The Equation of a Line From the Coordinates or Slope

1. In the equation |y=mx+b,| replace parameter |m| with the given slope.

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

3. Isolate parameter |b| to find the y-intecept’s value.

4. Write the straight line equation in the form |y=mx+b| with the values of the parameters |m| and |b|.

What is the equation of a line with a slope of |3{.}5|, that passes through point |(-6,-28)|?

1. Write the equation of a line, and replace |m| with |3{.}5|.||y = 3{.}5x + b||

2. Using the given point, 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 + 21 &=b \\ -7 &= b \end{align}||

4. Write the equation in its functional form with the parameters |3{.}5| and |b=-7|. ||y = 3{.}5 x - 7||

1. Determine the slope’s value using the following formula:
   ||m=\dfrac{\Delta y}{\Delta x}=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}||

2. In the equation |y=mx+b,| replace parameter |m| with the slope determined in step 1.

3. In the same equation, replace |x| and |y| with the coordinates |(x,y)| in one of the two given points (your choice).

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

5. Write the straight line equation in the form |y=mx+b| with the values of the parameters |m| and |b|.

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

1. First, determine the slope value. ||\text{Slope}=\dfrac{10-(-8)}{5-3}=\dfrac{18}{2}=9||

2. Write the equation of a line, and replace the parameter |m| with |9|. ||y = 9x + b||

3. Using a known point (here, we choose the point |(5,10)|), replace |y| with |10| and |x| with |5|. ||\begin{align} y &= 9x + b \\ 10 &= 9(5) + b \end{align}||

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

5. Write the equation of the line in its functional form with the parameters |m=9| and |b=-35|. ||y = 9x -35||

When the x- and y-intercepts are known, the symmetrical form can be used to find the equation of a linear function. Follow these steps:

1. Replace parameter |a| with the x-intercept.

2. Replace parameter |b| with the y-intercept.

3. Next, if needed, transform the equation into functional form or general form.

What is the equation of a line with an x-intercept of |5| and a y-intercept of |- 4|?

Steps 1 and 2: Replace the parameter |a| with |5| and the parameter |b| with |-4|. ||\dfrac{x}{5} - \dfrac{y}{4}=1||

Step 3: Transform the equation into general form or functional form.

1. Look for the common denominator between 5 and 4, thus, 20. To get 20, multiply the first fraction by 4 and the second by -5: ||\begin{align} \dfrac{x\color{blue}{\times 4}}{5\color{blue}{\times 4}}+\dfrac{\ \ \ y\color{blue}{\times -5}}{-4\color{blue}{\times -5}} &= 1 \\ \dfrac{4x}{20}-\dfrac{5y}{20} &= \dfrac{20}{20} \end{align}||

2. Since the same denominator is everywhere, simplify it by multiplying the equation by 20. The result is: ||4x -5y = 20||

3. Transform the equation obtained above into the general form by reducing the equation equality to zero, or into the functional form by isolating |y|: ||\begin{align} 4x -5y -20 &= 0\ \ \Longrightarrow\ \text{General form} \\\\ 4x - 20 &= 5y \\\\ \dfrac{4x}{5}-\dfrac{20}{5} &= \dfrac{5y}{5} \\\\ \dfrac{4x}{5}-4 &= y\ \ \Longrightarrow\ \text{Functional form} \end{align}||