The point-slope form of the equation of a line that passes through points $(8,4)$ and $(0,2)$ is $y-4=\frac{1}{4}(x-8)$. What is the slope-intercept form of the equation for this line?
A. $y=\frac{1}{4} x-12$
B. $y=\frac{1}{4} x-4$
C. $y=\frac{1}{4} x+2$
D. $y=\frac{1}{4} x+6$
Answer (1)
Distribute the 4 1 on the right side of the equation: y − 4 = 4 1 x − 2 .
Add 4 to both sides to isolate y : y = 4 1 x − 2 + 4 .
Simplify the equation: y = 4 1 x + 2 .
The slope-intercept form of the equation is y = 4 1 x + 2 .
Explanation
Understanding the Problem We are given the point-slope form of a line: y − 4 = 4 1 ( x − 8 ) . Our goal is to convert this equation into slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Distribute First, distribute the 4 1 on the right side of the equation:
y − 4 = 4 1 x − 4 1 ( 8 )
Simplify Simplify the right side:
y − 4 = 4 1 x − 2
Isolate y Next, add 4 to both sides of the equation to isolate y :
y = 4 1 x − 2 + 4
Slope-intercept form Finally, simplify to get the slope-intercept form:
y = 4 1 x + 2
So, the slope-intercept form of the equation is y = 4 1 x + 2 .
Examples
Understanding slope-intercept form is crucial in many real-world applications. For instance, imagine you're tracking the cost of a taxi ride. The initial fee is $2, and you pay an additional $0.25 per mile. The equation representing this scenario is y = 0.25 x + 2 , where y is the total cost and x is the number of miles traveled. This is a perfect example of slope-intercept form, where the slope (0.25) represents the cost per mile, and the y-intercept (2) represents the initial fee. By understanding this equation, you can easily calculate the cost of any taxi ride based on the distance traveled.
