Let's break down the problem step-by-step.

Given Information:

1. The total number of questions is 20.
2. The total points for the test is 100 points.
3. Each true-false question is worth 4 points.
4. Each multiple-choice question is worth 8 points.

Define Variables:

• Let $x$ be the number of true-false questions.
• Let $y$ be the number of multiple-choice questions.

Formulate Equations:

1. The total number of questions: $x + y = 20$

2. The total number of points: $4x + 8y = 100$

Solve the System of Equations:

Step 1: Simplify the Second Equation

$4x + 8y = 100$

Divide through by 4:

$x + 2y = 25$

Step 2: System of Equations

Now we have the following system:

1. $x + y = 20$
2. $x + 2y = 25$

Step 3: Subtract the Equations

$(x + 2y) - (x + y) = 25 - 20$

This simplifies to:

$y = 5$

Step 4: Solve for $x$

Substitute $y = 5$ into $x + y = 20$:

$x + 5 = 20$

$x = 15$

Solution:

• 15 true-false questions
• 5 multiple-choice questions

Verification:

1. Total number of questions: $15 + 5 = 20$ ✔
2. Total points: $4 \times 15 + 8 \times 5 = 60 + 40 = 100$ ✔

Do you want more details or have any questions?

Related Questions:

1. How would the solution change if the test had 25 questions instead of 20?
2. What if each true-false question was worth 5 points instead of 4?
3. How can you solve this problem graphically using the given grid?
4. Can you solve this using a different algebraic method?
5. What if the test had a different mix of question types and point values?

Tip:

In systems of equations, solving by substitution or elimination is a helpful approach for finding solutions efficiently.