The table represents a linear equation.
| x | y |
| --- | ---- |
| -4 | -11 |
| -2 | -6 |
| 6 | 14 |
| 10 | 24 |
Which equation correctly uses point ([tex] (-2,-6) [/tex]) to write the equation of this line in point-slope form?
[tex]y-6=\frac{5}{2}(x-2)[/tex]
[tex]y-6=\frac{2}{5}(x-2)[/tex]
[tex]y+6=\frac{2}{5}(x+2)[/tex]
[tex]y+6=\frac{5}{2}(x+2)[/tex]
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: y − ( − 6 ) = 2 5 ( x − ( − 2 )) .
Simplify the equation: y + 6 = 2 5 ( x + 2 ) .
The correct equation in point-slope form is: y + 6 = 2 5 ( x + 2 ) .
Explanation
Understanding the Problem We are given a table of values representing 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 calculate 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 .
Substituting into Point-Slope Form Now, we substitute the given point ( − 2 , − 6 ) and the calculated slope m = 2 5 into the point-slope form equation: y − y 1 = m ( x − x 1 ) y − ( − 6 ) = 2 5 ( x − ( − 2 )) y + 6 = 2 5 ( x + 2 ) This is the point-slope form of the equation using the point ( − 2 , − 6 ) .
Finding the Correct Equation Comparing our equation y + 6 = 2 5 ( x + 2 ) with the given options, we find that it matches the fourth option: 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 a savings account is growing (the slope) and the amount in the account at a specific time (a 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 its constant acceleration, you can predict its velocity at any other time using the point-slope form.
To derive the point-slope form using the point (-2, -6) and the slope 2 5 , the equation is y + 6 = 2 5 ( x + 2 ) . This corresponds to the fourth option provided. Thus, the correct choice is the equation: y + 6 = 2 5 ( x + 2 ) .
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