Write the equation of the line that is parallel to [tex]y=\frac{2}{3} x+1[/tex] and passes through the point [tex](-6,-1)[/tex].
Answer (2)
The line parallel to y = 3 2 x + 1 has the same slope, which is 3 2 .
Substitute the point ( − 6 , − 1 ) into the equation y = 3 2 x + b to find the y-intercept b .
Solve for b : − 1 = 3 2 ( − 6 ) + b , which gives b = 3 .
The equation of the line is y = 3 2 x + 3 .
Explanation
Understanding the Problem We are given a line y = 3 2 x + 1 and a point ( − 6 , − 1 ) . We need to find the equation of a line that is parallel to the given line and passes through the given point.
Finding the Slope Since the line we are looking for is parallel to y = 3 2 x + 1 , it will have the same slope. The slope of the given line is 3 2 . Therefore, the equation of the line we are looking for will be of the form y = 3 2 x + b , where b is the y-intercept.
Finding the y-intercept We know that the line passes through the point ( − 6 , − 1 ) . We can substitute these coordinates into the equation y = 3 2 x + b to solve for b :
− 1 = 3 2 ( − 6 ) + b − 1 = − 4 + b Adding 4 to both sides, we get: b = 3
Writing the Equation of the Line Now that we have the slope 3 2 and the y-intercept 3 , we can write the equation of the line: y = 3 2 x + 3
Final Answer Therefore, the equation of the line that is parallel to y = 3 2 x + 1 and passes through the point ( − 6 , − 1 ) is y = 3 2 x + 3 .
Examples
Imagine you're designing a ramp for a skateboard park. You want the ramp to have the same steepness (slope) as another ramp, but it needs to start at a different point. This problem helps you find the equation of the new ramp, ensuring it's parallel to the existing one and passes through the desired starting point. Understanding parallel lines is crucial in various fields like architecture, engineering, and even computer graphics, where maintaining consistent angles and directions is essential.
The equation of the line that is parallel to y = 3 2 x + 1 and passes through the point ( − 6 , − 1 ) is y = 3 2 x + 3 . To find this, we used the slope of the given line and substituted the coordinates of the point to determine the y-intercept. Finally, we combined the slope and y-intercept to write the equation of the line.
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