Standard Form to Point Slope Form

How to convert from Ax + By = C to y - y1 = m(x-x1)

Video Tutorial on Standard to Point Slope Form

Example Problem

Example

Convert 3x + 5y = 15 to point slope form.

Step 1

Isolate the $$\red y$$ term.

$ 3x + \red { 5 y} = 15 $ Step 1
Step 2

Multiply all terms by the multiplicative inverse of the coefficient of y.

$$ 5y = -3x + 15 \\ \red{\frac 1 5} \cdot 5y = \red{\frac 1 5} \cdot -3x + \red{\frac 1 5 } \cdot 15 \\ \frac{5y}{5} = \frac{-3x}{5} + \frac{15}{5} $$
Step 3

Simplify.

$$ \frac{5y}{5} = \frac{-3x}{5} + \frac{15}{5} \\ y = - \frac 3 5 x + 3 $$
Step 4

Substitute a convenient value for x into your equation and then solve for y.

You're doing this to get the values of $$ ( \blue{x_1}, \blue{y_1}) $$ for the point slope formula : $$ y - \blue y_1 = m(x - \blue x_1) $$

Remember that you can pick any value that you want. You're just choosing a value for $$x$$ and then finding its associated $$y $$ value.

Let's choose $$x = \blue 5$$. Yes, you could choose x = 0 and make your life really easy! After you solve for the y value, then .

$$ y = - \frac 3 5 x + 3 \\ \text{ Let's choose } x = \blue 5 \\ y = -\frac 3 5 \cdot \blue 5 + 3 \\ y = -3 + 3 \\ y = \blue 0 \\ \text{ Or, using } x = \blue 0 \\ y = - \frac 3 5 x + 3 \\ y = -\frac 3 5 \cdot \blue 0 + 3 \\ y = \blue 3 $$
Step 5

Substitute the x value you picked and y value you solved for into the general form of point slope formula.

$$ \text{ Using } x = \blue 5 \\ y -y_1 = m(x - x_1) \\ \boxed { y - \blue 0 = -\frac 3 5 (x - \blue 5) } \\ \text{ Or, using } x = \blue 0 \\ y -y_1 = m(x - x_1) \\ \boxed { y - \blue 3 = -\frac 3 5 (x - \blue 0) } \\ $$

Note that both of the above equations are equivalent. They both are valid. Neither equation is 'better'.

Practice Converting Equations

Practice 1

Convert $$ 3y - 2x = -12$$ to point slope form.

Step 1

Isolate the $$\red y $$ term.

$$ \red { 3y} - 2x = -12 $$
Step 1
Step 2

Multiply all terms by the multiplicative inverse of the coefficient of y.

$$ 3y = 2x -12 \\ \red { \frac 1 3 } \cdot 3y = \red { \frac 1 3 } \cdot 2x - \red { \frac 1 3 }\cdot 12 \\ \frac{ 3y}{3} = \frac{2x}{3} - \frac {12}{3} $$
Step 3

Simplify.

$$ \frac{ 3y}{3} = \frac{2x}{3} - \frac {12}{3} \\ y = \frac 2 3 x - 4 $$
Step 4

Substitute a convenient value for x into your equation and then solve for y.

You're doing this to get the values of $$ ( \blue{x_1}, \blue{y_1}) $$ for the point slope formula : $$ y - \blue y_1 = m(x - \blue x_1) $$

Remember that you can pick any value that you want. You're just choosing a value for $$x$$ and then finding its associated $$y $$ value.

Let's choose $$x = \blue 3$$. Yes, you could choose x = 0 and make your life really easy! After you solve for the y value, then .

$$ y = \frac 2 3 x - 4 \\ \text{ Using } x = \blue 3 \\ y = \frac 2 3 \cdot \blue 3 - 4 \\ y = 2 -4 \\ y = - 2 \\ \text{ Or, using } x = \blue 0 \\ y = \frac 2 3 \cdot \blue 0 - 4 \\ y = -4 $$
Step 5

Substitute the x value you picked and y value you solved for into the general form of point slope formula.

$$ \text{ Using } x = \blue 3 \\ y -y_1 = m(x - x_1) \\ \boxed { y + \blue 2 = \frac 2 3 (x - \blue 3) } \\ \text{ Or, using } x = \blue 0 \\ y -y_1 = m(x - x_1) \\ \boxed { y + \blue 4 = \frac 2 3 (x - \blue 0) } \\ $$

Note that both of the above equations are equivalent. They both are valid. Neither equation is 'better'.

Practice 2

Convert $$ 4y - 5x = 20 $$ to point slope form.

Step 1

Isolate the $$\red y$$ term.

$$ \red {4y} - 5x = 20 $$
Step 1
Step 2

Multiply all terms by the multiplicative inverse of the coefficient of y.

$$ 4y - 5x = 20 \\ \red{ \frac 1 4 }\cdot 4y = \red{ \frac 1 4 } \cdot 5x + \red{ \frac 1 4 } \cdot 20 \\ \frac{4y}{4} = \frac{5x}{4} + \frac{20}{4} $$
Step 3

Simplify.

Step 3
Step 4

Substitute a convenient value of x into your equation and solve for y (You're doing this to get the point x1, y1). Let's choose x = 3. Yes, you could choose x = 0 and make your life really easy!Then substitute the x and y values that found into the point slope formula.

Last step
Practice 3

Convert 2x + 3y = 18 to point slope form.

Step 1

Isolate the 'y' term.

Step 1
Step 2

Multiply all terms by the multiplicative inverse of the coefficient of y.

Step 2
Step 3

Simplify.

Step 3
Step 4

Substitute a convenient value of x into your equation and solve for y (You're doing this to get the point x1, y1). Let's choose x = 3. Yes, you could choose x = 0 and make your life really easy!Then substitute the x and y values that found into the point slope formula.

Last step

More challenging problems

Practice 4

Convert the equation Equation to point slope form.

Step 1

Isolate the 'y' term.

Step 1
Step 2

Multiply all terms by the multiplicative inverse of the coefficient of y.

Step 2
Step 3

Simplify.

Step 3
Step 4

Substitute a convenient value of x into your equation and solve for y (You're doing this to get the point x1, y1). Let's choose x = 3. Yes, you could choose x = 0 and make your life really easy!Then substitute the x and y values that found into the point slope formula.

Last step
Practice 5

Convert Equation to point slope form.

Step 1

Isolate the 'y' term.

Step 1
Step 2

Multiply all terms by the multiplicative inverse of the coefficient of y.

Step 2
Step 3

Simplify.

Step 3
Step 4

Substitute a convenient value of x into your equation and solve for y (You're doing this to get the point x1, y1). Let's choose x = 3. Yes, you could choose x = 0 and make your life really easy!Then substitute the x and y values that found into the point slope formula.

Last step
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