Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form.
Passing through $(8,3)$ with $x$-intercept 3
Write an equation for the line in point-slope form.
y-3=\frac{3}{5}(x-8)
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
Write an equation for the line in slope-intercept form.
$\square$
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
Answer (2)
Verify the slope using the points (8, 3) and (3, 0): m = 5 3 .
Distribute 5 3 in the point-slope form: y − 3 = 5 3 x − 5 24 .
Add 3 to both sides to isolate y: y = 5 3 x − 5 24 + 3 .
Simplify to slope-intercept form: y = 5 3 x − 5 9 .
y = 5 3 x − 5 9
Explanation
Understanding the Problem We are given a line that passes through the point ( 8 , 3 ) and has an x -intercept of 3. This means the line also passes through the point ( 3 , 0 ) . We are given the point-slope form of the line as y − 3 = 5 3 ( x − 8 ) . Our goal is to find the equation of this line in slope-intercept form, which is y = m x + b , where m is the slope and b is the y -intercept.
Verifying the Slope First, let's verify the slope given in the point-slope form. The slope m can be calculated using the two points ( 8 , 3 ) and ( 3 , 0 ) as follows: m = x 2 − x 1 y 2 − y 1 = 3 − 8 0 − 3 = − 5 − 3 = 5 3 So, the slope is indeed 5 3 .
Converting to Slope-Intercept Form Now, let's convert the point-slope form equation to slope-intercept form. The given point-slope form is: y − 3 = 5 3 ( x − 8 ) Distribute the 5 3 on the right side: y − 3 = 5 3 x − 5 3 × 8 y − 3 = 5 3 x − 5 24 Isolate y by adding 3 to both sides: y = 5 3 x − 5 24 + 3 To combine the constants, we need a common denominator. Since 3 = 5 15 , we have: y = 5 3 x − 5 24 + 5 15 Combine the constants: y = 5 3 x − 5 9
Final Answer The equation of the line in slope-intercept form is y = 5 3 x − 5 9 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, imagine you're tracking the distance a cyclist travels over time. If the cyclist maintains a constant speed, the relationship between time and distance can be modeled using a linear equation. The slope represents the cyclist's speed, and the y-intercept could represent the starting point. By knowing the equation, you can predict how far the cyclist will travel after a certain amount of time, or how long it will take to reach a specific destination. This principle applies to various scenarios, such as calculating fuel consumption, predicting project timelines, or analyzing financial growth.
The line passing through (8, 3) and (3, 0) has a slope of 5 3 . Its equation in point-slope form is y − 3 = 5 3 ( x − 8 ) and in slope-intercept form is y = 5 3 x − 5 9 .
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