Solve (a - 2y)/(a + y) - (y ^ 2 - 5ay)/(a ^ 2 - y ^ 2) | 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

$$\frac{a-2y}{a+y}-\frac{y^{2}-5ay}{a^{2}-y^{2}}$$

Evaluate

$\frac{y+a}{a-y}$

Short Solution Steps

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

$$\frac{a-2y}{a+y}-\frac{y^{2}-5ay}{\left(y+a\right)\left(-y+a\right)}$$

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

$$\frac{\left(a-2y\right)\left(-y+a\right)}{\left(y+a\right)\left(-y+a\right)}-\frac{y^{2}-5ay}{\left(y+a\right)\left(-y+a\right)}$$

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

$$\frac{\left(a-2y\right)\left(-y+a\right)-\left(y^{2}-5ay\right)}{\left(y+a\right)\left(-y+a\right)}$$

Do the multiplications in $\left(a-2y\right)\left(-y+a\right)-\left(y^{2}-5ay\right)$.

$$\frac{-ay+a^{2}+2y^{2}-2ya-y^{2}+5ay}{\left(y+a\right)\left(-y+a\right)}$$

Combine like terms in $-ay+a^{2}+2y^{2}-2ya-y^{2}+5ay$.

$$\frac{y^{2}+2ay+a^{2}}{\left(y+a\right)\left(-y+a\right)}$$

Factor the expressions that are not already factored in $\frac{y^{2}+2ay+a^{2}}{\left(y+a\right)\left(-y+a\right)}$.

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

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

$$\frac{y+a}{-y+a}$$

Expand

$\frac{y+a}{a-y}$

Short Solution Steps

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

$$\frac{a-2y}{a+y}-\frac{y^{2}-5ay}{\left(y+a\right)\left(-y+a\right)}$$

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

$$\frac{\left(a-2y\right)\left(-y+a\right)}{\left(y+a\right)\left(-y+a\right)}-\frac{y^{2}-5ay}{\left(y+a\right)\left(-y+a\right)}$$

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

$$\frac{\left(a-2y\right)\left(-y+a\right)-\left(y^{2}-5ay\right)}{\left(y+a\right)\left(-y+a\right)}$$

Do the multiplications in $\left(a-2y\right)\left(-y+a\right)-\left(y^{2}-5ay\right)$.

$$\frac{-ay+a^{2}+2y^{2}-2ya-y^{2}+5ay}{\left(y+a\right)\left(-y+a\right)}$$

Combine like terms in $-ay+a^{2}+2y^{2}-2ya-y^{2}+5ay$.

$$\frac{y^{2}+2ay+a^{2}}{\left(y+a\right)\left(-y+a\right)}$$

Factor the expressions that are not already factored in $\frac{y^{2}+2ay+a^{2}}{\left(y+a\right)\left(-y+a\right)}$.

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

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

$$\frac{y+a}{-y+a}$$