Precalculus Examples

$12 x + 6 y + 2 = 0$ , $3 y = - 15 x - 7$

Step 1

Divide each term in $3 y = - 15 x - 7$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 y = - 15 x - 7$ by $3$ .

$\frac{3 y}{3} = \frac{- 15 x}{3} + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{- 15 x}{3} + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 15 x}{3} + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

$y = \frac{- 15 x}{3} + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

$y = \frac{- 15 x}{3} + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Cancel the common factor of $- 15$ and $3$ .

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Step 1.3.1.1.1

Factor $3$ out of $- 15 x$ .

$y = \frac{3 (- 5 x)}{3} + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

Step 1.3.1.1.2

Cancel the common factors.

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Step 1.3.1.1.2.1

Factor $3$ out of $3$ .

$y = \frac{3 (- 5 x)}{3 (1)} + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

Step 1.3.1.1.2.2

Cancel the common factor.

$y = \frac{3 (- 5 x)}{3 \cdot 1} + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

Step 1.3.1.1.2.3

Rewrite the expression.

$y = \frac{- 5 x}{1} + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

Step 1.3.1.1.2.4

Divide $- 5 x$ by $1$ .

$y = - 5 x + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

$y = - 5 x + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

$y = - 5 x + \frac{- 7}{3}$

$12 x + 6 y + 2 = 0$

Step 1.3.1.2

Move the negative in front of the fraction.

$y = - 5 x - \frac{7}{3}$

$12 x + 6 y + 2 = 0$

$y = - 5 x - \frac{7}{3}$

$12 x + 6 y + 2 = 0$

$y = - 5 x - \frac{7}{3}$

$12 x + 6 y + 2 = 0$

$y = - 5 x - \frac{7}{3}$

$12 x + 6 y + 2 = 0$

Step 2

Replace all occurrences of $y$ with $- 5 x - \frac{7}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $12 x + 6 y + 2 = 0$ with $- 5 x - \frac{7}{3}$ .

$12 x + 6 (- 5 x - \frac{7}{3}) + 2 = 0$

$y = - 5 x - \frac{7}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $12 x + 6 (- 5 x - \frac{7}{3}) + 2$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$12 x + 6 (- 5 x) + 6 (- \frac{7}{3}) + 2 = 0$

$y = - 5 x - \frac{7}{3}$

Step 2.2.1.1.2

Multiply $- 5$ by $6$ .

$12 x - 30 x + 6 (- \frac{7}{3}) + 2 = 0$

$y = - 5 x - \frac{7}{3}$

Step 2.2.1.1.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.3.1

Move the leading negative in $- \frac{7}{3}$ into the numerator.

$12 x - 30 x + 6 (\frac{- 7}{3}) + 2 = 0$

$y = - 5 x - \frac{7}{3}$

Step 2.2.1.1.3.2

Factor $3$ out of $6$ .

$12 x - 30 x + 3 (2) (\frac{- 7}{3}) + 2 = 0$

$y = - 5 x - \frac{7}{3}$

Step 2.2.1.1.3.3

Cancel the common factor.

$12 x - 30 x + 3 \cdot (2 (\frac{- 7}{3})) + 2 = 0$

$y = - 5 x - \frac{7}{3}$

Step 2.2.1.1.3.4

Rewrite the expression.

$12 x - 30 x + 2 \cdot - 7 + 2 = 0$

$y = - 5 x - \frac{7}{3}$

$12 x - 30 x + 2 \cdot - 7 + 2 = 0$

$y = - 5 x - \frac{7}{3}$

Step 2.2.1.1.4

Multiply $2$ by $- 7$ .

$12 x - 30 x - 14 + 2 = 0$

$y = - 5 x - \frac{7}{3}$

$12 x - 30 x - 14 + 2 = 0$

$y = - 5 x - \frac{7}{3}$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $30 x$ from $12 x$ .

$- 18 x - 14 + 2 = 0$

$y = - 5 x - \frac{7}{3}$

Step 2.2.1.2.2

Add $- 14$ and $2$ .

$- 18 x - 12 = 0$

$y = - 5 x - \frac{7}{3}$

$- 18 x - 12 = 0$

$y = - 5 x - \frac{7}{3}$

$- 18 x - 12 = 0$

$y = - 5 x - \frac{7}{3}$

$- 18 x - 12 = 0$

$y = - 5 x - \frac{7}{3}$

$- 18 x - 12 = 0$

$y = - 5 x - \frac{7}{3}$

Step 3

Solve for $x$ in $- 18 x - 12 = 0$ .

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Step 3.1

Add $12$ to both sides of the equation.

$- 18 x = 12$

$y = - 5 x - \frac{7}{3}$

Step 3.2

Divide each term in $- 18 x = 12$ by $- 18$ and simplify.

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Step 3.2.1

Divide each term in $- 18 x = 12$ by $- 18$ .

$\frac{- 18 x}{- 18} = \frac{12}{- 18}$

$y = - 5 x - \frac{7}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 18$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 18 x}{- 18} = \frac{12}{- 18}$

$y = - 5 x - \frac{7}{3}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{- 18}$

$y = - 5 x - \frac{7}{3}$

$x = \frac{12}{- 18}$

$y = - 5 x - \frac{7}{3}$

$x = \frac{12}{- 18}$

$y = - 5 x - \frac{7}{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $12$ and $- 18$ .

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Step 3.2.3.1.1

Factor $6$ out of $12$ .

$x = \frac{6 (2)}{- 18}$

$y = - 5 x - \frac{7}{3}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $6$ out of $- 18$ .

$x = \frac{6 \cdot 2}{6 \cdot - 3}$

$y = - 5 x - \frac{7}{3}$

Step 3.2.3.1.2.2

Cancel the common factor.

$x = \frac{6 \cdot 2}{6 \cdot - 3}$

$y = - 5 x - \frac{7}{3}$

Step 3.2.3.1.2.3

Rewrite the expression.

$x = \frac{2}{- 3}$

$y = - 5 x - \frac{7}{3}$

$x = \frac{2}{- 3}$

$y = - 5 x - \frac{7}{3}$

$x = \frac{2}{- 3}$

$y = - 5 x - \frac{7}{3}$

Step 3.2.3.2

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

$y = - 5 x - \frac{7}{3}$

$x = - \frac{2}{3}$

$y = - 5 x - \frac{7}{3}$

$x = - \frac{2}{3}$

$y = - 5 x - \frac{7}{3}$

$x = - \frac{2}{3}$

$y = - 5 x - \frac{7}{3}$

Step 4

Replace all occurrences of $x$ with $- \frac{2}{3}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - 5 x - \frac{7}{3}$ with $- \frac{2}{3}$ .

$y = - 5 (- \frac{2}{3}) - \frac{7}{3}$

$x = - \frac{2}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 5 (- \frac{2}{3}) - \frac{7}{3}$ .

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Step 4.2.1.1

Multiply $- 5 (- \frac{2}{3})$ .

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Step 4.2.1.1.1

Multiply $- 1$ by $- 5$ .

$y = 5 (\frac{2}{3}) - \frac{7}{3}$

$x = - \frac{2}{3}$

Step 4.2.1.1.2

Combine $5$ and $\frac{2}{3}$ .

$y = \frac{5 \cdot 2}{3} - \frac{7}{3}$

$x = - \frac{2}{3}$

Step 4.2.1.1.3

Multiply $5$ by $2$ .

$y = \frac{10}{3} - \frac{7}{3}$

$x = - \frac{2}{3}$

$y = \frac{10}{3} - \frac{7}{3}$

$x = - \frac{2}{3}$

Step 4.2.1.2

Combine the numerators over the common denominator.

$y = \frac{10 - 7}{3}$

$x = - \frac{2}{3}$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Subtract $7$ from $10$ .

$y = \frac{3}{3}$

$x = - \frac{2}{3}$

Step 4.2.1.3.2

Divide $3$ by $3$ .

$y = 1$

$x = - \frac{2}{3}$

$y = 1$

$x = - \frac{2}{3}$

$y = 1$

$x = - \frac{2}{3}$

$y = 1$

$x = - \frac{2}{3}$

$y = 1$

$x = - \frac{2}{3}$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- \frac{2}{3}, 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{2}{3}, 1)$

Equation Form:

$x = - \frac{2}{3}, y = 1$

Step 7