Which equation represents a line which is parallel to the line [tex]y=-\frac{2}{3} x+1[/tex]?
A. [tex]2 y-3 x=-4[/tex]
B. [tex]2 x+3 y=-18[/tex]
C. [tex]3 x+2 y=16[/tex]
D. [tex]2 x-3 y=6[/tex]
Answer (2)
To find a line parallel to y = − 3 2 x + 1 , we look for an equation with the same slope. By converting each option to slope-intercept form, we identify 2 x + 3 y = − 18 as having the same slope of − 3 2 . Therefore, the answer is 2 x + 3 y = − 18 .
Explanation
Understanding the Problem We are given the equation of a line y = − 3 2 x + 1 and asked to find which of the given equations represents a line parallel to it. Parallel lines have the same slope. Therefore, we need to find the equation that, when written in slope-intercept form ( y = m x + b ), has a slope of − 3 2 .
Analyzing the Options Let's examine each option:
2 y − 3 x = − 4 Rewrite in slope-intercept form: 2 y = 3 x − 4 y = 2 3 x − 2 The slope is 2 3 .
2 x + 3 y = − 18 Rewrite in slope-intercept form: 3 y = − 2 x − 18 y = − 3 2 x − 6 The slope is − 3 2 .
3 x + 2 y = 16 Rewrite in slope-intercept form: 2 y = − 3 x + 16 y = − 2 3 x + 8 The slope is − 2 3 .
2 x − 3 y = 6 Rewrite in slope-intercept form: − 3 y = − 2 x + 6 y = 3 2 x − 2 The slope is 3 2 .
Finding the Parallel Line The equation 2 x + 3 y = − 18 has a slope of − 3 2 , which is the same as the slope of the given line y = − 3 2 x + 1 . Therefore, the line 2 x + 3 y = − 18 is parallel to the given line.
Examples
Understanding parallel lines is crucial in architecture and design. For example, when designing a building, architects need to ensure that walls are parallel to each other for structural stability and aesthetic appeal. Similarly, in urban planning, streets are often laid out in a grid pattern with parallel lines to facilitate traffic flow and efficient land use. The concept of parallel lines also extends to more complex designs, such as creating perspective in drawings or aligning elements in a graphical user interface.
The line parallel to y = − 3 2 x + 1 is represented by Option B: 2 x + 3 y = − 18 , since it has the same slope of − 3 2 .
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