Write the equation of the line that is parallel to the given line and has the following information.
27) [tex]$y=2 x+3$[/tex] and passes through [tex]$(1,9)$[/tex]
28) [tex]$y=-\frac{3}{4} x+1$[/tex] and passes through [tex]$(8,-1)$[/tex]
Answer (1)
The equation of a line parallel to y = 2 x + 3 and passing through ( 1 , 9 ) is found using the point-slope form and converted to slope-intercept form.
The slope of the parallel line is the same as the given line, which is 2 .
Using the point-slope form y − 9 = 2 ( x − 1 ) , we convert it to slope-intercept form: y = 2 x + 7 .
The equation of a line parallel to y = − 4 3 x + 1 and passing through ( 8 , − 1 ) is found similarly.
The slope of the parallel line is the same as the given line, which is − 4 3 .
Using the point-slope form y + 1 = − 4 3 ( x − 8 ) , we convert it to slope-intercept form: y = − 4 3 x + 5 .
The equations of the lines are y = 2 x + 7 and y = − 4 3 x + 5 .
Explanation
Understanding the Problem We are given two problems. In each, we need to find the equation of a line that is parallel to a given line and passes through a specific point. The key idea here is that parallel lines have the same slope. We will use the slope-intercept form of a line, y = m x + b , and the point-slope form, y − y 1 = m ( x − x 1 ) , to find the equations.
Finding the Equation for Problem 27 For problem 27, the given line is y = 2 x + 3 . The slope of this line is m = 2 . Since the parallel line has the same slope, its slope is also 2 . The parallel line passes through the point ( 1 , 9 ) . Using the point-slope form, we have y − 9 = 2 ( x − 1 ) .
Converting to Slope-Intercept Form (Problem 27) Now, we convert the equation to slope-intercept form:
y − 9 = 2 x − 2
y = 2 x − 2 + 9
y = 2 x + 7 .
Finding the Equation for Problem 28 For problem 28, the given line is y = − 4 3 x + 1 . The slope of this line is m = − 4 3 . Since the parallel line has the same slope, its slope is also − 4 3 . The parallel line passes through the point ( 8 , − 1 ) . Using the point-slope form, we have y − ( − 1 ) = − 4 3 ( x − 8 ) .
Converting to Slope-Intercept Form (Problem 28) Now, we convert the equation to slope-intercept form:
y + 1 = − 4 3 x + 4 3 × 8
y + 1 = − 4 3 x + 6
y = − 4 3 x + 6 − 1
y = − 4 3 x + 5 .
Final Answer Therefore, the equation of the line parallel to y = 2 x + 3 and passing through ( 1 , 9 ) is y = 2 x + 7 , and the equation of the line parallel to y = − 4 3 x + 1 and passing through ( 8 , − 1 ) is y = − 4 3 x + 5 .
Examples
Understanding parallel lines is crucial in various real-world applications, such as designing roads or buildings. For instance, when city planners design parallel streets, they ensure consistent traffic flow and efficient use of space. Similarly, architects use parallel lines in building designs to create stable and aesthetically pleasing structures. The principles of parallel lines also apply in fields like computer graphics, where they are used to create realistic perspectives and 3D models.
