Precalculus Examples

$2 x + y = - 2$ , $x + 2 y = 2$

Solve the system of equations.

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

$2 x + y = - 2$

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

Simplify.

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Multiply $- 2$ by $2$ to get $- 4$ .

$2 x + y = - 2$

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

Simplify the left side.

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Multiply $- 2$ by $x + 2 y$ to get $(- 2) (x + 2 y)$ .

$2 x + y = - 2$

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

Apply the distributive property.

$2 x + y = - 2$

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

Multiply $2$ by $- 2$ to get $- 4$ .

$2 x + y = - 2$

$- 2 x - 4 y = - 4$

$2 x + y = - 2$

$- 2 x - 4 y = - 4$

$2 x + y = - 2$

$- 2 x - 4 y = - 4$

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

		$2$	$x$	$+$		$y$	$=$	$-$	$2$
$+$	$-$	$2$	$x$	$-$	$4$	$y$	$=$	$-$	$4$
				$-$	$3$	$y$	$=$	$-$	$6$

Divide each term by $- 3$ and simplify.

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Divide each term in $- 3 y = - 6$ by $- 3$ .

$- \frac{3 y}{- 3} = - \frac{6}{- 3}$

Simplify the left side of the equation by cancelling the common factors.

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Reduce the expression by cancelling the common factors.

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Factor $3$ out of $- 3$ .

$- \frac{3 (y)}{3 (- 1)} = - \frac{6}{- 3}$

Cancel the common factor.

$- \frac{3 y}{3 \cdot - 1} = - \frac{6}{- 3}$

Rewrite the expression.

$- \frac{y}{- 1} = - \frac{6}{- 3}$

Move the negative one from the denominator of $\frac{y}{- 1}$ .

$- (- 1 \cdot y) = - \frac{6}{- 3}$

Simplify the expression.

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Multiply $- 1$ by $y$ to get $- 1 y$ .

$- (- 1 y) = - \frac{6}{- 3}$

Rewrite $- 1 y$ as $- y$ .

$y = - \frac{6}{- 3}$

Divide $6$ by $- 3$ to get $- 2$ .

$y = 2$

Substitute the value found for $y$ into one of the original equations, then solve for $x$ .

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Substitute the value found for $y$ into one of the original equations to solve for $x$ .

$2 x + (2) = - 2$

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

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Since $2$ does not contain the variable to solve for, move it to the right side of the equation by subtracting $2$ from both sides.

$2 x = - 2 - 2$

Subtract $2$ from $- 2$ to get $- 4$ .

$2 x = - 4$

Divide each term by $2$ and simplify.

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Divide each term in $2 x = - 4$ by $2$ .

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

Divide $x$ by $1$ to get $x$ .

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

Simplify the right side of the equation.

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Divide $4$ by $2$ to get $2$ .

$x = - 1 \cdot 2$

Multiply $- 1$ by $2$ to get $- 2$ .

$x = - 2$

The solution to the independent system of equations can be represented as a point.

$(- 2, 2)$

Since the system has a point of intersection, the system is independent.

Independent

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		$2$	$x$	$+$		$y$	$=$	$-$	$2$
$+$	$-$	$2$	$x$	$-$	$4$	$y$	$=$	$-$	$4$
				$-$	$3$	$y$	$=$	$-$	$6$

		$2$	$x$	$+$		$y$	$=$	$-$	$2$
$+$	$-$	$2$	$x$	$-$	$4$	$y$	$=$	$-$	$4$
				$-$	$3$	$y$	$=$	$-$	$6$

Precalculus Examples

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		$2$	$x$	$+$		$y$	$=$	$-$	$2$
$+$	$-$	$2$	$x$	$-$	$4$	$y$	$=$	$-$	$4$
				$-$	$3$	$y$	$=$	$-$	$6$