Precalculus Examples

$5 x - y = 16$ , $2 x + 3 y = 3$

Step 1

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

$(3) \cdot (5 x - y) = (3) (16)$

$2 x + 3 y = 3$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(3) \cdot (5 x - y)$ .

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Step 2.1.1.1

Apply the distributive property.

$3 (5 x) + 3 (- y) = (3) (16)$

$2 x + 3 y = 3$

Step 2.1.1.2

Multiply.

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Step 2.1.1.2.1

Multiply $5$ by $3$ .

$15 x + 3 (- y) = (3) (16)$

$2 x + 3 y = 3$

Step 2.1.1.2.2

Multiply $- 1$ by $3$ .

$15 x - 3 y = (3) (16)$

$2 x + 3 y = 3$

$15 x - 3 y = (3) (16)$

$2 x + 3 y = 3$

$15 x - 3 y = (3) (16)$

$2 x + 3 y = 3$

$15 x - 3 y = (3) (16)$

$2 x + 3 y = 3$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $3$ by $16$ .

$15 x - 3 y = 48$

$2 x + 3 y = 3$

$15 x - 3 y = 48$

$2 x + 3 y = 3$

$15 x - 3 y = 48$

$2 x + 3 y = 3$

Step 3

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

	$1$	$5$	$x$	$-$	$3$	$y$	$=$	$4$	$8$
$+$		$2$	$x$	$+$	$3$	$y$	$=$		$3$
	$1$	$7$	$x$				$=$	$5$	$1$

Step 4

Divide each term in $17 x = 51$ by $17$ and simplify.

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Step 4.1

Divide each term in $17 x = 51$ by $17$ .

$\frac{17 x}{17} = \frac{51}{17}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $17$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{17 x}{17} = \frac{51}{17}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{51}{17}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $51$ by $17$ .

$x = 3$

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 $y$ .

$15 (3) - 3 y = 48$

Step 5.2

Multiply $15$ by $3$ .

$45 - 3 y = 48$

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 $45$ from both sides of the equation.

$- 3 y = 48 - 45$

Step 5.3.2

Subtract $45$ from $48$ .

$- 3 y = 3$

Step 5.4

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

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Step 5.4.1

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

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

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.

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

Step 5.4.2.1.2

Divide $y$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $3$ by $- 3$ .

$y = - 1$

Step 6

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

$(3, - 1)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(3, - 1)$

Equation Form:

$x = 3, y = - 1$

Step 8

	$1$	$5$	$x$	$-$	$3$	$y$	$=$	$4$	$8$
$+$		$2$	$x$	$+$	$3$	$y$	$=$		$3$
	$1$	$7$	$x$				$=$	$5$	$1$

	$1$	$5$	$x$	$-$	$3$	$y$	$=$	$4$	$8$
$+$		$2$	$x$	$+$	$3$	$y$	$=$		$3$
	$1$	$7$	$x$				$=$	$5$	$1$

	$1$	$5$	$x$	$-$	$3$	$y$	$=$	$4$	$8$
$+$		$2$	$x$	$+$	$3$	$y$	$=$		$3$
	$1$	$7$	$x$				$=$	$5$	$1$