The table represents a linear equation.
| x | y |
| --- | ---- |
| -4 | -11 |
| -2 | -6 |
| 6 | 14 |
| 10 | 24 |
Which equation correctly uses point $(-2,-6)$ to write the equation of this line in point-slope form?
A. $y-6=\frac{5}{2}(x-2)$
B. $y-6=\frac{2}{5}(x-2)$
C. $y+6=\frac{2}{5}(x+2)$
D. $y+6=\frac{5}{2}(x+2)$
Answer (2)
Calculate the slope of the line using two points from the table: m = 2 5 .
Substitute the point ( − 2 , − 6 ) and the slope into the point-slope form equation: y − ( − 6 ) = 2 5 ( x − ( − 2 )) .
Simplify the equation: y + 6 = 2 5 ( x + 2 ) .
The correct equation is y + 6 = 2 5 ( x + 2 ) .
Explanation
Understanding the Problem We are given a table of values for a linear equation and asked to find the correct point-slope form of the equation using the point ( − 2 , − 6 ) . 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 of the line.
Calculating the Slope First, we need to find the slope of the line. We can use any two points from the table. Let's use the points ( − 4 , − 11 ) and ( − 2 , − 6 ) . The slope m is calculated as follows: m = x 2 − x 1 y 2 − y 1 = − 2 − ( − 4 ) − 6 − ( − 11 ) = − 2 + 4 − 6 + 11 = 2 5 So, the slope of the line is 2 5 .
Applying Point-Slope Form Now, we use the point-slope form with the point ( − 2 , − 6 ) and the slope 2 5 :
y − y 1 = m ( x − x 1 ) y − ( − 6 ) = 2 5 ( x − ( − 2 )) y + 6 = 2 5 ( x + 2 ) This matches one of the given options.
Final Answer Therefore, the correct equation in point-slope form is y + 6 = 2 5 ( x + 2 ) .
Examples
Point-slope form is useful in many real-world scenarios. For example, if you know the rate at which you are saving money (slope) and how much you have saved at a particular time (a point), you can use the point-slope form to write an equation that models your savings over time. This allows you to predict how much money you will have saved at any given point in the future. Similarly, in physics, if you know the velocity of an object at a certain time and its constant acceleration, you can use point-slope form to determine its velocity at any other time.
The point-slope form using the point ( − 2 , − 6 ) and the calculated slope 2 5 results in the equation y + 6 = 2 5 ( x + 2 ) . Therefore, the correct answer is option D. This form is particularly useful for representing linear relationships efficiently.
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