The table represents a linear equation.
| x | y |
| --- | --- |
| -10 | 8 |
| -5 | 7 |
| 10 | 4 |
| 15 | 3 |
What is the equation of this line in point-slope form?
A. [tex]$y-8=-0.15(x-10)$[/tex]
B. [tex]$y+8=-0.15(x-10)$[/tex]
C. [tex]$y-8=-0.2(x+10)$[/tex]
D. [tex]$y+8=-0.2(x-10)$[/tex]
Answer (1)
Calculate the slope using two points from the table: m = − 5 − ( − 10 ) 7 − 8 = − 0.2 .
Use the point-slope form of a linear equation: y − y 1 = m ( x − x 1 ) .
Substitute the slope and a point (e.g., ( − 10 , 8 ) ) into the point-slope form: y − 8 = − 0.2 ( x + 10 ) .
The equation of the line in point-slope form is y − 8 = − 0.2 ( x + 10 ) .
Explanation
Understanding the Problem We are given a table of x and y values that represent a linear equation. Our goal is to find the equation of this line in point-slope form. The point-slope form of a linear equation is given by y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line.
Calculating the Slope First, we need to calculate the slope of the line. We can use any two points from the table to find the slope. Let's use the points ( − 10 , 8 ) and ( − 5 , 7 ) . The slope m is calculated as: m = x 2 − x 1 y 2 − y 1 = − 5 − ( − 10 ) 7 − 8 = 5 − 1 = − 0.2
Finding the Point-Slope Form Now that we have the slope, m = − 0.2 , we can use the point-slope form of the equation. Let's use the point ( − 10 , 8 ) . Substituting the slope and the point into the point-slope form, we get: y − 8 = − 0.2 ( x − ( − 10 ))
Simplifying, we have: y − 8 = − 0.2 ( x + 10 ) This matches one of the given options.
Alternative Point-Slope Form Alternatively, we can use another point from the table, such as ( − 5 , 7 ) . Substituting this point and the slope into the point-slope form, we get: y − 7 = − 0.2 ( x − ( − 5 ))
Simplifying, we have: y − 7 = − 0.2 ( x + 5 ) This is also a valid point-slope form of the equation, but it's not among the given options.
Final Answer Comparing our result y − 8 = − 0.2 ( x + 10 ) with the given options, we find that it matches the third option. Therefore, the equation of the line in point-slope form is: y − 8 = − 0.2 ( x + 10 )
Stating the Answer The equation of the line in point-slope form is y − 8 = − 0.2 ( x + 10 ) .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, consider a taxi service that charges a fixed fee plus a per-mile rate. If the fixed fee is $5 and the per-mile rate is 0.2 , t h e t o t a l cos t y f or a r i d eo f x mi l esc anb ere p rese n t e d b y a l in e a re q u a t i o n y = 0.2x + 5$. This equation allows you to predict the cost of any ride based on the distance traveled. Similarly, linear equations are used in physics to describe motion with constant velocity, in economics to model supply and demand curves, and in computer science for linear regression analysis.
