Use the given conditions to write an equation for the line in point-slope form and slope-intercept form.
Passing through $(-5,-5)$ and $(5,7)$
Type the point-slope form of the equation of the line.
$\square$
(Use integers or simplified fractions for any numbers in the equation.)
Answer (2)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = 5 6 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with the point ( − 5 , − 5 ) to get y + 5 = 5 6 ( x + 5 ) .
Convert the point-slope form to slope-intercept form by solving for y , resulting in y = 5 6 x + 1 .
The equation of the line in point-slope form is y + 5 = 5 6 ( x + 5 ) .
Explanation
Problem Analysis We are given two points through which the line passes: ( − 5 , − 5 ) and ( 5 , 7 ) . Our goal is to find the equation of this line in both point-slope form and slope-intercept form. Let's start by finding the slope of the line.
Calculating the Slope The slope m of a line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = x 2 − x 1 y 2 − y 1 In our case, ( x 1 , y 1 ) = ( − 5 , − 5 ) and ( x 2 , y 2 ) = ( 5 , 7 ) . Plugging these values into the formula, we get: m = 5 − ( − 5 ) 7 − ( − 5 ) = 5 + 5 7 + 5 = 10 12 = 5 6 So, the slope of the line is 5 6 .
Writing the Point-Slope Form The point-slope form of the equation of a line 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. We can use either of the given points. Let's use ( − 5 , − 5 ) . Then, the point-slope form is: y − ( − 5 ) = 5 6 ( x − ( − 5 )) Simplifying, we get: y + 5 = 5 6 ( x + 5 ) This is the equation of the line in point-slope form.
Converting to Slope-Intercept Form Now, let's convert the point-slope form to slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Starting from the point-slope form: y + 5 = 5 6 ( x + 5 ) Distribute the 5 6 on the right side: y + 5 = 5 6 x + 5 6 ( 5 ) y + 5 = 5 6 x + 6 Subtract 5 from both sides to solve for y :
y = 5 6 x + 6 − 5 y = 5 6 x + 1 This is the equation of the line in slope-intercept form.
Final Answer The equation of the line in point-slope form is y + 5 = 5 6 ( x + 5 ) , and the equation of the line in slope-intercept form is y = 5 6 x + 1 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, imagine you're tracking the growth of a plant. If the plant grows at a constant rate, you can use a linear equation to model its height over time. The slope represents the growth rate, and the y-intercept represents the initial height of the plant. Similarly, in economics, linear equations can model supply and demand curves, helping to predict market equilibrium. These equations are also fundamental in physics for describing motion with constant velocity.
The equation of the line in point-slope form is y + 5 = 5 6 ( x + 5 ) , and in slope-intercept form, it is y = 5 6 x + 1 .
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