Question

Write an equation for the line parallel to the given line that contains C.
Cleft parenthesis negative 3 comma 8 right parenthesis​; y equals two sevenths x plus 7
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Part 1
The equation of the parallel line is
  
enter your response here.
​(Use integers or fractions for any numbers in the equation. Simplify your​ answer.)

Answer 1

To find the equation of a line that is parallel to a given line and passes through a specific point, we can follow these steps:

1. **Identify the slope of the given line**: The equation given is \(y = \frac{2}{7}x + 7\). The slope (m) of this line is \(\frac{2}{7}\).

2. **Use the slope for the parallel line**: Since parallel lines have the same slope, the slope of the line we're looking for will also be \(\frac{2}{7}\).

3. **Use the point (C)**: We need to find the equation of the line that goes through the point \(C(-3, 8)\) with the slope \(\frac{2}{7}\). We can use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Where \((x_1, y_1)\) is the point and \(m\) is the slope.

Substituting in the values:
\[
y - 8 = \frac{2}{7}(x - (-3))
\]
Simplifying:
\[
y - 8 = \frac{2}{7}(x + 3)
\]

4. **Distributing the slope**:
\[
y - 8 = \frac{2}{7}x + \frac{6}{7}
\]

5. **Adding 8 to both sides**:
\[
y = \frac{2}{7}x + \frac{6}{7} + 8
\]

To combine the constants \(\frac{6}{7} + 8\), we can convert \(8\) to a fraction:
\(8 = \frac{56}{7}\)

Therefore:
\[
y = \frac{2}{7}x + \frac{6}{7} + \frac{56}{7}
\]

Combining the fractions:
\[
y = \frac{2}{7}x + \frac{62}{7}
\]

6. **Final Equation**:
The equation of the parallel line is:
\[
y = \frac{2}{7}x + \frac{62}{7}
\]

So, the final answer is:
\[
\boxed{y = \frac{2}{7}x + \frac{62}{7}}
\]