Solve (2(x-a)/c-a+1)-(2b-3a+c)(x-a)/(c-a)(b-a)

$$( \frac { 2 ( x - a ) } { c - a } + 1 ) - \frac { ( 2 b - 3 a + c ) ( x - a ) } { ( c - a ) ( b - a ) }$$

Evaluate

$$\frac{b-x}{b-a}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{c-a}{c-a}.

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

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

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

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

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

Combine like terms in 2x-2a+c-a.

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

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

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

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

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

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

$$\frac{-2xa+2xb+3a^{2}-3ab-ca+cb-2bx+2ba+3ax-3a^{2}-cx+ca}{\left(-a+b\right)\left(-a+c\right)}$$

Combine like terms in -2xa+2xb+3a^{2}-3ab-ca+cb-2bx+2ba+3ax-3a^{2}-cx+ca.

$$\frac{xa-ab+cb-cx}{\left(-a+b\right)\left(-a+c\right)}$$

Factor the expressions that are not already factored in \frac{xa-ab+cb-cx}{\left(-a+b\right)\left(-a+c\right)}.

$$\frac{\left(a-c\right)\left(x-b\right)}{\left(b-a\right)\left(c-a\right)}$$

Extract the negative sign in a-c.

$$\frac{-\left(c-a\right)\left(x-b\right)}{\left(b-a\right)\left(c-a\right)}$$

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

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

To find the opposite of x-b, find the opposite of each term.

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

The opposite of -b is b.

$$\frac{-x+b}{-a+b}$$

Expand

$$\frac{b-x}{b-a}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{c-a}{c-a}.

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

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

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

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

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

Combine like terms in 2x-2a+c-a.

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

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

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

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

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

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

$$\frac{-2xa+2xb+3a^{2}-3ab-ca+cb-2bx+2ba+3ax-3a^{2}-cx+ca}{\left(-a+b\right)\left(-a+c\right)}$$

Combine like terms in -2xa+2xb+3a^{2}-3ab-ca+cb-2bx+2ba+3ax-3a^{2}-cx+ca.

$$\frac{xa-ab+cb-cx}{\left(-a+b\right)\left(-a+c\right)}$$

Factor the expressions that are not already factored in \frac{xa-ab+cb-cx}{\left(-a+b\right)\left(-a+c\right)}.

$$\frac{\left(a-c\right)\left(x-b\right)}{\left(b-a\right)\left(c-a\right)}$$

Extract the negative sign in a-c.

$$\frac{-\left(c-a\right)\left(x-b\right)}{\left(b-a\right)\left(c-a\right)}$$

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

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

To find the opposite of x-b, find the opposite of each term.

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

The opposite of -b is b.

$$\frac{-x+b}{-a+b}$$