Linear Algebra Examples

$3 x + 7 y + 2 = 0$ , $12 x + 28 y - 6 = 0$

Step 1

Solve for $x$ in $3 x + 7 y + 2 = 0$ .

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Step 1.1

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

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Step 1.1.1

Subtract $7 y$ from both sides of the equation.

$3 x + 2 = - 7 y$

$12 x + 28 y - 6 = 0$

Step 1.1.2

Subtract $2$ from both sides of the equation.

$3 x = - 7 y - 2$

$12 x + 28 y - 6 = 0$

$3 x = - 7 y - 2$

$12 x + 28 y - 6 = 0$

Step 1.2

Divide each term in $3 x = - 7 y - 2$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = - 7 y - 2$ by $3$ .

$\frac{3 x}{3} = \frac{- 7 y}{3} + \frac{- 2}{3}$

$12 x + 28 y - 6 = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 7 y}{3} + \frac{- 2}{3}$

$12 x + 28 y - 6 = 0$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7 y}{3} + \frac{- 2}{3}$

$12 x + 28 y - 6 = 0$

$x = \frac{- 7 y}{3} + \frac{- 2}{3}$

$12 x + 28 y - 6 = 0$

$x = \frac{- 7 y}{3} + \frac{- 2}{3}$

$12 x + 28 y - 6 = 0$

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.

$x = - \frac{7 y}{3} + \frac{- 2}{3}$

$12 x + 28 y - 6 = 0$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{7 y}{3} - \frac{2}{3}$

$12 x + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$12 x + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$12 x + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$12 x + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$12 x + 28 y - 6 = 0$

Step 2

Replace all occurrences of $x$ with $- \frac{7 y}{3} - \frac{2}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $12 x + 28 y - 6 = 0$ with $- \frac{7 y}{3} - \frac{2}{3}$ .

$12 (- \frac{7 y}{3} - \frac{2}{3}) + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $12 (- \frac{7 y}{3} - \frac{2}{3}) + 28 y - 6$ .

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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.

$12 (- \frac{7 y}{3}) + 12 (- \frac{2}{3}) + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{7 y}{3}$ into the numerator.

$12 (\frac{- 7 y}{3}) + 12 (- \frac{2}{3}) + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.1.2.2

Factor $3$ out of $12$ .

$3 (4) (\frac{- 7 y}{3}) + 12 (- \frac{2}{3}) + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.1.2.3

Cancel the common factor.

$3 \cdot (4 (\frac{- 7 y}{3})) + 12 (- \frac{2}{3}) + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.1.2.4

Rewrite the expression.

$4 (- 7 y) + 12 (- \frac{2}{3}) + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$4 (- 7 y) + 12 (- \frac{2}{3}) + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.1.3

Multiply $- 7$ by $4$ .

$- 28 y + 12 (- \frac{2}{3}) + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.1.4

Cancel the common factor of $3$ .

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Step 2.2.1.1.4.1

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

$- 28 y + 12 (\frac{- 2}{3}) + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.1.4.2

Factor $3$ out of $12$ .

$- 28 y + 3 (4) (\frac{- 2}{3}) + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.1.4.3

Cancel the common factor.

$- 28 y + 3 \cdot (4 (\frac{- 2}{3})) + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.1.4.4

Rewrite the expression.

$- 28 y + 4 \cdot - 2 + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$- 28 y + 4 \cdot - 2 + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.1.5

Multiply $4$ by $- 2$ .

$- 28 y - 8 + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$- 28 y - 8 + 28 y - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Combine the opposite terms in $- 28 y - 8 + 28 y - 6$ .

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Step 2.2.1.2.1.1

Add $- 28 y$ and $28 y$ .

$0 - 8 - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.2.1.2

Subtract $8$ from $0$ .

$- 8 - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$- 8 - 6 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 2.2.1.2.2

Subtract $6$ from $- 8$ .

$- 14 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$- 14 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$- 14 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$- 14 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

$- 14 = 0$

$x = - \frac{7 y}{3} - \frac{2}{3}$

Step 3

Since $- 14 = 0$ is not true, there is no solution.

No solution

Step 4