The point-slope form of the equation of the line that passes through $(-9,-2)$ and $(1,3)$ is $y-3=\frac{1}{2}(x-1)$. What is the slope-intercept form of the equation for this line?
A. $y=\frac{1}{2} x+2$
B. $y=\frac{1}{2} x-4$
C. $y=\frac{1}{2} x+\frac{5}{2}$
D. $y=\frac{1}{2} x-\frac{7}{2}$
Answer (2)
Distribute the slope: y − 3 = 2 1 ( x − 1 ) becomes y − 3 = 2 1 x − 2 1 .
Isolate y : Add 3 to both sides to get y = 2 1 x − 2 1 + 3 .
Simplify: Combine the constants to get y = 2 1 x + 2 5 .
The slope-intercept form of the equation is y = 2 1 x + 2 5 .
Explanation
Understanding the Problem We are given the point-slope form of a line: y − 3 = 2 1 ( x − 1 ) . Our goal is to convert this to slope-intercept form, which looks like y = m x + b , where m is the slope and b is the y-intercept.
Distributing the Slope To convert from point-slope form to slope-intercept form, we need to isolate y on the left side of the equation. First, distribute the 2 1 on the right side:
y − 3 = 2 1 x − 2 1
Isolating y Next, add 3 to both sides of the equation to isolate y :
y = 2 1 x − 2 1 + 3
Finding a Common Denominator Now, we need to simplify the right side by combining the constants. We can rewrite 3 as 2 6 to have a common denominator:
y = 2 1 x − 2 1 + 2 6
Combining Constants Combine the constants:
y = 2 1 x + 2 5
Final Answer The equation is now in slope-intercept form. The slope is 2 1 and the y-intercept is 2 5 . Therefore, the slope-intercept form of the equation is y = 2 1 x + 2 5 .
Examples
Understanding slope-intercept form is crucial in many real-world applications. For instance, imagine you are tracking the cost of a taxi ride. The initial fee could be the y-intercept, and the cost per mile would be the slope. If the initial fee is $2.50 and the cost per mile is 0.50 , t h ee q u a t i o n y = 0.50x + 2.50$ represents the total cost of the ride. By understanding this equation, you can easily calculate the cost for any distance traveled.
The slope-intercept form of the equation is y = 2 1 x + 2 5 , which corresponds to option C. The slope is 2 1 and the y-intercept is 2 5 .
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