Questions: The equation of line g is y = 8/9 x + 2. Line h is parallel to line g and passes through (9,4). What is the equation of line h? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

The equation of line g is y = 8/9 x + 2. Line h is parallel to line g and passes through (9,4). What is the equation of line h? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.
Transcript text: The equation of line $g$ is $y=\frac{8}{9} x+2$. Line $h$ is parallel to line $g$ and passes through $(9,4)$. What is the equation of line $h$ ? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.
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Solution

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Solution Steps

To find the equation of line \( h \) that is parallel to line \( g \) and passes through the point \((9, 4)\), we need to follow these steps:

  1. Identify the slope of line \( g \). Since line \( h \) is parallel to line \( g \), it will have the same slope.
  2. Use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the point through which the line passes.
  3. Convert the equation to slope-intercept form, \( y = mx + b \).
Step 1: Identify the Slope of Line \( g \)

The equation of line \( g \) is given by: \[ y = \frac{8}{9}x + 2 \] The slope of line \( g \) is \( \frac{8}{9} \).

Step 2: Use the Point-Slope Form

Since line \( h \) is parallel to line \( g \), it will have the same slope. Therefore, the slope of line \( h \) is also \( \frac{8}{9} \). Line \( h \) passes through the point \( (9, 4) \).

Using the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope and \( (x_1, y_1) \) is the point through which the line passes, we substitute \( m = \frac{8}{9} \), \( x_1 = 9 \), and \( y_1 = 4 \): \[ y - 4 = \frac{8}{9}(x - 9) \]

Step 3: Convert to Slope-Intercept Form

To convert the equation to slope-intercept form \( y = mx + b \), we simplify the equation: \[ y - 4 = \frac{8}{9}x - \frac{8}{9} \cdot 9 \] \[ y - 4 = \frac{8}{9}x - 8 \] \[ y = \frac{8}{9}x - 8 + 4 \] \[ y = \frac{8}{9}x - 4 \]

Final Answer

The equation of line \( h \) in slope-intercept form is: \[ \boxed{y = \frac{8}{9}x - 4} \]

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