A line has a $y$-intercept of -1 and is parallel to the line passing through $(2,3)$ and $(4,7)$. Write its equation in slope-intercept form.
Answer (1)
Calculate the slope of the line passing through (2,3) and (4,7): m = 4 − 2 7 − 3 = 2 .
Since the line is parallel, it has the same slope: m = 2 .
The y-intercept is given as -1: b = − 1 .
Write the equation in slope-intercept form: y = 2 x − 1 , so the final answer is y = 2 x − 1 .
Explanation
Understanding the Problem We are given that a line has a y -intercept of -1 and is parallel to the line passing through the points ( 2 , 3 ) and ( 4 , 7 ) . We need to find the equation of this line in slope-intercept form. The slope-intercept form of a line is given by y = m x + b , where m is the slope and b is the y -intercept.
Calculating the Slope First, we need to find the slope of the line passing through the points ( 2 , 3 ) and ( 4 , 7 ) . The slope, m , is given by the formula: m = x 2 − x 1 y 2 − y 1 Substituting the given points, we have: m = 4 − 2 7 − 3 = 2 4 = 2 So, the slope of the line passing through ( 2 , 3 ) and ( 4 , 7 ) is 2.
Determining the Parallel Slope Since the line we want to find is parallel to the line passing through ( 2 , 3 ) and ( 4 , 7 ) , it has the same slope. Therefore, the slope of our line is also 2.
Identifying the y-intercept We are given that the y -intercept of our line is -1. This means that b = − 1 .
Writing the Equation Now we can write the equation of the line in slope-intercept form, y = m x + b . We have m = 2 and b = − 1 , so the equation is: y = 2 x + ( − 1 ) y = 2 x − 1 Thus, the equation of the line in slope-intercept form is y = 2 x − 1 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, consider a taxi service that charges a fixed fee plus a per-mile rate. If the initial fee is $3 and the per-mile rate is $2, the total cost y for a ride of x miles can be modeled by the equation y = 2 x + 3 . This is a linear equation in slope-intercept form, where the slope represents the per-mile rate and the y-intercept represents the initial fee. By understanding slope-intercept form, one can easily determine the cost of a ride for any given distance.
