Precalculus Examples

$3 x - y = 1$ , $5 x + 3 y = 11$

Step 1

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

$(3) \cdot (3 x - y) = (3) (1)$

$5 x + 3 y = 11$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(3) \cdot (3 x - y)$ .

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Step 2.1.1.1

Apply the distributive property.

$3 (3 x) + 3 (- y) = (3) (1)$

$5 x + 3 y = 11$

Step 2.1.1.2

Multiply.

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Step 2.1.1.2.1

Multiply $3$ by $3$ .

$9 x + 3 (- y) = (3) (1)$

$5 x + 3 y = 11$

Step 2.1.1.2.2

Multiply $- 1$ by $3$ .

$9 x - 3 y = (3) (1)$

$5 x + 3 y = 11$

$9 x - 3 y = (3) (1)$

$5 x + 3 y = 11$

$9 x - 3 y = (3) (1)$

$5 x + 3 y = 11$

$9 x - 3 y = (3) (1)$

$5 x + 3 y = 11$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $3$ by $1$ .

$9 x - 3 y = 3$

$5 x + 3 y = 11$

$9 x - 3 y = 3$

$5 x + 3 y = 11$

$9 x - 3 y = 3$

$5 x + 3 y = 11$

Step 3

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

		$9$	$x$	$-$	$3$	$y$	$=$		$3$
$+$		$5$	$x$	$+$	$3$	$y$	$=$	$1$	$1$
	$1$	$4$	$x$				$=$	$1$	$4$

Step 4

Divide each term in $14 x = 14$ by $14$ and simplify.

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Step 4.1

Divide each term in $14 x = 14$ by $14$ .

$\frac{14 x}{14} = \frac{14}{14}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $14$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{14 x}{14} = \frac{14}{14}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{14}{14}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $14$ by $14$ .

$x = 1$

Step 5

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

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Step 5.1

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

$9 (1) - 3 y = 3$

Step 5.2

Multiply $9$ by $1$ .

$9 - 3 y = 3$

Step 5.3

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

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Step 5.3.1

Subtract $9$ from both sides of the equation.

$- 3 y = 3 - 9$

Step 5.3.2

Subtract $9$ from $3$ .

$- 3 y = - 6$

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $- 3$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $y$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $- 6$ by $- 3$ .

$y = 2$

Step 6

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

$(1, 2)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(1, 2)$

Equation Form:

$x = 1, y = 2$

Step 8

		$9$	$x$	$-$	$3$	$y$	$=$		$3$
$+$		$5$	$x$	$+$	$3$	$y$	$=$	$1$	$1$
	$1$	$4$	$x$				$=$	$1$	$4$

		$9$	$x$	$-$	$3$	$y$	$=$		$3$
$+$		$5$	$x$	$+$	$3$	$y$	$=$	$1$	$1$
	$1$	$4$	$x$				$=$	$1$	$4$

		$9$	$x$	$-$	$3$	$y$	$=$		$3$
$+$		$5$	$x$	$+$	$3$	$y$	$=$	$1$	$1$
	$1$	$4$	$x$				$=$	$1$	$4$