Algebra Examples

What is the equation of the line that passes through the point $(6, - 6)$ and has a slope of $- \frac{5}{3}$ ?

Step 1

Find the value of $b$ using the formula for the equation of a line.

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Step 1.1

Use the formula for the equation of a line to find $b$ .

$y = m x + b$

Step 1.2

Substitute the value of $m$ into the equation.

$y = (- \frac{5}{3}) \cdot x + b$

Step 1.3

Substitute the value of $x$ into the equation.

$y = (- \frac{5}{3}) \cdot (6) + b$

Step 1.4

Substitute the value of $y$ into the equation.

$- 6 = (- \frac{5}{3}) \cdot (6) + b$

Step 1.5

Find the value of $b$ .

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Step 1.5.1

Rewrite the equation as $- \frac{5}{3} \cdot 6 + b = - 6$ .

$- \frac{5}{3} \cdot 6 + b = - 6$

Step 1.5.2

Simplify each term.

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Step 1.5.2.1

Cancel the common factor of $3$ .

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Step 1.5.2.1.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$\frac{- 5}{3} \cdot 6 + b = - 6$

Step 1.5.2.1.2

Factor $3$ out of $6$ .

$\frac{- 5}{3} \cdot (3 (2)) + b = - 6$

Step 1.5.2.1.3

Cancel the common factor.

$\frac{- 5}{3} \cdot (3 \cdot 2) + b = - 6$

Step 1.5.2.1.4

Rewrite the expression.

$- 5 \cdot 2 + b = - 6$

Step 1.5.2.2

Multiply $- 5$ by $2$ .

$- 10 + b = - 6$

Step 1.5.3

Move all terms not containing $b$ to the right side of the equation.

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Step 1.5.3.1

Add $10$ to both sides of the equation.

$b = - 6 + 10$

Step 1.5.3.2

Add $- 6$ and $10$ .

$b = 4$

Step 2

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

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

Step 3