Solve (a + 2x)/(3a - 3x) - (3c - a)/(2a - 2c) + (a ^ 2 - cx)/(a ^ 2 - ac + cx - ax) | 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

$$\frac{a+2x}{3a-3x}-\frac{3c-a}{2a-2c}+\frac{a^{2}-cx}{a^{2}-ac+cx-ax}$$

Evaluate

$\frac{x+11a}{6\left(a-x\right)}$

Short Solution Steps

Factor $3a-3x$. Factor $2a-2c$.

$$\frac{a+2x}{3\left(-x+a\right)}-\frac{3c-a}{2\left(a-c\right)}+\frac{a^{2}-cx}{a^{2}-ac+cx-ax}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $3\left(-x+a\right)$ and $2\left(a-c\right)$ is $6\left(a-c\right)\left(-x+a\right)$. Multiply $\frac{a+2x}{3\left(-x+a\right)}$ times $\frac{2\left(a-c\right)}{2\left(a-c\right)}$. Multiply $\frac{3c-a}{2\left(a-c\right)}$ times $\frac{3\left(-x+a\right)}{3\left(-x+a\right)}$.

$$\frac{\left(a+2x\right)\times 2\left(a-c\right)}{6\left(a-c\right)\left(-x+a\right)}-\frac{\left(3c-a\right)\times 3\left(-x+a\right)}{6\left(a-c\right)\left(-x+a\right)}+\frac{a^{2}-cx}{a^{2}-ac+cx-ax}$$

Since $\frac{\left(a+2x\right)\times 2\left(a-c\right)}{6\left(a-c\right)\left(-x+a\right)}$ and $\frac{\left(3c-a\right)\times 3\left(-x+a\right)}{6\left(a-c\right)\left(-x+a\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a+2x\right)\times 2\left(a-c\right)-\left(3c-a\right)\times 3\left(-x+a\right)}{6\left(a-c\right)\left(-x+a\right)}+\frac{a^{2}-cx}{a^{2}-ac+cx-ax}$$

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

$$\frac{2a^{2}-2ac+4xa-4xc+9cx-9ca-3ax+3a^{2}}{6\left(a-c\right)\left(-x+a\right)}+\frac{a^{2}-cx}{a^{2}-ac+cx-ax}$$

Combine like terms in $2a^{2}-2ac+4xa-4xc+9cx-9ca-3ax+3a^{2}$.

$$\frac{5a^{2}-11ac+xa+5xc}{6\left(a-c\right)\left(-x+a\right)}+\frac{a^{2}-cx}{a^{2}-ac+cx-ax}$$

Factor $a^{2}-ac+cx-ax$.

$$\frac{5a^{2}-11ac+xa+5xc}{6\left(a-c\right)\left(-x+a\right)}+\frac{a^{2}-cx}{\left(a-c\right)\left(-x+a\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $6\left(a-c\right)\left(-x+a\right)$ and $\left(a-c\right)\left(-x+a\right)$ is $6\left(a-c\right)\left(-x+a\right)$. Multiply $\frac{a^{2}-cx}{\left(a-c\right)\left(-x+a\right)}$ times $\frac{6}{6}$.

$$\frac{5a^{2}-11ac+xa+5xc}{6\left(a-c\right)\left(-x+a\right)}+\frac{6\left(a^{2}-cx\right)}{6\left(a-c\right)\left(-x+a\right)}$$

Since $\frac{5a^{2}-11ac+xa+5xc}{6\left(a-c\right)\left(-x+a\right)}$ and $\frac{6\left(a^{2}-cx\right)}{6\left(a-c\right)\left(-x+a\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{5a^{2}-11ac+xa+5xc+6\left(a^{2}-cx\right)}{6\left(a-c\right)\left(-x+a\right)}$$

Do the multiplications in $5a^{2}-11ac+xa+5xc+6\left(a^{2}-cx\right)$.

$$\frac{5a^{2}-11ac+xa+5xc+6a^{2}-6cx}{6\left(a-c\right)\left(-x+a\right)}$$

Combine like terms in $5a^{2}-11ac+xa+5xc+6a^{2}-6cx$.

$$\frac{11a^{2}-11ac+xa-xc}{6\left(a-c\right)\left(-x+a\right)}$$

Factor the expressions that are not already factored in $\frac{11a^{2}-11ac+xa-xc}{6\left(a-c\right)\left(-x+a\right)}$.

$$\frac{\left(a-c\right)\left(x+11a\right)}{6\left(a-c\right)\left(-x+a\right)}$$

Cancel out $a-c$ in both numerator and denominator.

$$\frac{x+11a}{6\left(-x+a\right)}$$

Expand $6\left(-x+a\right)$.

$$\frac{x+11a}{-6x+6a}$$

Expand

$-\frac{x+11a}{6\left(x-a\right)}$

Short Solution Steps

Factor $3a-3x$. Factor $2a-2c$.

$$\frac{a+2x}{3\left(-x+a\right)}-\frac{3c-a}{2\left(a-c\right)}+\frac{a^{2}-cx}{a^{2}-ac+cx-ax}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $3\left(-x+a\right)$ and $2\left(a-c\right)$ is $6\left(a-c\right)\left(-x+a\right)$. Multiply $\frac{a+2x}{3\left(-x+a\right)}$ times $\frac{2\left(a-c\right)}{2\left(a-c\right)}$. Multiply $\frac{3c-a}{2\left(a-c\right)}$ times $\frac{3\left(-x+a\right)}{3\left(-x+a\right)}$.

$$\frac{\left(a+2x\right)\times 2\left(a-c\right)}{6\left(a-c\right)\left(-x+a\right)}-\frac{\left(3c-a\right)\times 3\left(-x+a\right)}{6\left(a-c\right)\left(-x+a\right)}+\frac{a^{2}-cx}{a^{2}-ac+cx-ax}$$

Since $\frac{\left(a+2x\right)\times 2\left(a-c\right)}{6\left(a-c\right)\left(-x+a\right)}$ and $\frac{\left(3c-a\right)\times 3\left(-x+a\right)}{6\left(a-c\right)\left(-x+a\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a+2x\right)\times 2\left(a-c\right)-\left(3c-a\right)\times 3\left(-x+a\right)}{6\left(a-c\right)\left(-x+a\right)}+\frac{a^{2}-cx}{a^{2}-ac+cx-ax}$$

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

$$\frac{2a^{2}-2ac+4xa-4xc+9cx-9ca-3ax+3a^{2}}{6\left(a-c\right)\left(-x+a\right)}+\frac{a^{2}-cx}{a^{2}-ac+cx-ax}$$

Combine like terms in $2a^{2}-2ac+4xa-4xc+9cx-9ca-3ax+3a^{2}$.

$$\frac{5a^{2}-11ac+xa+5xc}{6\left(a-c\right)\left(-x+a\right)}+\frac{a^{2}-cx}{a^{2}-ac+cx-ax}$$

Factor $a^{2}-ac+cx-ax$.

$$\frac{5a^{2}-11ac+xa+5xc}{6\left(a-c\right)\left(-x+a\right)}+\frac{a^{2}-cx}{\left(a-c\right)\left(-x+a\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $6\left(a-c\right)\left(-x+a\right)$ and $\left(a-c\right)\left(-x+a\right)$ is $6\left(a-c\right)\left(-x+a\right)$. Multiply $\frac{a^{2}-cx}{\left(a-c\right)\left(-x+a\right)}$ times $\frac{6}{6}$.

$$\frac{5a^{2}-11ac+xa+5xc}{6\left(a-c\right)\left(-x+a\right)}+\frac{6\left(a^{2}-cx\right)}{6\left(a-c\right)\left(-x+a\right)}$$

Since $\frac{5a^{2}-11ac+xa+5xc}{6\left(a-c\right)\left(-x+a\right)}$ and $\frac{6\left(a^{2}-cx\right)}{6\left(a-c\right)\left(-x+a\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{5a^{2}-11ac+xa+5xc+6\left(a^{2}-cx\right)}{6\left(a-c\right)\left(-x+a\right)}$$

Do the multiplications in $5a^{2}-11ac+xa+5xc+6\left(a^{2}-cx\right)$.

$$\frac{5a^{2}-11ac+xa+5xc+6a^{2}-6cx}{6\left(a-c\right)\left(-x+a\right)}$$

Combine like terms in $5a^{2}-11ac+xa+5xc+6a^{2}-6cx$.

$$\frac{11a^{2}-11ac+xa-xc}{6\left(a-c\right)\left(-x+a\right)}$$

Factor the expressions that are not already factored in $\frac{11a^{2}-11ac+xa-xc}{6\left(a-c\right)\left(-x+a\right)}$.

$$\frac{\left(a-c\right)\left(x+11a\right)}{6\left(a-c\right)\left(-x+a\right)}$$

Cancel out $a-c$ in both numerator and denominator.

$$\frac{x+11a}{6\left(-x+a\right)}$$

Expand $6\left(-x+a\right)$.

$$\frac{x+11a}{-6x+6a}$$