Finding the Equation of a Linear Function

To find the equation of a line from the rate of change and a point, follow these steps.

1. In the equation |y=ax+b|, replace the parameter |a| by the rate of change given.

2. In the same equation, replace |x| and |y| by the |(x,y)| coordinates of the given point.

3. Isolate parameter |b| to find the value of the |y|-intercept.

4. Write the equation of the line in the form |y=ax+b| with the values of the parameters |a| and |b|.

What is the equation of the line that has a rate of change of |3.5| and that passes through the point |(-6,-28)|?

1. Replace |a| with |3.5| in the equation of the line ||y = 3.5x + b||

2. Replace |y| with |-28| and |x| with |-6| ||\begin{align} y &= 3.5x + b \\ -28 &= 3.5(-6) + b \end{align}||

3. Isolate the parameter |b| ||\begin{align} -28 &= 3.5(-6) + b \\ -28 &= -21 + b \\ -28 \color{red}{+21} &= -21 \color{red}{+21} + b \\ -7 &= b \end{align}||

4. Write the equation of the line with the parameters |a=3.5| and |b=-7| ||y = 3.5 x - 7||

When looking for the equation of a line from the coordinates of two points, follow these steps.

1. Determine the value of the rate of change using the following formula. ||a = \dfrac{\Delta y}{\Delta x} = \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}||

2. In the equation |y=ax+b,| replace the parameter |a| by the rate of change determined in step 1.

3. In this same equation, replace |x| and |y| by the |(x,y)| coordinates of one of the two points (choose one).

4. Isolate the parameter |b| to find the value of the |y|-intercept.

5. Write the equation of the line in the form |y=ax+b| with the values of the parameters |a| and |b.|

What is the equation of the line that passes through the following points: |(3,-8)| and |(5,10)|?

1. Determine the value of the rate of change |(a)| ||\begin{align} a = \dfrac{\Delta y}{\Delta x} &= \dfrac{y_2-y_1}{x_2-x_1} \\ &= \dfrac{10-(-8)}{5-3}\\ &=\dfrac{18}{2} \\ &=9 \end{align}||

2. Replace parameter |a| with |9| in the equation of the line ||y=9x+b||

3. Replace |x| and |y| with the |(x,y)| coordinates of one of the two points given

   Choose the point |(5,10).| Then replace |y| with |10| and |x| with |5.| ||\begin{align} y &= 9x + b \\ 10 &= 9(5) + b \end{align}||

4. Isolate the parameter |b| ||\begin{align} 10 &= 9(5) + b \\ 10 &= 45 + b \\ 10 \color{red}{- 45} &= 45 \color{red}{- 45} +b \\ -35 &= b  \end{align}||

5. Write the equation of the line with the parameters |a=9| and |b=-35| ||y = 9x -35||