QuestionThe equation of line $a$ is $y=\frac{-8}{3} x+\frac{9}{8}$ . Line $b$ is perpendicular to $a$ . What is the slope of line $b$ ?
Simplify your answer and write it as a proper fraction, improper fraction, or integer.

Studdy Solution

STEP 1

1. The equation of line $a$ is given as $y = \frac{-8}{3}x + \frac{9}{8}$ .
2. Line $b$ is perpendicular to line $a$ .
3. We need to find the slope of line $b$ .

STEP 2

1. Identify the slope of line $a$ .
2. Determine the relationship between the slopes of perpendicular lines.
3. Calculate the slope of line $b$ .

STEP 3

Identify the slope of line $a$ .
The equation of line $a$ is in the slope-intercept form $y = mx + c$ , where $m$ is the slope. Therefore, the slope of line $a$ is $m_a = \frac{-8}{3}$ .

STEP 4

Determine the relationship between the slopes of perpendicular lines.
If two lines are perpendicular, the product of their slopes is $-1$ . Therefore, if $m_a$ is the slope of line $a$ and $m_b$ is the slope of line $b$ , then:
$m_a \times m_b = -1$

STEP 5

Calculate the slope of line $b$ .
Using the relationship $m_a \times m_b = -1$ , substitute $m_a = \frac{-8}{3}$ :
$\frac{-8}{3} \times m_b = -1$
To find $m_b$ , divide both sides by $\frac{-8}{3}$ :
$m_b = \frac{-1}{\frac{-8}{3}}$
Simplify the expression by multiplying by the reciprocal:
$m_b = \frac{-1 \times 3}{-8}$
$m_b = \frac{3}{8}$
The slope of line $b$ is:
$\boxed{\frac{3}{8}}$

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