Select the correct answer.
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 the point ( − 3 , − 1 ) to get y + 1 = 3 2 ( x + 3 ) .
Simplify the equation to the standard form 2 x − 3 y = − 3 .
The equation of the line is 2 x − 3 y = − 3 .
Explanation
Understanding the Problem 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.
Calculating the Slope First, we need to calculate the slope of the line. The slope, m , is given by the formula: m = x 2 − x 1 y 2 − y 1
Slope Calculation Result Using the given points, we have: m = 3 − ( − 3 ) 3 − ( − 1 ) = 3 + 3 3 + 1 = 6 4 = 3 2
Using Point-Slope Form Now that we have the slope, we can use the point-slope form of a line, which is: y − y 1 = m ( x − x 1 )
Substituting Values We can use either point to find the equation. Let's use the point ( − 3 , − 1 ) . Plugging in the values, we get: y − ( − 1 ) = 3 2 ( x − ( − 3 ))
Simplifying the Equation Simplifying the equation, we have: y + 1 = 3 2 ( x + 3 )
Eliminating the Fraction To get rid of the fraction, we multiply both sides by 3: 3 ( y + 1 ) = 2 ( x + 3 ) 3 y + 3 = 2 x + 6
Rearranging to Standard Form Now, we rearrange the equation to the standard form A x + B y = C :
2 x − 3 y = 3 − 6 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 .
Examples
In real life, determining the equation of a line can be useful in various scenarios, such as predicting trends or modeling relationships between two variables. For example, if you have data points showing the relationship between the number of hours studied and the exam score, you can find the equation of the line that best fits the data. This equation can then be used to predict the exam score for a given number of study hours, or vice versa. Understanding linear equations helps in making informed decisions and predictions based on available data.
