Find the equation of the line passing through the points $(2,11)$ and $(-8,-19)$.
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Answer (2)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 8 − 2 − 19 − 11 = 3 .
Substitute the slope and one of the points into the equation y = m x + b to solve for the y-intercept b . Using the point ( 2 , 11 ) , we get 11 = 3 ( 2 ) + b , which gives b = 5 .
Write the equation of the line using the slope and y-intercept: y = 3 x + 5 .
The equation of the line is y = 3 x + 5 .
Explanation
Understanding the Problem We are given two points, ( 2 , 11 ) and ( − 8 , − 19 ) , and we want to find the equation of the line that passes through them. 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 of the line. The slope, m , can be calculated using 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 given points. In our case, ( x 1 , y 1 ) = ( 2 , 11 ) and ( x 2 , y 2 ) = ( − 8 , − 19 ) . Plugging these values into the formula, we get: m = − 8 − 2 − 19 − 11 = − 10 − 30 = 3 So, the slope of the line is 3.
Finding the y-intercept Now that we have the slope, we can find the y-intercept, b . We can use the point-slope form of a line, y = m x + b , and substitute one of the given points and the slope into the equation. Let's use the point ( 2 , 11 ) :
11 = 3 ( 2 ) + b 11 = 6 + b Subtracting 6 from both sides, we get: b = 11 − 6 = 5 So, the y-intercept is 5.
Writing the Equation of the Line Now we have the slope, m = 3 , and the y-intercept, b = 5 . We can write the equation of the line as: y = 3 x + 5 Thus, the equation of the line passing through the points ( 2 , 11 ) and ( − 8 , − 19 ) is y = 3 x + 5 .
Examples
Imagine you are tracking the growth of a plant. After 2 weeks, the plant is 11 inches tall, and after 8 weeks, it's 19 inches tall. Using the equation of a line, you can model the plant's growth and predict its height at any given week. This linear equation helps in understanding the rate of growth and making future predictions based on the current data.
The equation of the line passing through the points (2, 11) and (-8, -19) is given by y = 3 x + 5 . This equation is derived by calculating the slope and y-intercept using the coordinates of the given points.
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