Algebra Examples

$2 x - 5 y = 10$ $3 x + y = 15$

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.

$2 x - 5 y = 10$

$(5) \cdot (3 x + y) = (5) (15)$

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

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Step 1.2.1.1.1

Apply the distributive property.

$2 x - 5 y = 10$

$5 (3 x) + 5 y = (5) (15)$

Step 1.2.1.1.2

Multiply $3$ by $5$ .

$2 x - 5 y = 10$

$15 x + 5 y = (5) (15)$

$2 x - 5 y = 10$

$15 x + 5 y = (5) (15)$

$2 x - 5 y = 10$

$15 x + 5 y = (5) (15)$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Multiply $5$ by $15$ .

$2 x - 5 y = 10$

$15 x + 5 y = 75$

$2 x - 5 y = 10$

$15 x + 5 y = 75$

$2 x - 5 y = 10$

$15 x + 5 y = 75$

Step 1.3

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

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

Step 1.4

Divide each term in $17 x = 85$ by $17$ and simplify.

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Step 1.4.1

Divide each term in $17 x = 85$ by $17$ .

$\frac{17 x}{17} = \frac{85}{17}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $17$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{17 x}{17} = \frac{85}{17}$

Step 1.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{85}{17}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Divide $85$ by $17$ .

$x = 5$

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 $y$ .

$2 (5) - 5 y = 10$

Step 1.5.2

Multiply $2$ by $5$ .

$10 - 5 y = 10$

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

Subtract $10$ from both sides of the equation.

$- 5 y = 10 - 10$

Step 1.5.3.2

Subtract $10$ from $10$ .

$- 5 y = 0$

Step 1.5.4

Divide each term in $- 5 y = 0$ by $- 5$ and simplify.

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Step 1.5.4.1

Divide each term in $- 5 y = 0$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{0}{- 5}$

Step 1.5.4.2

Simplify the left side.

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Step 1.5.4.2.1

Cancel the common factor of $- 5$ .

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Step 1.5.4.2.1.1

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{0}{- 5}$

Step 1.5.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{- 5}$

Step 1.5.4.3

Simplify the right side.

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Step 1.5.4.3.1

Divide $0$ by $- 5$ .

$y = 0$

Step 1.6

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

$(5, 0)$

Step 2

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

Independent

Step 3

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

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

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