Examples

4 x - 9 y = 21

$4 x - 9 y = 21$ ,

12 x - 27 y = 63

$12 x - 27 y = 63$

Step 1

Solve the system of equations.

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Step 1.1

Multiply each equation by the value that makes the coefficients of

x

$x$ opposite.

(- 3) \cdot (4 x - 9 y) = (- 3) (21)

$(- 3) \cdot (4 x - 9 y) = (- 3) (21)$

12 x - 27 y = 63

$12 x - 27 y = 63$

Step 1.2

Simplify.

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Step 1.2.1

Simplify the left side.

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Step 1.2.1.1

Simplify

(- 3) \cdot (4 x - 9 y)

$(- 3) \cdot (4 x - 9 y)$ .

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Step 1.2.1.1.1

Apply the distributive property.

- 3 (4 x) - 3 (- 9 y) = (- 3) (21)

$- 3 (4 x) - 3 (- 9 y) = (- 3) (21)$

12 x - 27 y = 63

$12 x - 27 y = 63$

Step 1.2.1.1.2

Multiply.

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Step 1.2.1.1.2.1

Multiply

4

$4$ by

- 3

$- 3$ .

- 12 x - 3 (- 9 y) = (- 3) (21)

$- 12 x - 3 (- 9 y) = (- 3) (21)$

12 x - 27 y = 63

$12 x - 27 y = 63$

Step 1.2.1.1.2.2

Multiply

- 9

$- 9$ by

- 3

$- 3$ .

- 12 x + 27 y = (- 3) (21)

$- 12 x + 27 y = (- 3) (21)$

12 x - 27 y = 63

$12 x - 27 y = 63$

- 12 x + 27 y = (- 3) (21)

$- 12 x + 27 y = (- 3) (21)$

12 x - 27 y = 63

$12 x - 27 y = 63$

- 12 x + 27 y = (- 3) (21)

$- 12 x + 27 y = (- 3) (21)$

12 x - 27 y = 63

$12 x - 27 y = 63$

- 12 x + 27 y = (- 3) (21)

$- 12 x + 27 y = (- 3) (21)$

12 x - 27 y = 63

$12 x - 27 y = 63$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Multiply

- 3

$- 3$ by

21

$21$ .

- 12 x + 27 y = - 63

$- 12 x + 27 y = - 63$

12 x - 27 y = 63

$12 x - 27 y = 63$

- 12 x + 27 y = - 63

$- 12 x + 27 y = - 63$

12 x - 27 y = 63

$12 x - 27 y = 63$

- 12 x + 27 y = - 63

$- 12 x + 27 y = - 63$

12 x - 27 y = 63

$12 x - 27 y = 63$

Step 1.3

Add the two equations together to eliminate

x

$x$ from the system.

	$-$ $-$	$1$ $1$	$2$ $2$	$x$ $x$	$+$ $+$	$2$ $2$	$7$ $7$	$y$ $y$	$=$ $=$	$-$ $-$	$6$ $6$	$3$ $3$
$+$ $+$		$1$ $1$	$2$ $2$	$x$ $x$	$-$ $-$	$2$ $2$	$7$ $7$	$y$ $y$	$=$ $=$		$6$ $6$	$3$ $3$
								$0$ $0$	$=$ $=$			$0$ $0$

Step 1.4

Since

0 = 0

$0 = 0$ , the equations intersect at an infinite number of points.

Infinite number of solutions

Step 1.5

Solve one of the equations for

y

$y$ .

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Step 1.5.1

Add

12 x

$12 x$ to both sides of the equation.

27 y = - 63 + 12 x

$27 y = - 63 + 12 x$

Step 1.5.2

Divide each term in

27 y = - 63 + 12 x

$27 y = - 63 + 12 x$ by

27

$27$ and simplify.

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Step 1.5.2.1

Divide each term in

27 y = - 63 + 12 x

$27 y = - 63 + 12 x$ by

27

$27$ .

\frac{27 y}{27} = \frac{- 63}{27} + \frac{12 x}{27}

$\frac{27 y}{27} = \frac{- 63}{27} + \frac{12 x}{27}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of

27

$27$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

\frac{27 y}{27} = \frac{- 63}{27} + \frac{12 x}{27}

$\frac{27 y}{27} = \frac{- 63}{27} + \frac{12 x}{27}$

Step 1.5.2.2.1.2

Divide

y

$y$ by

1

$1$ .

y = \frac{- 63}{27} + \frac{12 x}{27}

$y = \frac{- 63}{27} + \frac{12 x}{27}$

y = \frac{- 63}{27} + \frac{12 x}{27}

$y = \frac{- 63}{27} + \frac{12 x}{27}$

y = \frac{- 63}{27} + \frac{12 x}{27}

$y = \frac{- 63}{27} + \frac{12 x}{27}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Simplify each term.

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Step 1.5.2.3.1.1

Cancel the common factor of

- 63

$- 63$ and

27

$27$ .

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Step 1.5.2.3.1.1.1

Factor

9

$9$ out of

- 63

$- 63$ .

y = \frac{9 (- 7)}{27} + \frac{12 x}{27}

$y = \frac{9 (- 7)}{27} + \frac{12 x}{27}$

Step 1.5.2.3.1.1.2

Cancel the common factors.

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Step 1.5.2.3.1.1.2.1

Factor

9

$9$ out of

27

$27$ .

y = \frac{9 \cdot - 7}{9 \cdot 3} + \frac{12 x}{27}

$y = \frac{9 \cdot - 7}{9 \cdot 3} + \frac{12 x}{27}$

Step 1.5.2.3.1.1.2.2

Cancel the common factor.

y = \frac{9 \cdot - 7}{9 \cdot 3} + \frac{12 x}{27}

$y = \frac{9 \cdot - 7}{9 \cdot 3} + \frac{12 x}{27}$

Step 1.5.2.3.1.1.2.3

Rewrite the expression.

y = \frac{- 7}{3} + \frac{12 x}{27}

$y = \frac{- 7}{3} + \frac{12 x}{27}$

y = \frac{- 7}{3} + \frac{12 x}{27}

$y = \frac{- 7}{3} + \frac{12 x}{27}$

y = \frac{- 7}{3} + \frac{12 x}{27}

$y = \frac{- 7}{3} + \frac{12 x}{27}$

Step 1.5.2.3.1.2

Move the negative in front of the fraction.

y = - \frac{7}{3} + \frac{12 x}{27}

$y = - \frac{7}{3} + \frac{12 x}{27}$

Step 1.5.2.3.1.3

Cancel the common factor of

12

$12$ and

27

$27$ .

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Step 1.5.2.3.1.3.1

Factor

3

$3$ out of

12 x

$12 x$ .

y = - \frac{7}{3} + \frac{3 (4 x)}{27}

$y = - \frac{7}{3} + \frac{3 (4 x)}{27}$

Step 1.5.2.3.1.3.2

Cancel the common factors.

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Step 1.5.2.3.1.3.2.1

Factor

3

$3$ out of

27

$27$ .

y = - \frac{7}{3} + \frac{3 (4 x)}{3 (9)}

$y = - \frac{7}{3} + \frac{3 (4 x)}{3 (9)}$

Step 1.5.2.3.1.3.2.2

Cancel the common factor.

y = - \frac{7}{3} + \frac{3 (4 x)}{3 \cdot 9}

$y = - \frac{7}{3} + \frac{3 (4 x)}{3 \cdot 9}$

Step 1.5.2.3.1.3.2.3

Rewrite the expression.

y = - \frac{7}{3} + \frac{4 x}{9}