Linear Algebra Examples

$99 x + 10.99 y = 331$ , $x + y = 115$

Step 1

Subtract $y$ from both sides of the equation.

$x = 115 - y$

$99 x + 10.99 y = 331$

Step 2

Replace all occurrences of $x$ with $115 - y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $99 x + 10.99 y = 331$ with $115 - y$ .

$99 (115 - y) + 10.99 y = 331$

$x = 115 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $99 (115 - y) + 10.99 y$ .

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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.

$99 \cdot 115 + 99 (- y) + 10.99 y = 331$

$x = 115 - y$

Step 2.2.1.1.2

Multiply $99$ by $115$ .

$11385 + 99 (- y) + 10.99 y = 331$

$x = 115 - y$

Step 2.2.1.1.3

Multiply $- 1$ by $99$ .

$11385 - 99 y + 10.99 y = 331$

$x = 115 - y$

$11385 - 99 y + 10.99 y = 331$

$x = 115 - y$

Step 2.2.1.2

Add $- 99 y$ and $10.99 y$ .

$11385 - 88.01 y = 331$

$x = 115 - y$

$11385 - 88.01 y = 331$

$x = 115 - y$

$11385 - 88.01 y = 331$

$x = 115 - y$

$11385 - 88.01 y = 331$

$x = 115 - y$

Step 3

Solve for $y$ in $11385 - 88.01 y = 331$ .

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Step 3.1

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

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Step 3.1.1

Subtract $11385$ from both sides of the equation.

$- 88.01 y = 331 - 11385$

$x = 115 - y$

Step 3.1.2

Subtract $11385$ from $331$ .

$- 88.01 y = - 11054$

$x = 115 - y$

$- 88.01 y = - 11054$

$x = 115 - y$

Step 3.2

Divide each term in $- 88.01 y = - 11054$ by $- 88.01$ and simplify.

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Step 3.2.1

Divide each term in $- 88.01 y = - 11054$ by $- 88.01$ .

$\frac{- 88.01 y}{- 88.01} = \frac{- 11054}{- 88.01}$

$x = 115 - y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 88.01$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 88.01 y}{- 88.01} = \frac{- 11054}{- 88.01}$

$x = 115 - y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 11054}{- 88.01}$

$x = 115 - y$

$y = \frac{- 11054}{- 88.01}$

$x = 115 - y$

$y = \frac{- 11054}{- 88.01}$

$x = 115 - y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 11054$ by $- 88.01$ .

$y = 125.5993637$

$x = 115 - y$

$y = 125.5993637$

$x = 115 - y$

$y = 125.5993637$

$x = 115 - y$

$y = 125.5993637$

$x = 115 - y$

Step 4

Replace all occurrences of $y$ with $125.5993637$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 115 - y$ with $125.5993637$ .

$x = 115 - (125.5993637)$

$y = 125.5993637$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $115 - (125.5993637)$ .

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Step 4.2.1.1

Multiply $- 1$ by $125.5993637$ .

$x = 115 - 125.5993637$

$y = 125.5993637$

Step 4.2.1.2

Subtract $125.5993637$ from $115$ .

$x = - 10.5993637$

$y = 125.5993637$

$x = - 10.5993637$

$y = 125.5993637$

$x = - 10.5993637$

$y = 125.5993637$

$x = - 10.5993637$

$y = 125.5993637$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 10.5993637, 125.5993637)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 10.5993637, 125.5993637)$

Equation Form:

$x = - 10.5993637, y = 125.5993637$

Step 7