What is the correct order of steps to find the equation in slope intercept form when you know the slope is -3 and a point on the line is $(1,-2)$?
[tex]
\begin{array}{l}
y=-3 x+1 \\
y-(-2)=-3(x-1) \\
y+2=-3 x+3
\end{array}
[/tex]
Answer (2)
Use the point-slope form of the equation: y − y 1 = m ( x − x 1 ) .
Substitute the given slope m = − 3 and point ( 1 , − 2 ) into the point-slope form: y − ( − 2 ) = − 3 ( x − 1 ) .
Simplify the equation: y + 2 = − 3 x + 3 .
Isolate y to get the slope-intercept form: y = − 3 x + 1 .
The correct order is y − ( − 2 ) = − 3 ( x − 1 ) ; y + 2 = − 3 x + 3 ; y = − 3 x + 1 .
Explanation
Understanding the Problem We are given the slope of a line, m = − 3 , and a point on the line, ( 1 , − 2 ) . Our goal is to find the equation of the line in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. The provided equations are steps in the process, and we need to determine the correct order of these steps.
Applying Point-Slope Form The first step is to use the point-slope form of a linear equation, which is given by: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is a point on the line and m is the slope. We substitute the given values m = − 3 and ( x 1 , y 1 ) = ( 1 , − 2 ) into the point-slope form: y − ( − 2 ) = − 3 ( x − 1 ) This corresponds to the second equation in the given list.
Simplifying the Equation Next, we simplify the equation obtained in the previous step: y + 2 = − 3 ( x − 1 ) Distribute the − 3 on the right side: y + 2 = − 3 x + 3 This corresponds to the third equation in the given list.
Isolating y and Obtaining Slope-Intercept Form Finally, we isolate y to get the slope-intercept form: y = − 3 x + 3 − 2 y = − 3 x + 1 This corresponds to the first equation in the given list.
Determining the Correct Order Therefore, the correct order of the steps is:
y − ( − 2 ) = − 3 ( x − 1 )
y + 2 = − 3 x + 3
y = − 3 x + 1
Examples
Understanding linear equations is crucial in many real-world scenarios. 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. If the car's initial value is $20,000 and it depreciates by 2 , 000 e a c h ye a r , t h ee q u a t i o n w o u l d b e y = -2000x + 20000 , w h ere y i s t h ec a r ′ s v a l u e a f t er x$ years. This helps in predicting the car's value at any point in time.
To find the equation in slope-intercept form with a slope of -3 and a point (1, -2), we start with the point-slope form, then simplify and isolate y. The correct steps are: 1. y − ( − 2 ) = − 3 ( x − 1 ) , 2. y + 2 = − 3 x + 3 , 3. y = − 3 x + 1 .
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