Which equation represents the line that passes through $(-8,11)$ and $\left(4, \frac{7}{2}\right)$?
A. $y=-\frac{5}{8} x+6$
B. $y=-\frac{5}{8} x+16$
C. $y=-\frac{15}{2} x-49$
D. $y=-\frac{15}{2} x+71$
Answer (2)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 8 5 .
Use the point-slope form of the line equation y − y 1 = m ( x − x 1 ) with the point ( − 8 , 11 ) to get y − 11 = − 8 5 ( x + 8 ) .
Simplify the equation to slope-intercept form: y = − 8 5 x + 6 .
The equation of the line is y = − 8 5 x + 6 .
Explanation
Understanding the Problem We are given two points, ( − 8 , 11 ) and ( 4 , 2 7 ) , and we need to find the equation of the line that passes through these points. We are also given four possible equations for the line, and we need to determine which one is correct.
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 ) = ( − 8 , 11 ) and ( x 2 , y 2 ) = ( 4 , 2 7 ) .
Plugging in the values, we get: m = 4 − ( − 8 ) 2 7 − 11 = 4 + 8 2 7 − 2 22 = 12 − 2 15 = − 2 15 × 12 1 = − 24 15 = − 8 5 So, the slope of the line is − 8 5 .
Finding the Equation of the Line Now we use the point-slope form of the equation of a line, which is: y − y 1 = m ( x − x 1 ) Using the point ( − 8 , 11 ) and the slope m = − 8 5 , we have: y − 11 = − 8 5 ( x − ( − 8 )) y − 11 = − 8 5 ( x + 8 ) y − 11 = − 8 5 x − 8 5 ( 8 ) y − 11 = − 8 5 x − 5 y = − 8 5 x − 5 + 11 y = − 8 5 x + 6
Comparing with the Given Options The equation of the line is y = − 8 5 x + 6 . Comparing this with the given options, we see that it matches the first option.
Final Answer Therefore, the equation that represents the line passing through ( − 8 , 11 ) and ( 4 , 2 7 ) is: y = − 8 5 x + 6
Examples
Understanding linear equations is crucial in many real-world applications. For instance, consider a scenario where a taxi charges a fixed fee plus a variable amount per mile. If the taxi charges $6 for a ride of 0 miles and the rate decreases by $5/8 per mile driven, the equation derived in this problem helps determine the total cost for any distance. This concept extends to various fields, including economics, physics, and engineering, where linear relationships are used to model and predict outcomes.
The equation of the line passing through the points (-8, 11) and ( 4 , 2 7 ) is y = − 8 5 x + 6 . This matches choice A from the options provided. Thus, the final answer to the question is option A.
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