The table represents a linear equation.
| x | y |
| --- | --- |
| -10 | 8 |
| -5 | 7 |
| 10 | 4 |
| 15 | 3 |
Which equation shows how $(-10,8)$ can be used to write the equation of this line in point-slope form?
A. $y-8=-0.15(x-10)$
B. $y+8=-0.15(x-10)$
C. $y-8=-0.2(x+10)$
D. $y+8=-0.2(x-10)$
Answer (2)
Calculate the slope using two points from the table: m = − 5 − ( − 10 ) 7 − 8 = − 0.2 .
Substitute the point ( − 10 , 8 ) and the slope m = − 0.2 into the point-slope form equation: y − 8 = − 0.2 ( x − ( − 10 )) .
Simplify the equation: y − 8 = − 0.2 ( x + 10 ) .
The correct equation in point-slope form is: y − 8 = − 0.2 ( x + 10 ) .
Explanation
Understanding the Problem We are given a table of values representing a linear equation and asked to find the equation in point-slope form using the point ( − 10 , 8 ) . The point-slope form of a linear equation is given by y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope.
Calculating the Slope First, we need to calculate the slope of the line. We can use any two points from the table. Let's use the points ( − 10 , 8 ) and ( − 5 , 7 ) . The slope m is calculated as follows: m = x 2 − x 1 y 2 − y 1 = − 5 − ( − 10 ) 7 − 8 = 5 − 1 = − 0.2
Applying Point-Slope Form Now that we have the slope m = − 0.2 and the point ( − 10 , 8 ) , we can plug these values into the point-slope form equation: y − y 1 = m ( x − x 1 ) y − 8 = − 0.2 ( x − ( − 10 )) y − 8 = − 0.2 ( x + 10 )
Finding the Correct Equation Comparing this equation with the given options, we find that the correct equation is: y − 8 = − 0.2 ( x + 10 )
Examples
Point-slope form is useful in many real-world scenarios. For example, if you know the rate at which a savings account is growing (slope) and the amount in the account at a specific time (point), you can use the point-slope form to determine the amount in the account at any other time. Similarly, in physics, if you know the velocity of an object at a certain time and the constant acceleration (slope), you can determine the object's velocity at any other time.
Using the point ( − 10 , 8 ) and calculating the slope from two points in the table, we find the equation in point-slope form is y − 8 = − 0.2 ( x + 10 ) . Therefore, the correct answer is option C.
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