Which linear function represents the line given by the point-slope equation $y+7=-\frac{2}{3}(x+6)$?
A. $f(x)=-\frac{2}{3} x-11$
B. $f(x)=-\frac{2}{3} x-1$
C. $f(x)=-\frac{2}{3} x+3$
D. $f(x)=-\frac{2}{3} x+13$
Answer (2)
Distribute the slope: y + 7 = − 3 2 x − 3 2 ( 6 ) .
Simplify: y + 7 = − 3 2 x − 4 .
Isolate y : y = − 3 2 x − 4 − 7 .
Express as a linear function: f ( x ) = − 3 2 x − 11 . The final answer is f ( x ) = − 3 2 x − 11 .
Explanation
Understanding the Problem We are given the point-slope form of a linear equation: y + 7 = − 3 2 ( x + 6 ) . Our goal is to rewrite this equation in the slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Then, we can express it as a linear function f ( x ) = m x + b .
Distributing the Slope First, distribute the − 3 2 on the right side of the equation:
y + 7 = − 3 2 x − 3 2 ( 6 )
Simplifying the Equation Simplify the right side:
y + 7 = − 3 2 x − 4
Isolating y Subtract 7 from both sides to isolate y :
y = − 3 2 x − 4 − 7
Slope-Intercept Form Simplify to get the equation in slope-intercept form:
y = − 3 2 x − 11
The Linear Function Express the equation as a linear function:
f ( x ) = − 3 2 x − 11
So, the linear function that represents the given line is f ( x ) = − 3 2 x − 11 .
Examples
Linear functions are incredibly useful in everyday life. For example, imagine you are saving money. If you save a fixed amount each week, the total amount you've saved can be modeled using a linear function. If you start with $0 and save 15 p er w ee k , yo u rs a v in g sc anb ere p rese n t e d b y f(x) = 15x , w h ere x$ is the number of weeks. This helps you predict how much money you'll have saved after a certain period.
The linear function that represents the line given by the point-slope equation y + 7 = − 3 2 ( x + 6 ) is f ( x ) = − 3 2 x − 11 . The chosen multiple choice option is A: f ( x ) = − 3 2 x − 11 .
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