What is the equation of the line that goes through $(-3,-1)$ and $(3,3)$?
A. $3 x+2 y=15$
B. $3 y+2 x=15$
C. $3 x-2 y=3$
D. $2 x-3 y=-3$
Answer (1)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = 3 2 .
Use the point-slope form of the line y − y 1 = m ( x − x 1 ) with point ( − 3 , − 1 ) to get y + 1 = 3 2 ( x + 3 ) .
Simplify the equation to y = 3 2 x + 1 .
Convert to standard form 2 x − 3 y = − 3 , so the final answer is 2 x − 3 y = − 3 .
Explanation
Problem Analysis We are given two points, ( − 3 , − 1 ) and ( 3 , 3 ) , and we need to find the equation of the line that passes through these points.
Calculate the Slope First, we need to find the slope of the line. The slope, m , is given by the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points. Substituting the given points, we have: m = 3 − ( − 3 ) 3 − ( − 1 ) = 3 + 3 3 + 1 = 6 4 = 3 2
Apply Point-Slope Form Now that we have the slope, we can use the point-slope form of the equation of a line, which is: y − y 1 = m ( x − x 1 ) Using the point ( − 3 , − 1 ) and the slope m = 3 2 , we get: y − ( − 1 ) = 3 2 ( x − ( − 3 )) y + 1 = 3 2 ( x + 3 )
Simplify the Equation Next, we simplify the equation: y + 1 = 3 2 x + 3 2 ( 3 ) y + 1 = 3 2 x + 2 y = 3 2 x + 2 − 1 y = 3 2 x + 1
Convert to Standard Form To convert this to the standard form A x + B y = C , we multiply the entire equation by 3 to eliminate the fraction: 3 y = 2 x + 3 Rearranging the terms, we get: − 2 x + 3 y = 3 Multiplying by -1 to make the coefficient of x positive, we have: 2 x − 3 y = − 3
Final Answer Comparing this equation with the given options, we see that it matches option D: 2 x − 3 y = − 3 Therefore, the equation of the line is 2 x − 3 y = − 3 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, in economics, you can model the relationship between the price of a product and the quantity demanded using a linear equation. If you know two price-quantity data points, you can determine the equation of the line and predict the demand at other price points. Similarly, in physics, the relationship between distance, time, and constant velocity can be modeled using a linear equation, allowing you to predict the position of an object at any given time.
