Linear Algebra Examples

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

Step 1

Multiply both sides by $5$ .

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

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

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Simplify $\frac{x + 1}{3} \cdot 5$ .

Tap for more steps...

Step 2.2.1.1

Combine $\frac{x + 1}{3}$ and $5$ .

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

Step 2.2.1.2

Move $5$ to the left of $x + 1$ .

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

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Multiply both sides by $3$ .

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

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.1.1

Simplify $(x - 2) \cdot 3$ .

Tap for more steps...

Step 3.2.1.1.1

Apply the distributive property.

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

Step 3.2.1.1.2

Simplify the expression.

Tap for more steps...

Step 3.2.1.1.2.1

Move $3$ to the left of $x$ .

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

Step 3.2.1.1.2.2

Multiply $- 2$ by $3$ .

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

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $\frac{5 (x + 1)}{3} \cdot 3$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.2.1.1.1

Cancel the common factor.

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

Step 3.2.2.1.1.2

Rewrite the expression.

$3 x - 6 = 5 (x + 1)$

Step 3.2.2.1.2

Apply the distributive property.

$3 x - 6 = 5 x + 5 \cdot 1$

Step 3.2.2.1.3

Multiply $5$ by $1$ .

$3 x - 6 = 5 x + 5$

Step 3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.1

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 3.3.1.1

Subtract $5 x$ from both sides of the equation.

$3 x - 6 - 5 x = 5$

Step 3.3.1.2

Subtract $5 x$ from $3 x$ .

$- 2 x - 6 = 5$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

Add $6$ to both sides of the equation.

$- 2 x = 5 + 6$

Step 3.3.2.2

Add $5$ and $6$ .

$- 2 x = 11$

Step 3.3.3

Divide each term in $- 2 x = 11$ by $- 2$ and simplify.

Tap for more steps...

Step 3.3.3.1

Divide each term in $- 2 x = 11$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{11}{- 2}$

Step 3.3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{11}{- 2}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{- 2}$

Step 3.3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.3.1

Move the negative in front of the fraction.

$x = - \frac{11}{2}$

Step 4

The domain is the set of all valid $x$ values.

${x | x = - \frac{11}{2}}$

Step 5