Find the equation of the line passing through the points $(8,-16)$ and $(1,5)$.
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Answer (1)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 3 .
Substitute the slope and one point into the equation y = m x + b to solve for the y-intercept b , which gives b = 8 .
Write the equation of the line in the form y = m x + b using the calculated values of m and b .
The equation of the line is y = − 3 x + 8 .
Explanation
Problem Analysis We are given two points, ( 8 , − 16 ) and ( 1 , 5 ) , and we want to find the equation of the line that passes through these points. The equation of a line can be written in the form y = m x + b , where m is the slope and b is the y-intercept.
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 ) = ( 8 , − 16 ) and ( x 2 , y 2 ) = ( 1 , 5 ) . Plugging these values into the formula, we get: m = 1 − 8 5 − ( − 16 ) = 1 − 8 5 + 16 = − 7 21 = − 3 So, the slope of the line is m = − 3 .
Finding the Y-Intercept Now that we have the slope, we can use one of the points to find the y-intercept b . Let's use the point ( 1 , 5 ) . We plug the values x = 1 , y = 5 , and m = − 3 into the equation y = m x + b :
5 = ( − 3 ) ( 1 ) + b 5 = − 3 + b Adding 3 to both sides, we get: b = 5 + 3 = 8 So, the y-intercept is b = 8 .
Writing the Equation of the Line Now we have the slope m = − 3 and the y-intercept b = 8 . We can write the equation of the line as: y = − 3 x + 8
Final Answer Therefore, the equation of the line passing through the points ( 8 , − 16 ) and ( 1 , 5 ) is y = − 3 x + 8 .
Examples
Understanding linear equations 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 use a linear equation to model this depreciation. If the car was initially worth $20,000 and depreciates by 2 , 000 e a c h ye a r , t h ee q u a t i o n w o u l d b e y = -2000x + 20000 , w h ere y i s t h ec a r ′ s v a l u e a f t er x$ years. This allows you to predict the car's value at any point in time, aiding in financial planning or deciding when to sell the car.
