Precalculus Examples

$5 x + y = - 16$ , $7 x - 3 y = - 40$

Step 1

Solve the system of equations.

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Step 1.1

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

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

$7 x - 3 y = - 40$

Step 1.2

Simplify.

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Step 1.2.1

Simplify the left side.

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Step 1.2.1.1

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

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Step 1.2.1.1.1

Apply the distributive property.

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

$7 x - 3 y = - 40$

Step 1.2.1.1.2

Multiply $5$ by $3$ .

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

$7 x - 3 y = - 40$

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

$7 x - 3 y = - 40$

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

$7 x - 3 y = - 40$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Multiply $3$ by $- 16$ .

$15 x + 3 y = - 48$

$7 x - 3 y = - 40$

$15 x + 3 y = - 48$

$7 x - 3 y = - 40$

$15 x + 3 y = - 48$

$7 x - 3 y = - 40$

Step 1.3

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

	$1$	$5$	$x$	$+$	$3$	$y$	$=$	$-$	$4$	$8$
$+$		$7$	$x$	$-$	$3$	$y$	$=$	$-$	$4$	$0$
	$2$	$2$	$x$				$=$	$-$	$8$	$8$

Step 1.4

Divide each term in $22 x = - 88$ by $22$ and simplify.

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Step 1.4.1

Divide each term in $22 x = - 88$ by $22$ .

$\frac{22 x}{22} = \frac{- 88}{22}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $22$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{22 x}{22} = \frac{- 88}{22}$

Step 1.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 88}{22}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Divide $- 88$ by $22$ .

$x = - 4$

Step 1.5

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

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Step 1.5.1

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

$15 (- 4) + 3 y = - 48$

Step 1.5.2

Multiply $15$ by $- 4$ .

$- 60 + 3 y = - 48$

Step 1.5.3

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

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Step 1.5.3.1

Add $60$ to both sides of the equation.

$3 y = - 48 + 60$

Step 1.5.3.2

Add $- 48$ and $60$ .

$3 y = 12$

Step 1.5.4

Divide each term in $3 y = 12$ by $3$ and simplify.

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Step 1.5.4.1

Divide each term in $3 y = 12$ by $3$ .

$\frac{3 y}{3} = \frac{12}{3}$

Step 1.5.4.2

Simplify the left side.

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Step 1.5.4.2.1

Cancel the common factor of $3$ .

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Step 1.5.4.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{12}{3}$

Step 1.5.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{12}{3}$

Step 1.5.4.3

Simplify the right side.

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Step 1.5.4.3.1

Divide $12$ by $3$ .

$y = 4$

Step 1.6

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

$(- 4, 4)$

Step 2

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

Independent

Step 3

	$1$	$5$	$x$	$+$	$3$	$y$	$=$	$-$	$4$	$8$
$+$		$7$	$x$	$-$	$3$	$y$	$=$	$-$	$4$	$0$
	$2$	$2$	$x$				$=$	$-$	$8$	$8$

	$1$	$5$	$x$	$+$	$3$	$y$	$=$	$-$	$4$	$8$
$+$		$7$	$x$	$-$	$3$	$y$	$=$	$-$	$4$	$0$
	$2$	$2$	$x$				$=$	$-$	$8$	$8$

	$1$	$5$	$x$	$+$	$3$	$y$	$=$	$-$	$4$	$8$
$+$		$7$	$x$	$-$	$3$	$y$	$=$	$-$	$4$	$0$
	$2$	$2$	$x$				$=$	$-$	$8$	$8$