What is the equation of a line that passes through $(8,-5)$ and is parallel to the graphed line?
A. $y=\frac{3}{4} x+1$
B. $y=\frac{3}{4} x-11$
Answer (2)
Determine the slope of the parallel line: m = 4 3 .
Use the point-slope form with point ( 8 , − 5 ) : y − ( − 5 ) = 4 3 ( x − 8 ) .
Simplify the equation to slope-intercept form: y = 4 3 x − 11 .
The equation of the line is: y = 4 3 x − 11 .
Explanation
Understanding the Problem We are given a point ( 8 , − 5 ) and need to find the equation of a line that passes through this point and is parallel to a given line. The options provided suggest the slope of the parallel line is 4 3 .
Finding the Slope Since the line we are looking for is parallel to the given line, it will have the same slope. Therefore, the slope of our line is m = 4 3 . We also know that the line passes through the point ( 8 , − 5 ) . We can use the point-slope form of a linear equation to find the equation of the line.
Using Point-Slope Form The point-slope form of a linear equation is given by: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is a point on the line and m is the slope. In our case, ( x 1 , y 1 ) = ( 8 , − 5 ) and m = 4 3 . Substituting these values into the point-slope form, we get: y − ( − 5 ) = 4 3 ( x − 8 ) y + 5 = 4 3 ( x − 8 )
Simplifying the Equation Now, we simplify the equation to slope-intercept form ( y = m x + b ): y + 5 = 4 3 x − 4 3 × 8 y + 5 = 4 3 x − 6 Subtract 5 from both sides: y = 4 3 x − 6 − 5 y = 4 3 x − 11
Final Answer The equation of the line that passes through ( 8 , − 5 ) and is parallel to the given line is y = 4 3 x − 11 .
Examples
Imagine you're designing a ramp for a building. You know the ramp needs to pass through a specific point and maintain a certain slope to be accessible. This problem helps you determine the exact equation for the ramp's surface, ensuring it meets the required specifications. Understanding linear equations and slopes is crucial in various real-world applications, from construction to navigation.
The equation of the line that passes through the point ( 8 , − 5 ) and is parallel to the given line is y = 4 3 x − 11 . Therefore, the correct answer is option B. The reasoning involves using the slope of the parallel line and the point-slope form of a linear equation.
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