What is the equation in slope-intercept form of the line that passes through the points (-26, -11) and (39, 34)?
Understand the Problem
The question is asking for the equation of a line in slope-intercept form that passes through two given points. To solve this, we will first calculate the slope using the two points and then use the slope to find the y-intercept to form the complete equation.
Answer
The equation is $$ y = \frac{9}{13} x + 7 $$
Answer for screen readers
The equation of the line in slope-intercept form is
$$ y = \frac{9}{13} x + 7 $$
Steps to Solve
- Calculate the Slope of the Line
To find the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$, use the formula:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
For the points $(-26, -11)$ and $(39, 34)$, substitute:
$$ m = \frac{34 - (-11)}{39 - (-26)} = \frac{34 + 11}{39 + 26} = \frac{45}{65} = \frac{9}{13} $$
- Use One of the Points to Find the Y-Intercept
Using the slope $m = \frac{9}{13}$ and one of the points, say $(39, 34)$, we can use the equation of a line in slope-intercept form:
$$ y = mx + b $$
Substituting the point into the equation:
$$ 34 = \frac{9}{13} \cdot 39 + b $$
- Solve for the Y-Intercept, ( b )
Calculate the right side:
$$ 34 = \frac{351}{13} + b $$
Then, isolate ( b ):
$$ b = 34 - \frac{351}{13} $$
Convert ( 34 ) to have a common denominator:
$$ b = \frac{442}{13} - \frac{351}{13} = \frac{91}{13} $$
- Write the Equation in Slope-Intercept Form
Now that we have the slope ( m ) and ( b ):
$$ y = \frac{9}{13} x + \frac{91}{13} $$
Since this does not match any answer choices directly, let's convert the ( b ) to proper form.
- Identify the Equation for Final Review
The slope-intercept equation of the line that passes through both points can be arranged as follows:
$$ y = \frac{9}{13} x + 7 $$
as ( \frac{91}{13} ) simplifies to ( 7 ).
The equation of the line in slope-intercept form is
$$ y = \frac{9}{13} x + 7 $$
More Information
The slope of the line represents the rate of change of ( y ) with respect to ( x ). It indicates how steep the line is. The y-intercept is where the line crosses the y-axis.
Tips
- Forgetting to simplify the slope when calculating.
- Incorrectly substituting the y-intercept back into the equation.
- Mixing up the coordinates from the points used to calculate the slope.