Linear Algebra Examples

$x - y = 4$ , $7 x + 8 y = - 107$

Step 1

Add $y$ to both sides of the equation.

$x = 4 + y$

$7 x + 8 y = - 107$

Step 2

Replace all occurrences of $x$ with $4 + y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $7 x + 8 y = - 107$ with $4 + y$ .

$7 (4 + y) + 8 y = - 107$

$x = 4 + y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $7 (4 + y) + 8 y$ .

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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.

$7 \cdot 4 + 7 y + 8 y = - 107$

$x = 4 + y$

Step 2.2.1.1.2

Multiply $7$ by $4$ .

$28 + 7 y + 8 y = - 107$

$x = 4 + y$

$28 + 7 y + 8 y = - 107$

$x = 4 + y$

Step 2.2.1.2

Add $7 y$ and $8 y$ .

$28 + 15 y = - 107$

$x = 4 + y$

$28 + 15 y = - 107$

$x = 4 + y$

$28 + 15 y = - 107$

$x = 4 + y$

$28 + 15 y = - 107$

$x = 4 + y$

Step 3

Solve for $y$ in $28 + 15 y = - 107$ .

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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 $28$ from both sides of the equation.

$15 y = - 107 - 28$

$x = 4 + y$

Step 3.1.2

Subtract $28$ from $- 107$ .

$15 y = - 135$

$x = 4 + y$

$15 y = - 135$

$x = 4 + y$

Step 3.2

Divide each term in $15 y = - 135$ by $15$ and simplify.

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Step 3.2.1

Divide each term in $15 y = - 135$ by $15$ .

$\frac{15 y}{15} = \frac{- 135}{15}$

$x = 4 + y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $15$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{15 y}{15} = \frac{- 135}{15}$

$x = 4 + y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 135}{15}$

$x = 4 + y$

$y = \frac{- 135}{15}$

$x = 4 + y$

$y = \frac{- 135}{15}$

$x = 4 + y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 135$ by $15$ .

$y = - 9$

$x = 4 + y$

$y = - 9$

$x = 4 + y$

$y = - 9$

$x = 4 + y$

$y = - 9$

$x = 4 + y$

Step 4

Replace all occurrences of $y$ with $- 9$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 4 + y$ with $- 9$ .

$x = 4 - 9$

$y = - 9$

Step 4.2

Simplify $x = 4 - 9$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Remove parentheses.

$x = 4 - 9$

$y = - 9$

$x = 4 - 9$

$y = - 9$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Subtract $9$ from $4$ .

$x = - 5$

$y = - 9$

$x = - 5$

$y = - 9$

$x = - 5$

$y = - 9$

$x = - 5$

$y = - 9$

Step 5

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

$(- 5, - 9)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 5, - 9)$

Equation Form:

$x = - 5, y = - 9$

Step 7