Which of the following equations, when graphed, produces a line that passes through the points $(-3,-5)$ and $(4,-8.5)$?
A. $y=-0.5 x-6.5$
B. $y=0.5 x-3.5$
C. $y=2 x+1$
D. $y=-2 x-0.5$
Answer (2)
Calculate the slope using the formula: m = x 2 − x 1 y 2 − y 1 = 4 − ( − 3 ) − 8.5 − ( − 5 ) = − 0.5 .
Use the point-slope form with point ( − 3 , − 5 ) : y − ( − 5 ) = − 0.5 ( x − ( − 3 )) .
Convert to slope-intercept form: y + 5 = − 0.5 x − 1.5 ⇒ y = − 0.5 x − 6.5 .
The equation that passes through the given points is y = − 0.5 x − 6.5 .
Explanation
Problem Analysis We are given two points, ( − 3 , − 5 ) and ( 4 , − 8.5 ) , and we need to find the equation of the line that passes through these points. The equation should match one of the given options.
Calculate the Slope First, we need to calculate the slope ( m ) of the line using the formula: m = x 2 − x 1 y 2 − y 1 Substituting the given points into the formula: m = 4 − ( − 3 ) − 8.5 − ( − 5 ) = 4 + 3 − 8.5 + 5 = 7 − 3.5 = − 0.5
Use Point-Slope Form Now that we have the slope, we can use the point-slope form of a linear equation: y − y 1 = m ( x − x 1 ) Let's use the point ( − 3 , − 5 ) . Substituting the point and the slope into the point-slope form: y − ( − 5 ) = − 0.5 ( x − ( − 3 )) y + 5 = − 0.5 ( x + 3 )
Convert to Slope-Intercept Form Convert the point-slope form to slope-intercept form ( y = m x + b ): y + 5 = − 0.5 x − 1.5 y = − 0.5 x − 1.5 − 5 y = − 0.5 x − 6.5
Identify the Correct Equation Comparing the resulting equation with the given options, we find that it matches option A: y = − 0.5 x − 6.5
Final Answer The equation of the line that passes through the points ( − 3 , − 5 ) and ( 4 , − 8.5 ) is y = − 0.5 x − 6.5 .
Examples
Linear equations are used extensively in real life. For example, if you are tracking the depreciation of an asset over time, you might use a linear equation to model the decrease in value. If a machine is bought for $10,000 and depreciates linearly by 500 e a c h ye a r , t h ee q u a t i o n w o u l d b e V = -500t + 10000 , w h ere V i s t h e v a l u eo f t h e ma c hin e an d t$ is the time in years. This allows you to predict the value of the machine at any point in time.
The equation of the line that passes through the points ( − 3 , − 5 ) and ( 4 , − 8.5 ) is y = − 0.5 x − 6.5 , which corresponds to option A. Therefore, the correct answer is option A. This equation describes a line with a slope of -0.5 and a y-intercept of -6.5.
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