Find the equation of the line parallel to the line $y=4 x-2$ that passes through the point $(-1,5)$.
The slope of $y=4 x-2$ is $\square$
The slope of a line parallel to $y=4 x-2$ is $\square$.
The equation of the line parallel to $y=4 x-2$ that passes through the point $(-1,5)$ is $y=\square
Answer (2)
The slope of the line y = 4 x − 2 is 4 , which is also the slope of the parallel line. The equation of the parallel line that passes through the point ( − 1 , 5 ) is y = 4 x + 9 .
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The slope of the given line y = 4 x − 2 is 4 .
Parallel lines have the same slope, so the slope of the parallel line is also 4 .
Using the point-slope form with the point ( − 1 , 5 ) and slope 4 , we have y − 5 = 4 ( x + 1 ) .
Simplifying to slope-intercept form, the equation of the line is y = 4 x + 9 .
Explanation
Understanding the Problem The given line is y = 4 x − 2 . We need to find the equation of a line that is parallel to this line and passes through the point ( − 1 , 5 ) .
Finding the Slope of the Given Line The slope of the given line y = 4 x − 2 is the coefficient of x , which is 4 .
Determining the Slope of the Parallel Line Since parallel lines have the same slope, the slope of the line we want to find is also 4 .
Using the Point-Slope Form Now we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line. We have m = 4 and ( x 1 , y 1 ) = ( − 1 , 5 ) . Substituting these values into the point-slope form, we get:
y − 5 = 4 ( x − ( − 1 )) y − 5 = 4 ( x + 1 )
Simplifying to Slope-Intercept Form Now, we simplify the equation to get it into slope-intercept form, y = m x + b :
y − 5 = 4 x + 4 y = 4 x + 4 + 5 y = 4 x + 9
Final Answer Therefore, the equation of the line parallel to y = 4 x − 2 that passes through the point ( − 1 , 5 ) is y = 4 x + 9 .
The slope of y = 4 x − 2 is 4 .
The slope of a line parallel to y = 4 x − 2 is 4 .
The equation of the line parallel to y = 4 x − 2 that passes through the point ( − 1 , 5 ) is y = 4 x + 9 .
Examples
Understanding parallel lines is crucial in architecture and design. For instance, when designing a building, architects use parallel lines to ensure walls are aligned and structures are stable. If a wall needs to be parallel to another for aesthetic or structural reasons, knowing the slope of the existing wall and using the point-slope form can help determine the equation of the new wall, ensuring it aligns perfectly with the design.
