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 = − 8 5 .
Use the point-slope form with point ( − 8 , 11 ) : y − 11 = − 8 5 ( x + 8 ) .
Convert to slope-intercept form: y = − 8 5 x + 6 .
The equation of the line is: y = − 8 5 x + 6 .
Explanation
Problem Analysis We are given two points ( − 8 , 11 ) and ( 4 , 2 7 ) and four equations. We need to find the equation that represents the line passing through these two points.
Finding 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 ) .
Calculating the Slope Substitute the values into the formula: m = 4 − ( − 8 ) 2 7 − 11 = 4 + 8 2 7 − 2 22 = 12 − 2 15 = − 2 × 12 15 = − 24 15 = − 8 5 So, the slope of the line is − 8 5 .
Using Point-Slope Form Now we use the point-slope form of a line, which is given by: 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
Converting to Slope-Intercept Form Now, we convert the equation to slope-intercept form, which is y = m x + b :
y = − 8 5 x − 5 + 11 y = − 8 5 x + 6
Verification We can check if the point ( 4 , 2 7 ) satisfies the equation: 2 7 = − 8 5 ( 4 ) + 6 2 7 = − 8 20 + 6 2 7 = − 2 5 + 6 2 7 = − 2 5 + 2 12 2 7 = 2 7 The point satisfies the equation.
Final Equation The equation of the line is y = − 8 5 x + 6 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, in economics, you might use a linear equation to model the relationship between the price of a product and the quantity demanded. If you know two price-quantity points, you can determine the equation of the line and predict demand at other price points. Similarly, in physics, you can model the motion of an object moving at a constant velocity using a linear equation, where the slope represents the velocity and the y-intercept represents the initial position. These models help in making informed decisions and predictions in various fields.
The equation of the line that passes through the points ( − 8 , 11 ) and ( 4 , 2 7 ) is y = − 8 5 x + 6 , which corresponds to option A. This was determined by calculating the slope and using the point-slope form to derive the equation. Verifying that both points satisfy the equation confirms its correctness.
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