Question

Question 13 (Essay Worth 12 points)
(Evaluating Equations HC)

Given the equation 8m − 15 = 5m + 63 and the possible solution set S: {3, 26, 78, 126}:

Part A: Determine which integer(s) in the solution set makes the equation false. Show all work. (8 points)

Part B: Use a complete sentence to explain how you were able to determine which values make the equation false. (4 points)

Answer 1

To solve the equation \(8m - 15 = 5m + 63\) and evaluate the solution set \(S: \{3, 26, 78, 126\}\), we will follow the steps outlined in the question.

### Part A: Evaluate Each Integer in the Solution Set

1. **Substituting \(m = 3\)**:
\[
8(3) - 15 = 5(3) + 63
\]
\[
24 - 15 = 15 + 63
\]
\[
9 \neq 78 \quad \text{(False)}
\]

2. **Substituting \(m = 26\)**:
\[
8(26) - 15 = 5(26) + 63
\]
\[
208 - 15 = 130 + 63
\]
\[
193 = 193 \quad \text{(True)}
\]

3. **Substituting \(m = 78\)**:
\[
8(78) - 15 = 5(78) + 63
\]
\[
624 - 15 = 390 + 63
\]
\[
609 = 453 \quad \text{(False)}
\]

4. **Substituting \(m = 126\)**:
\[
8(126) - 15 = 5(126) + 63
\]
\[
1008 - 15 = 630 + 63
\]
\[
993 = 693 \quad \text{(False)}
\]

From our evaluations:
- \(m = 3\) is **False**.
- \(m = 26\) is **True**.
- \(m = 78\) is **False**.
- \(m = 126\) is **False**.

Thus, the integers in the solution set that make the equation false are: \(3\), \(78\), and \(126\).

### Part B: Explanation

To determine which values from the solution set make the equation false, I substituted each integer into the original equation \(8m - 15 = 5m + 63\). By performing the calculations for both sides of the equation for each value, I checked if the left-hand side matched the right-hand side. If the two sides were equal, the value was a solution (True); if they were not equal, the value did not satisfy the equation (False). This process allowed me to identify the integers that make the equation false: \(3\), \(78\), and \(126\).

Answer 2

more simple?