Pre-Algebra Examples

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

Step 1

To write $\frac{3 x - 5}{4 x + 3}$ as a fraction with a common denominator, multiply by $\frac{7 x - 2}{7 x - 2}$ .

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(4 x + 3) (7 x - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{3 x - 5}{4 x + 3}$ by $\frac{7 x - 2}{7 x - 2}$ .

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(7 x - 2) (4 x + 3)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Expand $(3 x - 5) (7 x - 2)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.2

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

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Step 5.2.1.2.1

Move $x$ .

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

Step 5.2.1.2.2

Multiply $x$ by $x$ .

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

Step 5.2.1.3

Multiply $3$ by $7$ .

$\frac{21 x^{2} + 3 x \cdot - 2 - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)}$

Step 5.2.1.4

Multiply $- 2$ by $3$ .

$\frac{21 x^{2} - 6 x - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)}$

Step 5.2.1.5

Multiply $7$ by $- 5$ .

$\frac{21 x^{2} - 6 x - 35 x - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)}$

Step 5.2.1.6

Multiply $- 5$ by $- 2$ .

$\frac{21 x^{2} - 6 x - 35 x + 10 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)}$

Step 5.2.2

Subtract $35 x$ from $- 6 x$ .

$\frac{21 x^{2} - 41 x + 10 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)}$

Step 5.3

Expand $(- 2 x + 1) (4 x + 3)$ using the FOIL Method.

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Step 5.3.1

Apply the distributive property.

$\frac{21 x^{2} - 41 x + 10 - 2 x (4 x + 3) + 1 (4 x + 3)}{(4 x + 3) (7 x - 2)}$

Step 5.3.2

Apply the distributive property.

$\frac{21 x^{2} - 41 x + 10 - 2 x (4 x) - 2 x \cdot 3 + 1 (4 x + 3)}{(4 x + 3) (7 x - 2)}$

Step 5.3.3

Apply the distributive property.

$\frac{21 x^{2} - 41 x + 10 - 2 x (4 x) - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)}$

Step 5.4

Simplify and combine like terms.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{21 x^{2} - 41 x + 10 - 2 \cdot 4 x \cdot x - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)}$

Step 5.4.1.2

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

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Step 5.4.1.2.1

Move $x$ .

$\frac{21 x^{2} - 41 x + 10 - 2 \cdot 4 (x \cdot x) - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)}$

Step 5.4.1.2.2

Multiply $x$ by $x$ .

$\frac{21 x^{2} - 41 x + 10 - 2 \cdot 4 x^{2} - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)}$

Step 5.4.1.3

Multiply $- 2$ by $4$ .

$\frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)}$

Step 5.4.1.4

Multiply $3$ by $- 2$ .

$\frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 6 x + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)}$

Step 5.4.1.5

Multiply $4 x$ by $1$ .

$\frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 6 x + 4 x + 1 \cdot 3}{(4 x + 3) (7 x - 2)}$

Step 5.4.1.6

Multiply $3$ by $1$ .

$\frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 6 x + 4 x + 3}{(4 x + 3) (7 x - 2)}$

Step 5.4.2

Add $- 6 x$ and $4 x$ .

$\frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 2 x + 3}{(4 x + 3) (7 x - 2)}$

Step 5.5

Subtract $8 x^{2}$ from $21 x^{2}$ .

$\frac{13 x^{2} - 41 x + 10 - 2 x + 3}{(4 x + 3) (7 x - 2)}$

Step 5.6

Subtract $2 x$ from $- 41 x$ .

$\frac{13 x^{2} - 43 x + 10 + 3}{(4 x + 3) (7 x - 2)}$

Step 5.7

Add $10$ and $3$ .

$\frac{13 x^{2} - 43 x + 13}{(4 x + 3) (7 x - 2)}$