Question
\( 1 \leftarrow \quad \) Determine \( b \) so that the slope of \( \overline{\mathrm{PQ}} \) is 6, where \( P \) and \( Q \) are the points \( (b, 4) \) and \( (10,34) \), respectively. What is the value of \( b \) ? \( b=\square \) (Simplify your answer. Type an integer or a fraction.)
Ask by Bird Macdonald. in the United States
Mar 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( b = 5 \)
Solution
To determine the value of \( b \) such that the slope of the line segment \( \overline{\mathrm{PQ}} \) is 6, we can use the formula for the slope between two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \):
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
In this case, the points are \( P(b, 4) \) and \( Q(10, 34) \). Thus, we can substitute the coordinates into the slope formula:
\[
6 = \frac{34 - 4}{10 - b}
\]
Now, let's simplify the right side:
\[
6 = \frac{30}{10 - b}
\]
Next, we can cross-multiply to eliminate the fraction:
\[
6(10 - b) = 30
\]
Expanding the left side gives:
\[
60 - 6b = 30
\]
Now, we can isolate \( b \) by first subtracting 60 from both sides:
\[
-6b = 30 - 60
\]
\[
-6b = -30
\]
Now, divide both sides by -6:
\[
b = \frac{-30}{-6} = 5
\]
Thus, the value of \( b \) is:
\[
b = 5
\]
So, the final answer is:
\( b = 5 \)
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Simplify this solution Extra Insights
To determine \( b \) so that the slope of line segment \( \overline{\mathrm{PQ}} \) is 6, we can use the slope formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \( P = (b, 4) \) and \( Q = (10, 34) \). Plugging in the coordinates into the formula, we get: \[ 6 = \frac{34 - 4}{10 - b} \] Simplifying, we have: \[ 6 = \frac{30}{10 - b} \] To eliminate the fraction, we can cross-multiply: \[ 6(10 - b) = 30 \] Now, expanding and solving for \( b \): \[ 60 - 6b = 30 \] Subtracting 60 from both sides gives: \[ -6b = 30 - 60 \] \[ -6b = -30 \] Dividing by -6 gives: \[ b = 5 \] So, the value of \( b \) is \( \boxed{5} \).
