What is the equation of the line that is parallel to the given line and has an $x$-intercept of -3?
$y=\frac{2}{3} x+3$
Answer (1)
The line is parallel to y = 3 2 x + 3 , so it has the same slope: m = 3 2 .
The equation of the line is y = 3 2 x + b .
The x -intercept is -3, so the line passes through ( − 3 , 0 ) . Substituting this point into the equation gives 0 = 3 2 ( − 3 ) + b , which simplifies to b = 2 .
The equation of the line is y = 3 2 x + 2 .
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is parallel to the line y = 3 2 x + 3 and has an x -intercept of -3. This means the line passes through the point ( − 3 , 0 ) .
Finding the Slope Since the line we are looking for is parallel to y = 3 2 x + 3 , it has the same slope, which is 3 2 . Therefore, the equation of the line is 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 ( − 3 , 0 ) . We can substitute these coordinates into the equation y = 3 2 x + b to solve for b :
0 = 3 2 ( − 3 ) + b 0 = − 2 + b b = 2
Writing the Equation of the Line Now that we have the slope 3 2 and the y -intercept 2 , we can write the equation of the line as y = 3 2 x + 2 .
Final Answer Therefore, the equation of the line that is parallel to y = 3 2 x + 3 and has an x -intercept of -3 is y = 3 2 x + 2 .
Examples
Understanding parallel lines and intercepts is crucial in various real-world applications. For instance, consider designing a road parallel to an existing one, ensuring they maintain a constant distance apart. The x -intercept can represent a starting point or a specific location on a coordinate system. By determining the equation of the parallel road, engineers can accurately plan construction while adhering to safety and distance regulations. This concept extends to mapping, navigation, and urban planning, where parallel lines and intercepts play a fundamental role in creating organized and efficient layouts.
