Solve (2)/(a-3)+(3)/(a+2)-(4a-7)/((a)^(2)-a-6) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{ 2 }{ a-3 } + \frac{ 3 }{ a+2 } - \frac{ 4a-7 }{ { a }^{ 2 } -a-6 }$$

Evaluate

$\frac{1}{a-3}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $a-3$ and $a+2$ is $\left(a-3\right)\left(a+2\right)$. Multiply $\frac{2}{a-3}$ times $\frac{a+2}{a+2}$. Multiply $\frac{3}{a+2}$ times $\frac{a-3}{a-3}$.

$$\frac{2\left(a+2\right)}{\left(a-3\right)\left(a+2\right)}+\frac{3\left(a-3\right)}{\left(a-3\right)\left(a+2\right)}-\frac{4a-7}{a^{2}-a-6}$$

Since $\frac{2\left(a+2\right)}{\left(a-3\right)\left(a+2\right)}$ and $\frac{3\left(a-3\right)}{\left(a-3\right)\left(a+2\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{2\left(a+2\right)+3\left(a-3\right)}{\left(a-3\right)\left(a+2\right)}-\frac{4a-7}{a^{2}-a-6}$$

Do the multiplications in $2\left(a+2\right)+3\left(a-3\right)$.

$$\frac{2a+4+3a-9}{\left(a-3\right)\left(a+2\right)}-\frac{4a-7}{a^{2}-a-6}$$

Combine like terms in $2a+4+3a-9$.

$$\frac{5a-5}{\left(a-3\right)\left(a+2\right)}-\frac{4a-7}{a^{2}-a-6}$$

Factor $a^{2}-a-6$.

$$\frac{5a-5}{\left(a-3\right)\left(a+2\right)}-\frac{4a-7}{\left(a-3\right)\left(a+2\right)}$$

Since $\frac{5a-5}{\left(a-3\right)\left(a+2\right)}$ and $\frac{4a-7}{\left(a-3\right)\left(a+2\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{5a-5-\left(4a-7\right)}{\left(a-3\right)\left(a+2\right)}$$

Do the multiplications in $5a-5-\left(4a-7\right)$.

$$\frac{5a-5-4a+7}{\left(a-3\right)\left(a+2\right)}$$

Combine like terms in $5a-5-4a+7$.

$$\frac{a+2}{\left(a-3\right)\left(a+2\right)}$$

Cancel out $a+2$ in both numerator and denominator.

$$\frac{1}{a-3}$$

Expand

$\frac{1}{a-3}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $a-3$ and $a+2$ is $\left(a-3\right)\left(a+2\right)$. Multiply $\frac{2}{a-3}$ times $\frac{a+2}{a+2}$. Multiply $\frac{3}{a+2}$ times $\frac{a-3}{a-3}$.

$$\frac{2\left(a+2\right)}{\left(a-3\right)\left(a+2\right)}+\frac{3\left(a-3\right)}{\left(a-3\right)\left(a+2\right)}-\frac{4a-7}{a^{2}-a-6}$$

Since $\frac{2\left(a+2\right)}{\left(a-3\right)\left(a+2\right)}$ and $\frac{3\left(a-3\right)}{\left(a-3\right)\left(a+2\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{2\left(a+2\right)+3\left(a-3\right)}{\left(a-3\right)\left(a+2\right)}-\frac{4a-7}{a^{2}-a-6}$$

Do the multiplications in $2\left(a+2\right)+3\left(a-3\right)$.

$$\frac{2a+4+3a-9}{\left(a-3\right)\left(a+2\right)}-\frac{4a-7}{a^{2}-a-6}$$

Combine like terms in $2a+4+3a-9$.

$$\frac{5a-5}{\left(a-3\right)\left(a+2\right)}-\frac{4a-7}{a^{2}-a-6}$$

Factor $a^{2}-a-6$.

$$\frac{5a-5}{\left(a-3\right)\left(a+2\right)}-\frac{4a-7}{\left(a-3\right)\left(a+2\right)}$$

Since $\frac{5a-5}{\left(a-3\right)\left(a+2\right)}$ and $\frac{4a-7}{\left(a-3\right)\left(a+2\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{5a-5-\left(4a-7\right)}{\left(a-3\right)\left(a+2\right)}$$

Do the multiplications in $5a-5-\left(4a-7\right)$.

$$\frac{5a-5-4a+7}{\left(a-3\right)\left(a+2\right)}$$

Combine like terms in $5a-5-4a+7$.

$$\frac{a+2}{\left(a-3\right)\left(a+2\right)}$$

Cancel out $a+2$ in both numerator and denominator.

$$\frac{1}{a-3}$$