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 (1)
Calculate the slope m using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 4 .
Substitute the slope and the coordinates of point J ( − 4 , − 5 ) into the equation y = m x + b .
Solve for b : − 5 = − 4 ( − 4 ) + b , which simplifies to − 5 = 16 + b .
Find the value of b : b = − 5 − 16 = − 21 , so the final answer is − 21 .
Explanation
Understanding the Problem We are given two points, J ( − 4 , − 5 ) and K ( − 6 , 3 ) , and we need to find the y-intercept, b , of the line passing through these points. The equation of the line is 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 calculated as the change in y divided by the change in x : m = x 2 − x 1 y 2 − y 1 Plugging in the coordinates of points J and K , we get: m = − 6 − ( − 4 ) 3 − ( − 5 ) = − 6 + 4 3 + 5 = − 2 8 = − 4
Using Slope-Intercept Form Now that we have the slope, m = − 4 , 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 J ( − 4 , − 5 ) :
− 5 = ( − 4 ) ( − 4 ) + b
Solving for b Now, solve for b :
− 5 = 16 + b b = − 5 − 16 b = − 21
Final Answer Therefore, the value of b is − 21 .
Examples
Understanding linear equations helps in many real-world scenarios. For example, if you are tracking the depreciation of an asset over time, you can use a linear equation to model the decrease in value. The slope represents the rate of depreciation, and the y-intercept represents the initial value of the asset. By knowing these values, you can predict the asset's value at any point in time. Similarly, in physics, understanding linear relationships between distance, speed, and time allows us to predict the position of an object at a given time, assuming constant speed.
