Precalculus Examples

x + 2 y = 4

$x + 2 y = 4$ ,

2 x + 4 y = 8

$2 x + 4 y = 8$

Step 1

Multiply each equation by the value that makes the coefficients of

x

$x$ opposite.

(- 2) \cdot (x + 2 y) = (- 2) (4)

$(- 2) \cdot (x + 2 y) = (- 2) (4)$

2 x + 4 y = 8

$2 x + 4 y = 8$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify

(- 2) \cdot (x + 2 y)

$(- 2) \cdot (x + 2 y)$ .

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Step 2.1.1.1

Apply the distributive property.

- 2 x - 2 (2 y) = (- 2) (4)

$- 2 x - 2 (2 y) = (- 2) (4)$

2 x + 4 y = 8

$2 x + 4 y = 8$

Step 2.1.1.2

Multiply

2

$2$ by

- 2

$- 2$ .

- 2 x - 4 y = (- 2) (4)

$- 2 x - 4 y = (- 2) (4)$

2 x + 4 y = 8

$2 x + 4 y = 8$

- 2 x - 4 y = (- 2) (4)

$- 2 x - 4 y = (- 2) (4)$

2 x + 4 y = 8

$2 x + 4 y = 8$

- 2 x - 4 y = (- 2) (4)

$- 2 x - 4 y = (- 2) (4)$

2 x + 4 y = 8

$2 x + 4 y = 8$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply

- 2

$- 2$ by

4

$4$ .

- 2 x - 4 y = - 8

$- 2 x - 4 y = - 8$

2 x + 4 y = 8

$2 x + 4 y = 8$

- 2 x - 4 y = - 8

$- 2 x - 4 y = - 8$

2 x + 4 y = 8

$2 x + 4 y = 8$

- 2 x - 4 y = - 8

$- 2 x - 4 y = - 8$

2 x + 4 y = 8

$2 x + 4 y = 8$

Step 3

Add the two equations together to eliminate

x

$x$ from the system.

	$-$ $-$	$2$ $2$	$x$ $x$	$-$ $-$	$4$ $4$	$y$ $y$	$=$ $=$	$-$ $-$	$8$ $8$
$+$ $+$		$2$ $2$	$x$ $x$	$+$ $+$	$4$ $4$	$y$ $y$	$=$ $=$		$8$ $8$
						$0$ $0$	$=$ $=$		$0$ $0$

Step 4

Since

0 = 0

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

Infinite number of solutions

Step 5

Solve one of the equations for

y

$y$ .

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Step 5.1

Add

2 x

$2 x$ to both sides of the equation.

- 4 y = - 8 + 2 x

$- 4 y = - 8 + 2 x$

Step 5.2

Divide each term in

- 4 y = - 8 + 2 x

$- 4 y = - 8 + 2 x$ by

- 4

$- 4$ and simplify.

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Step 5.2.1

Divide each term in

- 4 y = - 8 + 2 x

$- 4 y = - 8 + 2 x$ by

- 4

$- 4$ .

\frac{- 4 y}{- 4} = \frac{- 8}{- 4} + \frac{2 x}{- 4}

$\frac{- 4 y}{- 4} = \frac{- 8}{- 4} + \frac{2 x}{- 4}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

- 4

$- 4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{- 8}{- 4} + \frac{2 x}{- 4}$

Step 5.2.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{- 8}{- 4} + \frac{2 x}{- 4}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Divide

$- 8$ by

$- 4$ .

$y = 2 + \frac{2 x}{- 4}$

Step 5.2.3.1.2

Cancel the common factor of

$2$ and

$- 4$ .

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Step 5.2.3.1.2.1

Factor

$2$ out of

$2 x$ .

$y = 2 + \frac{2 (x)}{- 4}$

Step 5.2.3.1.2.2

Cancel the common factors.

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Step 5.2.3.1.2.2.1

Factor

$2$ out of

$- 4$ .

$y = 2 + \frac{2 x}{2 \cdot - 2}$

Step 5.2.3.1.2.2.2

Cancel the common factor.

$y = 2 + \frac{2 x}{2 \cdot - 2}$

Step 5.2.3.1.2.2.3

Rewrite the expression.

$y = 2 + \frac{x}{- 2}$

Step 5.2.3.1.3

Move the negative in front of the fraction.

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

Step 6

The solution is the set of ordered pairs that make

$y = 2 - \frac{x}{2}$ true.

$(x, 2 - \frac{x}{2})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(x, 2 - \frac{x}{2})$

Equation Form:

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

Step 8

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