Question

Systems of two equations to solve systems of three equations.

Students are baking shortbread, croissants, and pound cakes.
• One group of students used a total of 3,525 grams of flour to make
5 batches of shortbread, 4 pound cakes, and 2 batches of croissants.
• Another group of students used 1,020 grams of flour to make one
batch each of shortbread and croissants and one pound cake.
• There is the same amount of flour in 4 pound cakes as there is
in 2 batches of croissants.

1. Let s = the amount of flour in a batch of shortbread, let c = the
amount of flour in a batch of croissants, and let p = the amount
of flour in a pound cake. Write an equation for each of the
3 statements above.

2. Multiply one equation, then add or subtract it with another
equation you wrote in Exercise 1 to eliminate the variable p.

3. Use the third equation and your result from Exercise 2 to write a
system of two equations with the variables p and c. Solve your system
and find the values for p and c.

4. What is the value of s? Explain what you did.

Answer 1

Explanation:

Let s = the amount of flour in a batch of shortbread, let c = the amount of flour in a batch of croissants, and let p = the amount of flour in a pound cake.

The first statement can be written as 5s + 4p + 2c = 3525.

The second statement can be written as s + c + p = 1020.

The third statement can be written as 4p = 2c.

Multiplying the third equation by 2, we get 8p = 4c.

Subtracting this from the first equation, we get 5s + 4p + 2c - 8p = 3525 - 8p.

Simplifying, we get 5s - 6c = 3525 - 8p.

Now we have a system of two equations with the variables c and p:

5s - 6c = 3525 - 8p

s + c + p = 1020

We can use substitution to eliminate c. Solving the second equation for c, we get c = 1020 - s - p.

Substituting this into the first equation, we get 5s - 6(1020 - s - p) = 3525 - 8p.

Expanding the second equation and simplifying, we get 11s = 4545 - 8p.

Solving for s, we get s = (4545 - 8p) / 11.

Substituting the value of s into the second equation, we get c + p + (4545 - 8p) / 11 = 1020.

Expanding and simplifying, we get c + p + (4545 - 8p) / 11 = 1020.

Expanding, we get c + p = 1020 - (4545 - 8p) / 11.

Solving for c and p, we find that c = 664 grams and p = 356 grams.

Finally, the value of s is (4545 - 8 * 356) / 11 = 735 grams.