The slope-intercept form of the equation of a line that passes through point $(-3,8)$ is $y=-\frac{2}{3} x+6$. What is the point-slope form of the equation for this line?
A. $y-3=-\frac{2}{3}(x+8)$
B. $y+3=-\frac{2}{3}(x-8)$
C. $y+8=-\frac{2}{3}(x-3)$
D. $y-8=-\frac{2}{3}(x+3)$
Answer (1)
Recall the point-slope form: y − y 1 = m ( x − x 1 ) .
Identify the point and slope: ( x 1 , y 1 ) = ( − 3 , 8 ) and m = − 3 2 .
Substitute the values into the point-slope form: y − 8 = − 3 2 ( x − ( − 3 )) .
Simplify the equation: y − 8 = − 3 2 ( x + 3 ) .
The point-slope form of the equation is y − 8 = − 3 2 ( x + 3 ) .
Explanation
Understanding the Problem We are given a point ( − 3 , 8 ) and the slope-intercept form of the equation of a line y = − 3 2 x + 6 . We want to find the point-slope form of the equation for this line.
Recalling Point-Slope Form The point-slope form of a line is given by the equation y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope of the line.
Identifying the Slope and Point We are given the point ( − 3 , 8 ) , so x 1 = − 3 and y 1 = 8 . The slope-intercept form of the equation is y = − 3 2 x + 6 , which tells us that the slope of the line is m = − 3 2 .
Substituting Values Now, we substitute the values x 1 = − 3 , y 1 = 8 , and m = − 3 2 into the point-slope form equation:
y − 8 = − 3 2 ( x − ( − 3 ))
Simplifying the Equation Simplifying the equation, we get:
y − 8 = − 3 2 ( x + 3 )
Final Answer Comparing this equation with the given options, we see that the correct answer is y − 8 = − 3 2 ( x + 3 ) .
Examples
The point-slope form is useful in various real-world scenarios. For example, if you know the rate at which a savings account is growing (the slope) and the amount in the account at a specific time (a point), you can use the point-slope form to determine the amount in the account at any other time. Similarly, in physics, if you know the velocity of an object at a certain time and the constant acceleration (slope), you can find the velocity at any other time using the point-slope form.
