Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form.
Slope $=-\frac{1}{3}$, passing through $(4,-6)$
Write an equation for the line in point-slope form.
$\square$
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
Answer (2)
Use the point-slope form formula y − y 1 = m ( x − x 1 ) and substitute the given slope m = − 3 1 and point ( 4 , − 6 ) .
Simplify the point-slope equation to y + 6 = − 3 1 ( x − 4 ) .
Use the slope-intercept form formula y = m x + b and substitute the given slope m = − 3 1 .
Solve for the y-intercept b using the point ( 4 , − 6 ) , which gives b = − 3 14 , and write the slope-intercept equation as y = − 3 1 x − 3 14 .
y + 6 = − 3 1 ( x − 4 )
Explanation
Problem Analysis We are given the slope of a line, m = − 3 1 , and a point it passes through, ( 4 , − 6 ) . Our goal is to find the equation of this line in both point-slope form and slope-intercept form.
Point-Slope Form The point-slope form of a line is given by the equation: y − y 1 = m ( x − x 1 ) where m is the slope and ( x 1 , y 1 ) is a point on the line.
Substitute Values Substitute the given values m = − 3 1 and ( x 1 , y 1 ) = ( 4 , − 6 ) into the point-slope form: y − ( − 6 ) = − 3 1 ( x − 4 ) Simplify the equation: y + 6 = − 3 1 ( x − 4 )
Slope-Intercept Form The slope-intercept form of a line is given by the equation: y = m x + b where m is the slope and b is the y-intercept.
Find the y-intercept We already know the slope m = − 3 1 , so we can write: y = − 3 1 x + b To find b , substitute the point ( 4 , − 6 ) into the equation: − 6 = − 3 1 ( 4 ) + b Solve for b : − 6 = − 3 4 + b Add 3 4 to both sides: b = − 6 + 3 4 = − 3 18 + 3 4 = − 3 14
Write the equation Now substitute the values of m = − 3 1 and b = − 3 14 into the slope-intercept form: y = − 3 1 x − 3 14
Final Answer The equation of the line in point-slope form is: y + 6 = − 3 1 ( x − 4 ) The equation of the line in slope-intercept form is: y = − 3 1 x − 3 14
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you're tracking the depreciation of a car's value over time, you might use a linear equation to model the decrease in value each year. Similarly, in physics, the relationship between distance, speed, and time can be expressed using a linear equation. These equations help us make predictions and understand relationships between different variables in a simple and effective way.
The point-slope form of the line is y + 6 = − 3 1 ( x − 4 ) . The slope-intercept form of the line is y = − 3 1 x − 3 14 .
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