Question
11) \( (-5,-2) \) and \( (x, 6) \); slope: \( \frac{8}{3} \)
Ask by Daniels Bernard. in the United States
Jan 02,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = -2 \)
Solution
To find the value of \( x \) given the two points \((-5, -2)\) and \((x, 6)\) with a slope of \( \frac{8}{3} \), we'll use the **slope formula**:
\[
\text{Slope} (m) = \frac{y_2 - y_1}{x_2 - x_1}
\]
**Given:**
- Point 1: \((x_1, y_1) = (-5, -2)\)
- Point 2: \((x_2, y_2) = (x, 6)\)
- Slope: \( m = \frac{8}{3} \)
**Step-by-Step Solution:**
1. **Plug the known values into the slope formula:**
\[
\frac{8}{3} = \frac{6 - (-2)}{x - (-5)}
\]
2. **Simplify the numerator and the denominator:**
\[
\frac{8}{3} = \frac{6 + 2}{x + 5} \\
\frac{8}{3} = \frac{8}{x + 5}
\]
3. **Set up the equation:**
\[
\frac{8}{3} = \frac{8}{x + 5}
\]
4. **Cross-multiply to eliminate the fractions:**
\[
8 \cdot (x + 5) = 8 \cdot 3
\]
5. **Simplify both sides:**
\[
8x + 40 = 24
\]
6. **Solve for \( x \):**
\[
8x = 24 - 40 \\
8x = -16 \\
x = \frac{-16}{8} \\
x = -2
\]
**Answer:** \( x = -2 \)
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Simplify this solution Bonus Knowledge
To find the value of \( x \) using the slope formula, we can plug in the coordinates and the given slope. The slope formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \( (x_1, y_1) = (-5, -2) \) and \( (x_2, y_2) = (x, 6) \), with \( m = \frac{8}{3} \). Plugging in the values, we have: \[ \frac{8}{3} = \frac{6 - (-2)}{x - (-5)} \] This simplifies to: \[ \frac{8}{3} = \frac{6 + 2}{x + 5} = \frac{8}{x + 5} \] Cross-multiplying gives: \[ 8(x + 5) = 24 \] Expanding this results in: \[ 8x + 40 = 24 \] Solving for \( x \): \[ 8x = 24 - 40 \] \[ 8x = -16 \] \[ x = -2 \] So, the value of \( x \) is \( -2 \).
