Find an equation of the line described.
Through $(-7,-64)$ and $(6,53)$
(A) $y=\frac{1}{9} x-\frac{569}{9}$
(B) $y=-\frac{1}{9} x-\frac{583}{9}$
(C) $y=9 x-1$
(D) $y=-9 x-127$
Answer (2)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = 9 .
Use the point-slope form of the line equation y − y 1 = m ( x − x 1 ) with the point ( − 7 , − 64 ) to get y + 64 = 9 ( x + 7 ) .
Convert the equation to slope-intercept form y = m x + b , which simplifies to y = 9 x − 1 .
The equation of the line is y = 9 x − 1 .
Explanation
Understanding the Problem We are given two points, ( − 7 , − 64 ) and ( 6 , 53 ) , and we need to find the equation of the line that passes through these points. The equation of a line can be written in the slope-intercept form as 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 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. Plugging in the given values, we have: m = 6 − ( − 7 ) 53 − ( − 64 ) = 6 + 7 53 + 64 = 13 117 = 9
Using Point-Slope Form Now that we have the slope m = 9 , we can use the point-slope form of the equation of a line, which is: y − y 1 = m ( x − x 1 ) Using the point ( − 7 , − 64 ) , we have: y − ( − 64 ) = 9 ( x − ( − 7 )) y + 64 = 9 ( x + 7 ) y + 64 = 9 x + 63
Converting to Slope-Intercept Form Now, we convert the equation to slope-intercept form, y = m x + b , by solving for y :
y = 9 x + 63 − 64 y = 9 x − 1
Final Answer The equation of the line is y = 9 x − 1 . Comparing this with the given options, we see that it matches option (C).
Examples
Imagine you're tracking the growth of a plant. After 7 days, it's 64 cm tall, and after 6 days, it's 53 cm tall. The equation of the line we found can help you predict the plant's height on any given day, assuming it grows at a constant rate. This is a practical application of linear equations in understanding and predicting real-world phenomena.
The equation of the line passing through the points ( − 7 , − 64 ) and ( 6 , 53 ) is y = 9 x − 1 . This corresponds to option (C).
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