Algebra Examples

$\frac{3 x + 1}{x - 4} = \frac{6 x + 5}{2 x - 7}$

Step 1

Subtract $\frac{6 x + 5}{2 x - 7}$ from both sides of the equation.

$\frac{3 x + 1}{x - 4} - \frac{6 x + 5}{2 x - 7} = 0$

Step 2

Simplify $\frac{3 x + 1}{x - 4} - \frac{6 x + 5}{2 x - 7}$ .

Tap for more steps...

Step 2.1

To write $\frac{3 x + 1}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{2 x - 7}{2 x - 7}$ .

$\frac{3 x + 1}{x - 4} \cdot \frac{2 x - 7}{2 x - 7} - \frac{6 x + 5}{2 x - 7} = 0$

Step 2.2

To write $- \frac{6 x + 5}{2 x - 7}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{3 x + 1}{x - 4} \cdot \frac{2 x - 7}{2 x - 7} - \frac{6 x + 5}{2 x - 7} \cdot \frac{x - 4}{x - 4} = 0$

Step 2.3

Write each expression with a common denominator of $(x - 4) (2 x - 7)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.3.1

Multiply $\frac{3 x + 1}{x - 4}$ by $\frac{2 x - 7}{2 x - 7}$ .

$\frac{(3 x + 1) (2 x - 7)}{(x - 4) (2 x - 7)} - \frac{6 x + 5}{2 x - 7} \cdot \frac{x - 4}{x - 4} = 0$

Step 2.3.2

Multiply $\frac{6 x + 5}{2 x - 7}$ by $\frac{x - 4}{x - 4}$ .

$\frac{(3 x + 1) (2 x - 7)}{(x - 4) (2 x - 7)} - \frac{(6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.3.3

Reorder the factors of $(x - 4) (2 x - 7)$ .

$\frac{(3 x + 1) (2 x - 7)}{(2 x - 7) (x - 4)} - \frac{(6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{(3 x + 1) (2 x - 7) - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5

Simplify the numerator.

Tap for more steps...

Step 2.5.1

Expand $(3 x + 1) (2 x - 7)$ using the FOIL Method.

Tap for more steps...

Step 2.5.1.1

Apply the distributive property.

$\frac{3 x (2 x - 7) + 1 (2 x - 7) - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.1.2

Apply the distributive property.

$\frac{3 x (2 x) + 3 x \cdot - 7 + 1 (2 x - 7) - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.1.3

Apply the distributive property.

$\frac{3 x (2 x) + 3 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7 - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.2

Simplify and combine like terms.

Tap for more steps...

Step 2.5.2.1

Simplify each term.

Tap for more steps...

Step 2.5.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{3 \cdot 2 x \cdot x + 3 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7 - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.2.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.5.2.1.2.1

Move $x$ .

$\frac{3 \cdot 2 (x \cdot x) + 3 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7 - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.2.1.2.2

Multiply $x$ by $x$ .

$\frac{3 \cdot 2 x^{2} + 3 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7 - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.2.1.3

Multiply $3$ by $2$ .

$\frac{6 x^{2} + 3 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7 - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.2.1.4

Multiply $- 7$ by $3$ .

$\frac{6 x^{2} - 21 x + 1 (2 x) + 1 \cdot - 7 - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.2.1.5

Multiply $2 x$ by $1$ .

$\frac{6 x^{2} - 21 x + 2 x + 1 \cdot - 7 - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.2.1.6

Multiply $- 7$ by $1$ .

$\frac{6 x^{2} - 21 x + 2 x - 7 - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.2.2

Add $- 21 x$ and $2 x$ .

$\frac{6 x^{2} - 19 x - 7 - (6 x + 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.3

Apply the distributive property.

$\frac{6 x^{2} - 19 x - 7 + (- (6 x) - 1 \cdot 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.4

Multiply $6$ by $- 1$ .

$\frac{6 x^{2} - 19 x - 7 + (- 6 x - 1 \cdot 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.5

Multiply $- 1$ by $5$ .

$\frac{6 x^{2} - 19 x - 7 + (- 6 x - 5) (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.6

Expand $(- 6 x - 5) (x - 4)$ using the FOIL Method.

Tap for more steps...

Step 2.5.6.1

Apply the distributive property.

$\frac{6 x^{2} - 19 x - 7 - 6 x (x - 4) - 5 (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.6.2

Apply the distributive property.

$\frac{6 x^{2} - 19 x - 7 - 6 x \cdot x - 6 x \cdot - 4 - 5 (x - 4)}{(2 x - 7) (x - 4)} = 0$

Step 2.5.6.3

Apply the distributive property.

$\frac{6 x^{2} - 19 x - 7 - 6 x \cdot x - 6 x \cdot - 4 - 5 x - 5 \cdot - 4}{(2 x - 7) (x - 4)} = 0$

Step 2.5.7

Simplify and combine like terms.

Tap for more steps...

Step 2.5.7.1

Simplify each term.

Tap for more steps...

Step 2.5.7.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.5.7.1.1.1

Move $x$ .

$\frac{6 x^{2} - 19 x - 7 - 6 (x \cdot x) - 6 x \cdot - 4 - 5 x - 5 \cdot - 4}{(2 x - 7) (x - 4)} = 0$

Step 2.5.7.1.1.2

Multiply $x$ by $x$ .

$\frac{6 x^{2} - 19 x - 7 - 6 x^{2} - 6 x \cdot - 4 - 5 x - 5 \cdot - 4}{(2 x - 7) (x - 4)} = 0$

Step 2.5.7.1.2

Multiply $- 4$ by $- 6$ .

$\frac{6 x^{2} - 19 x - 7 - 6 x^{2} + 24 x - 5 x - 5 \cdot - 4}{(2 x - 7) (x - 4)} = 0$

Step 2.5.7.1.3

Multiply $- 5$ by $- 4$ .

$\frac{6 x^{2} - 19 x - 7 - 6 x^{2} + 24 x - 5 x + 20}{(2 x - 7) (x - 4)} = 0$

Step 2.5.7.2

Subtract $5 x$ from $24 x$ .

$\frac{6 x^{2} - 19 x - 7 - 6 x^{2} + 19 x + 20}{(2 x - 7) (x - 4)} = 0$

Step 2.5.8

Subtract $6 x^{2}$ from $6 x^{2}$ .

$\frac{- 19 x - 7 + 0 + 19 x + 20}{(2 x - 7) (x - 4)} = 0$

Step 2.5.9

Add $- 19 x$ and $0$ .

$\frac{- 19 x - 7 + 19 x + 20}{(2 x - 7) (x - 4)} = 0$

Step 2.5.10

Add $- 19 x$ and $19 x$ .

$\frac{0 - 7 + 20}{(2 x - 7) (x - 4)} = 0$

Step 2.5.11

Subtract $7$ from $0$ .

$\frac{- 7 + 20}{(2 x - 7) (x - 4)} = 0$

Step 2.5.12

Add $- 7$ and $20$ .

$\frac{13}{(2 x - 7) (x - 4)} = 0$

Step 3

Set the numerator equal to zero.

$13 = 0$

Step 4

Since $13 \neq 0$ , there are no solutions.

No solution