Solve {l}{x-3/5+x-4/7=6-2x-1/35}{y=x+5}{z=2x+6}{a=y}{b=z}{text{Solvefor}c,dtext{where}}{c=a}{d=b}

$$\left. \begin{array} { l } { \frac{x - 3}{5} + \frac{x - 4}{7} = 6 - \frac{2 x - 1}{35} }\\ { y = x + 5 }\\ { z = 2 x + 6 }\\ { a = y }\\ { b = z }\\ { \text{Solve for } c,d \text{ where} } \\ { c = a }\\ { d = b } \end{array} \right.$$

Solve for x, y, z, a, b, c, d

$$c=23$$

$$d=42$$

Solution Steps

Consider the first equation. Multiply both sides of the equation by 35, the least common multiple of 5,7,35.

$$7\left(x-3\right)+5\left(x-4\right)=210-\left(2x-1\right)$$

Use the distributive property to multiply 7 by x-3.

$$7x-21+5\left(x-4\right)=210-\left(2x-1\right)$$

Use the distributive property to multiply 5 by x-4.

$$7x-21+5x-20=210-\left(2x-1\right)$$

Combine 7x and 5x to get 12x.

$$12x-21-20=210-\left(2x-1\right)$$

Subtract 20 from -21 to get -41.

$$12x-41=210-\left(2x-1\right)$$

To find the opposite of 2x-1, find the opposite of each term.

$$12x-41=210-2x+1$$

Add 210 and 1 to get 211.

$$12x-41=211-2x$$

Add 2x to both sides.

$$12x-41+2x=211$$

Combine 12x and 2x to get 14x.

$$14x-41=211$$

Add 41 to both sides.

$$14x=211+41$$

Add 211 and 41 to get 252.

$$14x=252$$

Divide both sides by 14.

$$x=\frac{252}{14}$$

Divide 252 by 14 to get 18.

$$x=18$$

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

$$y=18+5$$

Add 18 and 5 to get 23.

$$y=23$$

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

$$z=2\times 18+6$$

Multiply 2 and 18 to get 36.

$$z=36+6$$

Add 36 and 6 to get 42.

$$z=42$$

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

$$a=23$$

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

$$b=42$$

Consider the equation (6). Insert the known values of variables into the equation.

$$c=23$$

Consider the equation (7). Insert the known values of variables into the equation.

$$d=42$$

The system is now solved.

$$x=18 y=23 z=42 a=23 b=42 c=23 d=42$$