A math test with 26 questions is worth 100 points. There are multiple choice questions that are worth 2 points each and fill in the blank questions that are worth 5 points each. How many of each type of question is there?

Let's define the variables to represent the number of each type of question:

- Let \( x \) be the number of multiple-choice questions (worth 2 points each).
- Let \( y \) be the number of fill-in-the-blank questions (worth 5 points each).

We have two pieces of information:

1. The total number of questions is 26.
   \[
   x + y = 26
   \]

2. The total number of points is 100.
   \[
   2x + 5y = 100
   \]

We now have a system of two equations with two variables:

\[
\begin{cases}
x + y = 26 \\
2x + 5y = 100
\end{cases}
\]

### Step 1: Solve the System of Equations

First, solve the first equation for \( x \):

\[
x = 26 - y
\]

Now, substitute this expression for \( x \) in the second equation:

\[
2(26 - y) + 5y = 100
\]

Simplify and solve for \( y \):

\[
52 - 2y + 5y = 100 \implies 3y = 48 \implies y = 16
\]

Now that we know \( y = 16 \), substitute it back into the equation for \( x \):

\[
x = 26 - y = 26 - 16 = 10
\]

### Conclusion

- There are **10 multiple-choice questions**.
- There are **16 fill-in-the-blank questions**.

### Verification

- Total number of questions: \(10 + 16 = 26\) (Correct)
- Total points: \(2 \times 10 + 5 \times 16 = 20 + 80 = 100\) (Correct)

Would you like a deeper explanation, or have any further questions?

Here are 5 related questions you might find interesting:

1. If the total points were 120 instead of 100, how many of each type of question would there be?
2. What if each multiple-choice question was worth 3 points? How would that change the number of each type?
3. How would the solution change if there were 30 questions in total?
4. If the test had 12 multiple-choice questions, how many fill-in-the-blank questions would there be?
5. How would the total score change if all fill-in-the-blank questions were worth 6 points instead?

**Tip:** When solving systems of equations, it's often helpful to isolate one variable first to simplify the problem.

Learn how to solve a system of equations involving multiple choice and fill-in-the-blank questions. Understand the distribution of 10 multiple-choice questions worth 2 points each and 16 fill-in-the-blank questions worth 5 points each, totaling 26 questions and 100 points. Suitable for students in Grades 7-9.

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