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 (2)
Identify the slope of the given line: 3 2 .
Use the same slope for the parallel line: y = 3 2 x + b .
Substitute the x -intercept ( − 3 , 0 ) into the equation to find b : 0 = 3 2 ( − 3 ) + b .
Solve for b and write the equation of the line: y = 3 2 x + 2 .
Explanation
Understanding the Problem We are given the equation of a line and asked to find the equation of a parallel line with a specific x -intercept. Parallel lines have the same slope, so we need to identify the slope of the given line. The x -intercept is the point where the line crosses the x -axis, meaning y = 0 at that point.
Finding the Slope The given line is y = 3 2 x + 3 . The slope of this line is 3 2 . Since we want a line parallel to this one, the slope of our new line will also be 3 2 .
Using the x-intercept The equation of our new line will be of the form y = 3 2 x + b , where b is the y -intercept. We are given that the x -intercept is -3, which means the line passes through the point ( − 3 , 0 ) .
Substituting the Point Substitute the point ( − 3 , 0 ) into the equation y = 3 2 x + b to solve for b :
0 = 3 2 ( − 3 ) + b
Simplifying Simplify the equation: 0 = − 2 + b
Solving for b Solve for b :
b = 2
Writing the Equation Now we have the slope 3 2 and the y -intercept 2 . The equation of the line is: y = 3 2 x + 2
Examples
Understanding parallel lines and intercepts is crucial in various real-world applications. For instance, consider city planning where streets need to be parallel to each other. If one street's equation is known, and the city planner needs another parallel street with a specific starting point (x-intercept), this problem-solving approach becomes essential. Similarly, in construction, parallel beams or structures often require precise calculations to ensure stability and alignment. Knowing how to determine the equation of a parallel line with a given intercept helps in these practical scenarios, ensuring accuracy and efficiency in design and execution.
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 . This is derived by keeping the same slope and substituting the given intercept to find the y -intercept. Thus, the final answer is y = 3 2 x + 2 .
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