Algebra Examples

$5 x + 2 y = 7$ $10 x + 4 y = 14$

Step 1

Multiply each equation by the value that makes the coefficients of $y$ opposite.

$(- 2) \cdot (5 x + 2 y) = (- 2) (7)$

$10 x + 4 y = 14$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(- 2) \cdot (5 x + 2 y)$ .

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Step 2.1.1.1

Apply the distributive property.

$- 2 (5 x) - 2 (2 y) = (- 2) (7)$

$10 x + 4 y = 14$

Step 2.1.1.2

Multiply.

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Step 2.1.1.2.1

Multiply $5$ by $- 2$ .

$- 10 x - 2 (2 y) = (- 2) (7)$

$10 x + 4 y = 14$

Step 2.1.1.2.2

Multiply $2$ by $- 2$ .

$- 10 x - 4 y = (- 2) (7)$

$10 x + 4 y = 14$

$- 10 x - 4 y = (- 2) (7)$

$10 x + 4 y = 14$

$- 10 x - 4 y = (- 2) (7)$

$10 x + 4 y = 14$

$- 10 x - 4 y = (- 2) (7)$

$10 x + 4 y = 14$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $- 2$ by $7$ .

$- 10 x - 4 y = - 14$

$10 x + 4 y = 14$

$- 10 x - 4 y = - 14$

$10 x + 4 y = 14$

$- 10 x - 4 y = - 14$

$10 x + 4 y = 14$

Step 3

Add the two equations together to eliminate $y$ from the system.

	$-$	$1$	$0$	$x$	$-$	$4$	$y$	$=$	$-$	$1$	$4$
$+$		$1$	$0$	$x$	$+$	$4$	$y$	$=$		$1$	$4$
							$0$	$=$			$0$

Step 4

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

Infinite number of solutions

Step 5

Solve one of the equations for $y$ .

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Step 5.1

Add $10 x$ to both sides of the equation.

$- 4 y = - 14 + 10 x$

Step 5.2

Divide each term in $- 4 y = - 14 + 10 x$ by $- 4$ and simplify.

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Step 5.2.1

Divide each term in $- 4 y = - 14 + 10 x$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{- 14}{- 4} + \frac{10 x}{- 4}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{- 14}{- 4} + \frac{10 x}{- 4}$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 14}{- 4} + \frac{10 x}{- 4}$

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

Cancel the common factor of $- 14$ and $- 4$ .

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Step 5.2.3.1.1.1

Factor $- 2$ out of $- 14$ .

$y = \frac{- 2 (7)}{- 4} + \frac{10 x}{- 4}$

Step 5.2.3.1.1.2

Cancel the common factors.

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Step 5.2.3.1.1.2.1

Factor $- 2$ out of $- 4$ .

$y = \frac{- 2 \cdot 7}{- 2 \cdot 2} + \frac{10 x}{- 4}$

Step 5.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 7}{- 2 \cdot 2} + \frac{10 x}{- 4}$

Step 5.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{7}{2} + \frac{10 x}{- 4}$

Step 5.2.3.1.2

Cancel the common factor of $10$ and $- 4$ .

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Step 5.2.3.1.2.1

Factor $2$ out of $10 x$ .

$y = \frac{7}{2} + \frac{2 (5 x)}{- 4}$

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 $- 4$ .

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

Step 5.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{7}{2} + \frac{2 (5 x)}{2 \cdot - 2}$

Step 5.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{7}{2} + \frac{5 x}{- 2}$

Step 5.2.3.1.3

Move the negative in front of the fraction.

$y = \frac{7}{2} - \frac{5 x}{2}$

Step 6

The solution is the set of ordered pairs that make $y = \frac{7}{2} - \frac{5 x}{2}$ true.

$(x, \frac{7}{2} - \frac{5 x}{2})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(x, \frac{7}{2} - \frac{5 x}{2})$

Equation Form:

$x = x, y = \frac{7}{2} - \frac{5 x}{2}$

Step 8

	$-$	$1$	$0$	$x$	$-$	$4$	$y$	$=$	$-$	$1$	$4$
$+$		$1$	$0$	$x$	$+$	$4$	$y$	$=$		$1$	$4$
							$0$	$=$			$0$

	$-$	$1$	$0$	$x$	$-$	$4$	$y$	$=$	$-$	$1$	$4$
$+$		$1$	$0$	$x$	$+$	$4$	$y$	$=$		$1$	$4$
							$0$	$=$			$0$

	$-$	$1$	$0$	$x$	$-$	$4$	$y$	$=$	$-$	$1$	$4$
$+$		$1$	$0$	$x$	$+$	$4$	$y$	$=$		$1$	$4$
							$0$	$=$			$0$