Linear Algebra Examples

$- 2 x + 3 y = 25$ , $- 4 x + 6 y = 50$

Step 1

Solve for $x$ in $- 2 x + 3 y = 25$ .

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Step 1.1

Subtract $3 y$ from both sides of the equation.

$- 2 x = 25 - 3 y$

$- 4 x + 6 y = 50$

Step 1.2

Divide each term in $- 2 x = 25 - 3 y$ by $- 2$ and simplify.

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Step 1.2.1

Divide each term in $- 2 x = 25 - 3 y$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{25}{- 2} + \frac{- 3 y}{- 2}$

$- 4 x + 6 y = 50$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{25}{- 2} + \frac{- 3 y}{- 2}$

$- 4 x + 6 y = 50$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{25}{- 2} + \frac{- 3 y}{- 2}$

$- 4 x + 6 y = 50$

$x = \frac{25}{- 2} + \frac{- 3 y}{- 2}$

$- 4 x + 6 y = 50$

$x = \frac{25}{- 2} + \frac{- 3 y}{- 2}$

$- 4 x + 6 y = 50$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{25}{2} + \frac{- 3 y}{- 2}$

$- 4 x + 6 y = 50$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$x = - \frac{25}{2} + \frac{3 y}{2}$

$- 4 x + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

$- 4 x + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

$- 4 x + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

$- 4 x + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

$- 4 x + 6 y = 50$

Step 2

Replace all occurrences of $x$ with $- \frac{25}{2} + \frac{3 y}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 4 x + 6 y = 50$ with $- \frac{25}{2} + \frac{3 y}{2}$ .

$- 4 (- \frac{25}{2} + \frac{3 y}{2}) + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 4 (- \frac{25}{2} + \frac{3 y}{2}) + 6 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$- 4 (- \frac{25}{2}) - 4 \frac{3 y}{2} + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{25}{2}$ into the numerator.

$- 4 (\frac{- 25}{2}) - 4 \frac{3 y}{2} + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2.1.1.2.2

Factor $2$ out of $- 4$ .

$2 (- 2) (\frac{- 25}{2}) - 4 \frac{3 y}{2} + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2.1.1.2.3

Cancel the common factor.

$2 \cdot (- 2 (\frac{- 25}{2})) - 4 \frac{3 y}{2} + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2.1.1.2.4

Rewrite the expression.

$- 2 \cdot - 25 - 4 \frac{3 y}{2} + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

$- 2 \cdot - 25 - 4 \frac{3 y}{2} + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2.1.1.3

Multiply $- 2$ by $- 25$ .

$50 - 4 \frac{3 y}{2} + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1

Factor $2$ out of $- 4$ .

$50 + 2 (- 2) (\frac{3 y}{2}) + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2.1.1.4.2

Cancel the common factor.

$50 + 2 \cdot (- 2 \frac{3 y}{2}) + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2.1.1.4.3

Rewrite the expression.

$50 - 2 (3 y) + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

$50 - 2 (3 y) + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2.1.1.5

Multiply $3$ by $- 2$ .

$50 - 6 y + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

$50 - 6 y + 6 y = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2.1.2

Combine the opposite terms in $50 - 6 y + 6 y$ .

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Step 2.2.1.2.1

Add $- 6 y$ and $6 y$ .

$50 + 0 = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 2.2.1.2.2

Add $50$ and $0$ .

$50 = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

$50 = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

$50 = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

$50 = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

$50 = 50$

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 3

Remove any equations from the system that are always true.

$x = - \frac{25}{2} + \frac{3 y}{2}$

Step 4