some arranging of the variables.
You get 1M= 8.83 - 2C. Presto! You can plug it in now to one of your original equations.
Step 3. Plug it in. I'll choose the first equation.
2(8.83-2C) + 3C= 14.47
Step 4. Multiply through.
17.66-4C+3C=14.47
Step 5. Clean it up.
-C=-3.19 which means C= 3.19
Alright I'm going to stop there. You should be able to find M with what I've given you.
To write and solve a system of equations to find the price of a single muffin and a single cup of coffee, let's first define the variables:
Let:
x = price of a single muffin
y = price of a single cup of coffee
Now we can set up the system of equations based on the given information:
Equation 1: For the first customer who purchased two muffins and three cups of coffee for $14.47:
2x + 3y = 14.47
Equation 2: For the second customer who ordered three muffins and five cups of coffee for $23.30:
3x + 5y = 23.30
To solve this system of equations, we can use any method of solving linear systems, such as substitution or elimination. Let's use the elimination method:
Multiply Equation 1 by 3 and Equation 2 by 2 to make the coefficients of x in both equations equal:
6x + 9y = 43.41
6x + 10y = 46.60
Now subtract Equation 1 from Equation 2:
(6x + 10y) - (6x + 9y) = 46.60 - 43.41
6x - 6x + 10y - 9y = 3.19
y = 3.19
Now, substitute the value of y back into Equation 1 or Equation 2 to solve for x:
2x + 3(3.19) = 14.47
2x + 9.57 = 14.47
2x = 14.47 - 9.57
2x = 4.90
x = 4.90/2
x = 2.45
Therefore, the price of a single muffin is $2.45 and the price of a single cup of coffee is $3.19.