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.
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.
Solution
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:
Identify the slope of line \( g \). Since line \( h \) is parallel to line \( g \), it will have the same slope.
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.
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:
\[ \boxed{y = \frac{8}{9}x - 4} \]