Select the correct answer.
What is the equation of the line that goes through the points $(-5,-1)$ and $(5,5)$?
A. $y=\frac{3}{5} x-8$
B. $y=\frac{2}{5} x+3$
C. $y=\frac{3}{5} x+2$
D. $y=\frac{2}{5} x-7
Answer (1)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = 5 3 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with the point ( 5 , 5 ) to get y − 5 = 5 3 ( x − 5 ) .
Convert to slope-intercept form by solving for y , resulting in y = 5 3 x + 2 .
The equation of the line is y = 5 3 x + 2 .
Explanation
Understanding the Problem We are given two points, ( − 5 , − 1 ) and ( 5 , 5 ) , and we need to find the equation of the line that passes through these points. The equation of a line can be written in the slope-intercept form as y = m x + b , where m is the slope and b is the y-intercept.
Calculating the Slope First, we need to calculate the slope ( m ) of the line using the formula: m = x 2 − x 1 y 2 − y 1 Substituting the coordinates of the given points, we have: m = 5 − ( − 5 ) 5 − ( − 1 ) = 5 + 5 5 + 1 = 10 6 = 5 3 So, the slope of the line is 5 3 .
Using Point-Slope Form Now that we have the slope, we can use the point-slope form of a line equation: y − y 1 = m ( x − x 1 ) We can use either point to find the equation. Let's use the point ( 5 , 5 ) . Substituting the slope and the coordinates into the point-slope equation, we get: y − 5 = 5 3 ( x − 5 ) Now, we convert this to slope-intercept form ( y = m x + b ) by solving for y :
Converting to Slope-Intercept Form y − 5 = 5 3 x − 5 3 × 5 y − 5 = 5 3 x − 3 y = 5 3 x − 3 + 5 y = 5 3 x + 2 So, the equation of the line is y = 5 3 x + 2 .
Selecting the Correct Answer Comparing the equation we found with the given options: A. y = 5 3 x − 8 B. y = 5 2 x + 3 C. y = 5 3 x + 2 D. y = 5 2 x − 7 The correct answer is C. y = 5 3 x + 2 .
Examples
Understanding linear equations is crucial in many real-world scenarios. For instance, imagine you're tracking the growth of a plant. If the plant grows at a constant rate, say 3 cm every 5 days, you can represent this growth using a linear equation. By plotting the plant's height over time, you can predict its future height based on the equation of the line. This concept extends to various fields, including finance, where linear equations can model simple interest or depreciation.
