Hills Nichols
10/12/2023 · Primary School
Write an equation of the line containing the given point and parallel to the given line. Express your answer in the form \( y=m x+b \). \( (5,9) ; x+8 y=5 \) The equation of the line is
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Step-by-step Solution
To find the equation of the line that is parallel to the given line \( x + 8y = 5 \) and passes through the point \( (5, 9) \), we first need to determine the slope of the given line.
1. **Convert the given line to slope-intercept form \( y = mx + b \)**:
\[
x + 8y = 5
\]
Subtract \( x \) from both sides:
\[
8y = -x + 5
\]
Now, divide by 8:
\[
y = -\frac{1}{8}x + \frac{5}{8}
\]
From this, we can see that the slope \( m \) of the given line is \( -\frac{1}{8} \).
2. **Since parallel lines have the same slope**, the slope of the line we want to find is also \( -\frac{1}{8} \).
3. **Use the point-slope form of the equation of a line**:
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is the point \( (5, 9) \) and \( m = -\frac{1}{8} \).
Plugging in the values:
\[
y - 9 = -\frac{1}{8}(x - 5)
\]
4. **Distribute and simplify**:
\[
y - 9 = -\frac{1}{8}x + \frac{5}{8}
\]
Now, add 9 to both sides:
\[
y = -\frac{1}{8}x + \frac{5}{8} + 9
\]
To combine \( \frac{5}{8} \) and 9, convert 9 to a fraction with a denominator of 8:
\[
9 = \frac{72}{8}
\]
So,
\[
y = -\frac{1}{8}x + \frac{5}{8} + \frac{72}{8} = -\frac{1}{8}x + \frac{77}{8}
\]
5. **Final equation**:
The equation of the line in the form \( y = mx + b \) is:
\[
y = -\frac{1}{8}x + \frac{77}{8}
\]
Thus, the equation of the line is:
\[
\boxed{y = -\frac{1}{8}x + \frac{77}{8}}
\]
Quick Answer
The equation of the line is \( y = -\frac{1}{8}x + \frac{77}{8} \).
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