Linear Algebra Examples

$6 x - 5 y = 3$ , $- 2 x + \frac{5}{3} y = 1$

Step 1

Solve for $x$ in $6 x - 5 y = 3$ .

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Step 1.1

Add $5 y$ to both sides of the equation.

$6 x = 3 + 5 y$

$- 2 x + \frac{5}{3} y = 1$

Step 1.2

Divide each term in $6 x = 3 + 5 y$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 x = 3 + 5 y$ by $6$ .

$\frac{6 x}{6} = \frac{3}{6} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{3}{6} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{6} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

$x = \frac{3}{6} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

$x = \frac{3}{6} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $3$ and $6$ .

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Step 1.2.3.1.1

Factor $3$ out of $3$ .

$x = \frac{3 (1)}{6} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $3$ out of $6$ .

$x = \frac{3 \cdot 1}{3 \cdot 2} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

Step 1.2.3.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot 1}{3 \cdot 2} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

Step 1.2.3.1.2.3

Rewrite the expression.

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 2 x + \frac{5}{3} y = 1$

Step 2

Replace all occurrences of $x$ with $\frac{1}{2} + \frac{5 y}{6}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 2 x + \frac{5}{3} y = 1$ with $\frac{1}{2} + \frac{5 y}{6}$ .

$- 2 (\frac{1}{2} + \frac{5 y}{6}) + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 2 (\frac{1}{2} + \frac{5 y}{6}) + \frac{5}{3} y$ .

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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.

$- 2 (\frac{1}{2}) - 2 \frac{5 y}{6} + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

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 $- 2$ .

$2 (- 1) (\frac{1}{2}) - 2 \frac{5 y}{6} + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 2.2.1.1.2.2

Cancel the common factor.

$2 \cdot (- 1 (\frac{1}{2})) - 2 \frac{5 y}{6} + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 2.2.1.1.2.3

Rewrite the expression.

$- 1 - 2 \frac{5 y}{6} + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 1 - 2 \frac{5 y}{6} + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 2.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.3.1

Factor $2$ out of $- 2$ .

$- 1 + 2 (- 1) (\frac{5 y}{6}) + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 2.2.1.1.3.2

Factor $2$ out of $6$ .

$- 1 + 2 \cdot (- 1 \frac{5 y}{2 \cdot 3}) + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 2.2.1.1.3.3

Cancel the common factor.

$- 1 + 2 \cdot (- 1 \frac{5 y}{2 \cdot 3}) + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 2.2.1.1.3.4

Rewrite the expression.

$- 1 - 1 \frac{5 y}{3} + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 1 - 1 \frac{5 y}{3} + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 2.2.1.1.4

Rewrite $- 1 \frac{5 y}{3}$ as $- \frac{5 y}{3}$ .

$- 1 - \frac{5 y}{3} + \frac{5}{3} y = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 2.2.1.1.5

Combine $\frac{5}{3}$ and $y$ .

$- 1 - \frac{5 y}{3} + \frac{5 y}{3} = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 1 - \frac{5 y}{3} + \frac{5 y}{3} = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 2.2.1.2

Combine the opposite terms in $- 1 - \frac{5 y}{3} + \frac{5 y}{3}$ .

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Step 2.2.1.2.1

Add $- \frac{5 y}{3}$ and $\frac{5 y}{3}$ .

$- 1 + 0 = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 2.2.1.2.2

Add $- 1$ and $0$ .

$- 1 = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 1 = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 1 = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 1 = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

$- 1 = 1$

$x = \frac{1}{2} + \frac{5 y}{6}$

Step 3

Since $- 1 = 1$ is not true, there is no solution.

No solution

Step 4