Linear Algebra Examples

$2 x + 3 y = 6$ , $4 x + 6 y = 24$

Step 1

Solve for $x$ in $2 x + 3 y = 6$ .

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Step 1.1

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

$2 x = 6 - 3 y$

$4 x + 6 y = 24$

Step 1.2

Divide each term in $2 x = 6 - 3 y$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 6 - 3 y$ by $2$ .

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

$4 x + 6 y = 24$

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.

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

$4 x + 6 y = 24$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$4 x + 6 y = 24$

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

$4 x + 6 y = 24$

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

$4 x + 6 y = 24$

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

Divide $6$ by $2$ .

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

$4 x + 6 y = 24$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$4 x + 6 y = 24$

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

$4 x + 6 y = 24$

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

$4 x + 6 y = 24$

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

$4 x + 6 y = 24$

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

$4 x + 6 y = 24$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $4 x + 6 y = 24$ with $3 - \frac{3 y}{2}$ .

$4 (3 - \frac{3 y}{2}) + 6 y = 24$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$4 \cdot 3 + 4 (- \frac{3 y}{2}) + 6 y = 24$

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

Step 2.2.1.1.2

Multiply $4$ by $3$ .

$12 + 4 (- \frac{3 y}{2}) + 6 y = 24$

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

Step 2.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.3.1

Move the leading negative in $- \frac{3 y}{2}$ into the numerator.

$12 + 4 (\frac{- 3 y}{2}) + 6 y = 24$

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

Step 2.2.1.1.3.2

Factor $2$ out of $4$ .

$12 + 2 (2) (\frac{- 3 y}{2}) + 6 y = 24$

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

Step 2.2.1.1.3.3

Cancel the common factor.

$12 + 2 \cdot (2 (\frac{- 3 y}{2})) + 6 y = 24$

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

Step 2.2.1.1.3.4

Rewrite the expression.

$12 + 2 (- 3 y) + 6 y = 24$

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

$12 + 2 (- 3 y) + 6 y = 24$

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

Step 2.2.1.1.4

Multiply $- 3$ by $2$ .

$12 - 6 y + 6 y = 24$

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

$12 - 6 y + 6 y = 24$

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

Step 2.2.1.2

Combine the opposite terms in $12 - 6 y + 6 y$ .

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Step 2.2.1.2.1

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

$12 + 0 = 24$

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

Step 2.2.1.2.2

Add $12$ and $0$ .

$12 = 24$

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

$12 = 24$

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

$12 = 24$

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

$12 = 24$

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

$12 = 24$

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

Step 3

Since $12 = 24$ is not true, there is no solution.

No solution

Step 4