Solve {l}{{(x-7)}{(x+3)}-2{(x+1)}{(x-4)}=-{(x-1)}{(x+3)}}{y=x}{z=y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { {(x - 7)} {(x + 3)} - 2 {(x + 1)} {(x - 4)} = -{(x - 1)} {(x + 3)} }\\ { y = x }\\ { z = y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z, a

$$a=4$$

Solution Steps

Consider the first equation. Use the distributive property to multiply x-7 by x+3 and combine like terms.

$$x^{2}-4x-21-2\left(x+1\right)\left(x-4\right)=\left(-\left(x-1\right)\right)\left(x+3\right)$$

To find the opposite of x-1, find the opposite of each term.

$$x^{2}-4x-21-2\left(x+1\right)\left(x-4\right)=\left(-x+1\right)\left(x+3\right)$$

Use the distributive property to multiply -x+1 by x+3 and combine like terms.

$$x^{2}-4x-21-2\left(x+1\right)\left(x-4\right)=-x^{2}-2x+3$$

Add x^{2} to both sides.

$$x^{2}-4x-21-2\left(x+1\right)\left(x-4\right)+x^{2}=-2x+3$$

Add 2x to both sides.

$$x^{2}-4x-21-2\left(x+1\right)\left(x-4\right)+x^{2}+2x=3$$

Use the distributive property to multiply -2 by x+1.

$$x^{2}-4x-21+\left(-2x-2\right)\left(x-4\right)+x^{2}+2x=3$$

Use the distributive property to multiply -2x-2 by x-4 and combine like terms.

$$x^{2}-4x-21-2x^{2}+6x+8+x^{2}+2x=3$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}-4x-21+6x+8+x^{2}+2x=3$$

Combine -4x and 6x to get 2x.

$$-x^{2}+2x-21+8+x^{2}+2x=3$$

Add -21 and 8 to get -13.

$$-x^{2}+2x-13+x^{2}+2x=3$$

Combine -x^{2} and x^{2} to get 0.

$$2x-13+2x=3$$

Combine 2x and 2x to get 4x.

$$4x-13=3$$

Add 13 to both sides.

$$4x=3+13$$

Add 3 and 13 to get 16.

$$4x=16$$

Divide both sides by 4.

$$x=\frac{16}{4}$$

Divide 16 by 4 to get 4.

$$x=4$$

Consider the second equation. Insert the known values of variables into the equation.

$$y=4$$

Consider the third equation. Insert the known values of variables into the equation.

$$z=4$$

Consider the fourth equation. Insert the known values of variables into the equation.

$$a=4$$

The system is now solved.

$$x=4 y=4 z=4 a=4$$