Line $JK$ passes through points $J(-4,-5)$ and $K(-6,3)$. 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 n using the formula n = x 2 − x 1 y 2 − y 1 , which gives n = − 6 − ( − 4 ) 3 − ( − 5 ) = − 4 .
Substitute the slope and the coordinates of point J ( − 4 , − 5 ) into the equation y = n x + b to get − 5 = − 4 ( − 4 ) + b .
Solve for b : − 5 = 16 + b , so b = − 5 − 16 = − 21 .
The value of b is − 21 .
Explanation
Understanding the Problem We are given two points, J ( − 4 , − 5 ) and K ( − 6 , 3 ) , and we want to find the y-intercept, b , of the line that passes through these points. The equation of the line is given in slope-intercept form as y = n x + b , where n is the slope and b is the y-intercept.
Calculating the Slope First, we need to find the slope, n , of the line. The slope is calculated as the change in y divided by the change in x , which is given by the formula: n = x 2 − x 1 y 2 − y 1 Using the coordinates of points J and K , we have: n = − 6 − ( − 4 ) 3 − ( − 5 ) = − 6 + 4 3 + 5 = − 2 8 = − 4 So, the slope of the line is n = − 4 .
Finding the y-intercept Now that we have the slope, we can use the slope-intercept form of the equation, y = n x + b , and substitute the coordinates of one of the points (either J or K ) to solve for b . Let's use point J ( − 4 , − 5 ) :
− 5 = ( − 4 ) ( − 4 ) + b − 5 = 16 + b Subtract 16 from both sides to solve for b :
b = − 5 − 16 = − 21 So, the y-intercept is b = − 21 .
Final Answer Therefore, the value of b is − 21 .
Examples
Imagine you're tracking the altitude of a descending airplane. You have two data points: at 4 minutes, the altitude is 5000 feet, and at 6 minutes, the altitude is 3000 feet. Using these points, you can determine the rate of descent (slope) and estimate the altitude at the start (y-intercept). This linear equation helps predict the plane's altitude at any given time, assuming a constant rate of descent. Understanding slope and y-intercept is crucial in various real-world applications, from predicting financial trends to understanding physics.
The value of the y-intercept, b , of the line passing through points J ( − 4 , − 5 ) and K ( − 6 , 3 ) is − 21 . This was calculated by first finding the slope of the line and then solving for b using the slope-intercept form of the linear equation. The final result is b = − 21 .
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