Linear Algebra Examples

$4 x - 10 y = 18$ , $- 2 x + 5 y = - 9$

Step 1

Solve for $x$ in $4 x - 10 y = 18$ .

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Step 1.1

Add $10 y$ to both sides of the equation.

$4 x = 18 + 10 y$

$- 2 x + 5 y = - 9$

Step 1.2

Divide each term in $4 x = 18 + 10 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = 18 + 10 y$ by $4$ .

$\frac{4 x}{4} = \frac{18}{4} + \frac{10 y}{4}$

$- 2 x + 5 y = - 9$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{18}{4} + \frac{10 y}{4}$

$- 2 x + 5 y = - 9$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{4} + \frac{10 y}{4}$

$- 2 x + 5 y = - 9$

$x = \frac{18}{4} + \frac{10 y}{4}$

$- 2 x + 5 y = - 9$

$x = \frac{18}{4} + \frac{10 y}{4}$

$- 2 x + 5 y = - 9$

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

Cancel the common factor of $18$ and $4$ .

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Step 1.2.3.1.1.1

Factor $2$ out of $18$ .

$x = \frac{2 (9)}{4} + \frac{10 y}{4}$

$- 2 x + 5 y = - 9$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 9}{2 \cdot 2} + \frac{10 y}{4}$

$- 2 x + 5 y = - 9$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 9}{2 \cdot 2} + \frac{10 y}{4}$

$- 2 x + 5 y = - 9$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{9}{2} + \frac{10 y}{4}$

$- 2 x + 5 y = - 9$

$x = \frac{9}{2} + \frac{10 y}{4}$

$- 2 x + 5 y = - 9$

$x = \frac{9}{2} + \frac{10 y}{4}$

$- 2 x + 5 y = - 9$

Step 1.2.3.1.2

Cancel the common factor of $10$ and $4$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $10 y$ .

$x = \frac{9}{2} + \frac{2 (5 y)}{4}$

$- 2 x + 5 y = - 9$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $4$ .

$x = \frac{9}{2} + \frac{2 (5 y)}{2 (2)}$

$- 2 x + 5 y = - 9$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = \frac{9}{2} + \frac{2 (5 y)}{2 \cdot 2}$

$- 2 x + 5 y = - 9$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 2 x + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 2 x + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 2 x + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 2 x + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 2 x + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 2 x + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 2 x + 5 y = - 9$

Step 2

Replace all occurrences of $x$ with $\frac{9}{2} + \frac{5 y}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 2 x + 5 y = - 9$ with $\frac{9}{2} + \frac{5 y}{2}$ .

$- 2 (\frac{9}{2} + \frac{5 y}{2}) + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 2 (\frac{9}{2} + \frac{5 y}{2}) + 5 y$ .

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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.

$- 2 (\frac{9}{2}) - 2 \frac{5 y}{2} + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{9}{2}) - 2 \frac{5 y}{2} + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 2.2.1.1.2.2

Cancel the common factor.

$2 \cdot (- 1 (\frac{9}{2})) - 2 \frac{5 y}{2} + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 2.2.1.1.2.3

Rewrite the expression.

$- 1 \cdot 9 - 2 \frac{5 y}{2} + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 1 \cdot 9 - 2 \frac{5 y}{2} + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 2.2.1.1.3

Multiply $- 1$ by $9$ .

$- 9 - 2 \frac{5 y}{2} + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

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 $- 2$ .

$- 9 + 2 (- 1) (\frac{5 y}{2}) + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 2.2.1.1.4.2

Cancel the common factor.

$- 9 + 2 \cdot (- 1 \frac{5 y}{2}) + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 2.2.1.1.4.3

Rewrite the expression.

$- 9 - 1 (5 y) + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 9 - 1 (5 y) + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 2.2.1.1.5

Multiply $5$ by $- 1$ .

$- 9 - 5 y + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 9 - 5 y + 5 y = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 2.2.1.2

Combine the opposite terms in $- 9 - 5 y + 5 y$ .

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Step 2.2.1.2.1

Add $- 5 y$ and $5 y$ .

$- 9 + 0 = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 2.2.1.2.2

Add $- 9$ and $0$ .

$- 9 = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 9 = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 9 = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 9 = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

$- 9 = - 9$

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 3

Remove any equations from the system that are always true.

$x = \frac{9}{2} + \frac{5 y}{2}$

Step 4