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:

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.