There are 45 questions on allison's english test. the multiple choice questions are worth 2 points each, and the short answers are worth 5 points each. the entire tea is worth 120 points.
how many short answer questions are there?

You can solve this problem by creating a system of equations and solving by substitution. This allows you to solve for one of the unknown values (in this case, the number of short answer and number of multiple choice) and you can then solve for both, if you would like.

The two variables in your equations standing in for these missing values represent the number of short answer (s) and the number of multiple choice questions (m).

First create an equation for the number of total questions on the test. The total number of multiple choice and short answer questions is 45:
m + s = 45
Then create an equation with these variables representing the number of points on the test. The point values for each question will be the variable coefficients (numbers that the variables are multiplied by in this problem).
2m + 5s = 120

Now you can take the first equation - since it is simpler, the variables don't have coefficients so it only requires subtraction to get one variable by itself - solve for one of the variables, and substitute the result into the second equation in place of that variable. You will then be able to solve for that variable!

First, solve for m in the first equation by moving the m to one side of the equals sign by itself. It then becomes 45 - s = m

Now put (45-s) into the second equation in place of m.

120 = 2(45-s) + 5s

Distribute the 2(45-s)

120 = 90 - 2s + 5s

Add the "s" expressions together. Subtract 90 from both sides.

30 = 3s

Divide by 3 and find that s = 10

There are 10 short answer questions on the test.

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