Algebra Examples

$4 x - 3 y = 8$ and $5 x - 2 y = - 11$

Step 1

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

$(- 2) \cdot (4 x - 3 y) = (- 2) (8)$

$(3) \cdot (5 x - 2 y) = (3) (- 11)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(- 2) \cdot (4 x - 3 y)$ .

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Step 2.1.1.1

Apply the distributive property.

$- 2 (4 x) - 2 (- 3 y) = (- 2) (8)$

$(3) \cdot (5 x - 2 y) = (3) (- 11)$

Step 2.1.1.2

Multiply.

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Step 2.1.1.2.1

Multiply $4$ by $- 2$ .

$- 8 x - 2 (- 3 y) = (- 2) (8)$

$(3) \cdot (5 x - 2 y) = (3) (- 11)$

Step 2.1.1.2.2

Multiply $- 3$ by $- 2$ .

$- 8 x + 6 y = (- 2) (8)$

$(3) \cdot (5 x - 2 y) = (3) (- 11)$

$- 8 x + 6 y = (- 2) (8)$

$(3) \cdot (5 x - 2 y) = (3) (- 11)$

$- 8 x + 6 y = (- 2) (8)$

$(3) \cdot (5 x - 2 y) = (3) (- 11)$

$- 8 x + 6 y = (- 2) (8)$

$(3) \cdot (5 x - 2 y) = (3) (- 11)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $- 2$ by $8$ .

$- 8 x + 6 y = - 16$

$(3) \cdot (5 x - 2 y) = (3) (- 11)$

$- 8 x + 6 y = - 16$

$(3) \cdot (5 x - 2 y) = (3) (- 11)$

Step 2.3

Simplify the left side.

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Step 2.3.1

Simplify $(3) \cdot (5 x - 2 y)$ .

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Step 2.3.1.1

Apply the distributive property.

$- 8 x + 6 y = - 16$

$3 (5 x) + 3 (- 2 y) = (3) (- 11)$

Step 2.3.1.2

Multiply.

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Step 2.3.1.2.1

Multiply $5$ by $3$ .

$- 8 x + 6 y = - 16$

$15 x + 3 (- 2 y) = (3) (- 11)$

Step 2.3.1.2.2

Multiply $- 2$ by $3$ .

$- 8 x + 6 y = - 16$

$15 x - 6 y = (3) (- 11)$

$- 8 x + 6 y = - 16$

$15 x - 6 y = (3) (- 11)$

$- 8 x + 6 y = - 16$

$15 x - 6 y = (3) (- 11)$

$- 8 x + 6 y = - 16$

$15 x - 6 y = (3) (- 11)$

Step 2.4

Simplify the right side.

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Step 2.4.1

Multiply $3$ by $- 11$ .

$- 8 x + 6 y = - 16$

$15 x - 6 y = - 33$

$- 8 x + 6 y = - 16$

$15 x - 6 y = - 33$

$- 8 x + 6 y = - 16$

$15 x - 6 y = - 33$

Step 3

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

	$-$	$8$	$x$	$+$	$6$	$y$	$=$	$-$	$1$	$6$
$+$	$1$	$5$	$x$	$-$	$6$	$y$	$=$	$-$	$3$	$3$
		$7$	$x$				$=$	$-$	$4$	$9$

Step 4

Divide each term in $7 x = - 49$ by $7$ and simplify.

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Step 4.1

Divide each term in $7 x = - 49$ by $7$ .

$\frac{7 x}{7} = \frac{- 49}{7}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $7$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 49}{7}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 49}{7}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 49$ by $7$ .

$x = - 7$

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 $y$ .

$- 8 \cdot - 7 + 6 y = - 16$

Step 5.2

Multiply $- 8$ by $- 7$ .

$56 + 6 y = - 16$

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 $56$ from both sides of the equation.

$6 y = - 16 - 56$

Step 5.3.2

Subtract $56$ from $- 16$ .

$6 y = - 72$

Step 5.4

Divide each term in $6 y = - 72$ by $6$ and simplify.

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Step 5.4.1

Divide each term in $6 y = - 72$ by $6$ .

$\frac{6 y}{6} = \frac{- 72}{6}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $6$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{- 72}{6}$

Step 5.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 72}{6}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $- 72$ by $6$ .

$y = - 12$

Step 6

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

$(- 7, - 12)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- 7, - 12)$

Equation Form:

$x = - 7, y = - 12$

Step 8

	$-$	$8$	$x$	$+$	$6$	$y$	$=$	$-$	$1$	$6$
$+$	$1$	$5$	$x$	$-$	$6$	$y$	$=$	$-$	$3$	$3$
		$7$	$x$				$=$	$-$	$4$	$9$

	$-$	$8$	$x$	$+$	$6$	$y$	$=$	$-$	$1$	$6$
$+$	$1$	$5$	$x$	$-$	$6$	$y$	$=$	$-$	$3$	$3$
		$7$	$x$				$=$	$-$	$4$	$9$

	$-$	$8$	$x$	$+$	$6$	$y$	$=$	$-$	$1$	$6$
$+$	$1$	$5$	$x$	$-$	$6$	$y$	$=$	$-$	$3$	$3$
		$7$	$x$				$=$	$-$	$4$	$9$