Line $CD$ passes through points $C(1,3)$ and $D(4,-3)$. If the equation of the line is written in slope-intercept form, $y=mx+b$, what is the value of $b$?
Answer (2)
Calculate the slope m using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 2 .
Substitute the slope m and the coordinates of point C ( 1 , 3 ) into the equation y = m x + b to solve for b .
Solve for b : 3 = − 2 ( 1 ) + b , which gives b = 5 .
The value of b is 5 .
Explanation
Problem Analysis We are given two points, C ( 1 , 3 ) and D ( 4 , − 3 ) , and we want to find the y-intercept, b , of the line passing through these points when the equation of the line is written in slope-intercept form, y = m x + b .
Calculating the Slope First, we need to find the slope, m , of the line. The slope is given by the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points. In our case, ( x 1 , y 1 ) = ( 1 , 3 ) and ( x 2 , y 2 ) = ( 4 , − 3 ) . Plugging these values into the formula, we get: m = 4 − 1 − 3 − 3 = 3 − 6 = − 2
Finding the y-intercept Now that we have the slope, m = − 2 , we can use the slope-intercept form of the equation, y = m x + b , and one of the points to solve for b . Let's use point C ( 1 , 3 ) . Substituting x = 1 , y = 3 , and m = − 2 into the equation, we get: 3 = ( − 2 ) ( 1 ) + b 3 = − 2 + b Adding 2 to both sides, we find: b = 3 + 2 = 5
Final Answer Therefore, the value of b is 5.
Examples
Understanding the slope-intercept form of a line is crucial in many real-world applications. For instance, imagine you are tracking the depreciation of a car's value over time. If the car's value decreases linearly, you can model this depreciation using the equation y = m x + b , where y is the car's value, x is the time in years, m is the annual depreciation rate (slope), and b is the initial value of the car (y-intercept). By knowing the car's value at two different times, you can determine the depreciation rate and predict its value at any point in the future. This concept extends to various scenarios, such as calculating the cost of a service based on a fixed fee and an hourly rate, or modeling the growth of a plant based on a constant growth rate and an initial height.
The value of b in the slope-intercept form of the equation for the line passing through points C ( 1 , 3 ) and D ( 4 , − 3 ) is 5 .
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