QuestionWrite an equation of the line containing the given point and parallel to the given line. $(5,6), x+7 y=5$
The equation of the line is $y=$ $\square$ (Simplify your answer. Use integers or fractions for any numbers in the expression.)

Studdy Solution

STEP 1

1. We need to find the equation of a line that passes through the point $(5, 6)$ .
2. The line is parallel to the given line $x + 7y = 5$ .
3. Parallel lines have the same slope.

STEP 2

1. Find the slope of the given line.
2. Use the point-slope form to write the equation of the new line.
3. Simplify the equation to slope-intercept form.

STEP 3

Find the slope of the given line.
First, rewrite the equation $x + 7y = 5$ in slope-intercept form $y = mx + b$ .
Subtract $x$ from both sides:
$7y = -x + 5$
Divide every term by 7:
$y = -\frac{1}{7}x + \frac{5}{7}$
The slope $m$ of the given line is $-\frac{1}{7}$ .

STEP 4

Use the point-slope form to write the equation of the new line.
The point-slope form is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is the given point.
Using the point $(5, 6)$ and the slope $-\frac{1}{7}$ :
$y - 6 = -\frac{1}{7}(x - 5)$

STEP 5

Simplify the equation to slope-intercept form.
Distribute the slope on the right side:
$y - 6 = -\frac{1}{7}x + \frac{5}{7}$
Add 6 to both sides to solve for $y$ :
$y = -\frac{1}{7}x + \frac{5}{7} + 6$
Convert 6 to a fraction with a denominator of 7:
$y = -\frac{1}{7}x + \frac{5}{7} + \frac{42}{7}$
Combine the constant terms:
$y = -\frac{1}{7}x + \frac{47}{7}$
The equation of the line is:
$y = -\frac{1}{7}x + \frac{47}{7}$

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