Solve -8=(-3)^4+((c/2-3d/4-9/4)-c+d+1)(-3)^3+(c/2-frac{3d}{4}-frac{9}{4})(-3)^2+c(-3)-d

$$- 8 = ( - 3 ) ^ { 4 } + ( ( \frac { c } { 2 } - \frac { 3 d } { 4 } - \frac { 9 } { 4 } ) - c + d + 1 ) ( - 3 ) ^ { 3 } + ( \frac { c } { 2 } - \frac { 3 d } { 4 } - \frac { 9 } { 4 } ) ( - 3 ) ^ { 2 } + c ( - 3 ) - d$$

Solve for c

$$c=\frac{29d}{30}-\frac{41}{6}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$-32=4\left(-3\right)^{4}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}-c+d+1\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Calculate -3 to the power of 4 and get 81.

$$-32=4\times 81+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}-c+d+1\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Multiply 4 and 81 to get 324.

$$-32=324+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}-c+d+1\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Combine \frac{c}{2} and -c to get -\frac{1}{2}c.

$$-32=324+4\left(-\frac{1}{2}c-\frac{3d}{4}-\frac{9}{4}+d+1\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Add -\frac{9}{4} and 1 to get -\frac{5}{4}.

$$-32=324+4\left(-\frac{1}{2}c-\frac{3d}{4}-\frac{5}{4}+d\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Since -\frac{3d}{4} and \frac{5}{4} have the same denominator, subtract them by subtracting their numerators.

$$-32=324+4\left(-\frac{1}{2}c+\frac{-3d-5}{4}+d\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Calculate -3 to the power of 3 and get -27.

$$-32=324+4\left(-\frac{1}{2}c+\frac{-3d-5}{4}+d\right)\left(-27\right)+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Multiply 4 and -27 to get -108.

$$-32=324-108\left(-\frac{1}{2}c+\frac{-3d-5}{4}+d\right)+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Use the distributive property to multiply -108 by -\frac{1}{2}c+\frac{-3d-5}{4}+d.

$$-32=324+54c-108\times \frac{-3d-5}{4}-108d+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Cancel out 4, the greatest common factor in 108 and 4.

$$-32=324+54c-27\left(-3d-5\right)-108d+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Use the distributive property to multiply -27 by -3d-5.

$$-32=324+54c+81d+135-108d+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Combine 81d and -108d to get -27d.

$$-32=324+54c-27d+135+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Add 324 and 135 to get 459.

$$-32=459+54c-27d+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{c}{2} times \frac{2}{2}.

$$-32=459+54c-27d+4\left(\frac{2c}{4}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Since \frac{2c}{4} and \frac{3d}{4} have the same denominator, subtract them by subtracting their numerators.

$$-32=459+54c-27d+4\left(\frac{2c-3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Since \frac{2c-3d}{4} and \frac{9}{4} have the same denominator, subtract them by subtracting their numerators.

$$-32=459+54c-27d+4\times \frac{2c-3d-9}{4}\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Calculate -3 to the power of 2 and get 9.

$$-32=459+54c-27d+4\times \frac{2c-3d-9}{4}\times 9+4c\left(-3\right)-4d$$

Multiply 4 and 9 to get 36.

$$-32=459+54c-27d+36\times \frac{2c-3d-9}{4}+4c\left(-3\right)-4d$$

Cancel out 4, the greatest common factor in 36 and 4.

$$-32=459+54c-27d+9\left(2c-3d-9\right)+4c\left(-3\right)-4d$$

Use the distributive property to multiply 9 by 2c-3d-9.

$$-32=459+54c-27d+18c-27d-81+4c\left(-3\right)-4d$$

Combine 54c and 18c to get 72c.

$$-32=459+72c-27d-27d-81+4c\left(-3\right)-4d$$

Combine -27d and -27d to get -54d.

$$-32=459+72c-54d-81+4c\left(-3\right)-4d$$

Subtract 81 from 459 to get 378.

$$-32=378+72c-54d+4c\left(-3\right)-4d$$

Multiply 4 and -3 to get -12.

$$-32=378+72c-54d-12c-4d$$

Combine 72c and -12c to get 60c.

$$-32=378+60c-54d-4d$$

Combine -54d and -4d to get -58d.

$$-32=378+60c-58d$$

Swap sides so that all variable terms are on the left hand side.

$$378+60c-58d=-32$$

Subtract 378 from both sides.

$$60c-58d=-32-378$$

Subtract 378 from -32 to get -410.

$$60c-58d=-410$$

Add 58d to both sides.

$$60c=-410+58d$$

The equation is in standard form.

$$60c=58d-410$$

Divide both sides by 60.

$$\frac{60c}{60}=\frac{58d-410}{60}$$

Dividing by 60 undoes the multiplication by 60.

$$c=\frac{58d-410}{60}$$

Divide -410+58d by 60.

$$c=\frac{29d}{30}-\frac{41}{6}$$

Solve for d

$$d=\frac{30c+205}{29}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$-32=4\left(-3\right)^{4}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}-c+d+1\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Calculate -3 to the power of 4 and get 81.

$$-32=4\times 81+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}-c+d+1\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Multiply 4 and 81 to get 324.

$$-32=324+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}-c+d+1\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Combine \frac{c}{2} and -c to get -\frac{1}{2}c.

$$-32=324+4\left(-\frac{1}{2}c-\frac{3d}{4}-\frac{9}{4}+d+1\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Add -\frac{9}{4} and 1 to get -\frac{5}{4}.

$$-32=324+4\left(-\frac{1}{2}c-\frac{3d}{4}-\frac{5}{4}+d\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Since -\frac{3d}{4} and \frac{5}{4} have the same denominator, subtract them by subtracting their numerators.

$$-32=324+4\left(-\frac{1}{2}c+\frac{-3d-5}{4}+d\right)\left(-3\right)^{3}+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Calculate -3 to the power of 3 and get -27.

$$-32=324+4\left(-\frac{1}{2}c+\frac{-3d-5}{4}+d\right)\left(-27\right)+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Multiply 4 and -27 to get -108.

$$-32=324-108\left(-\frac{1}{2}c+\frac{-3d-5}{4}+d\right)+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Use the distributive property to multiply -108 by -\frac{1}{2}c+\frac{-3d-5}{4}+d.

$$-32=324+54c-108\times \frac{-3d-5}{4}-108d+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Cancel out 4, the greatest common factor in 108 and 4.

$$-32=324+54c-27\left(-3d-5\right)-108d+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Use the distributive property to multiply -27 by -3d-5.

$$-32=324+54c+81d+135-108d+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Combine 81d and -108d to get -27d.

$$-32=324+54c-27d+135+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Add 324 and 135 to get 459.

$$-32=459+54c-27d+4\left(\frac{c}{2}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{c}{2} times \frac{2}{2}.

$$-32=459+54c-27d+4\left(\frac{2c}{4}-\frac{3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Since \frac{2c}{4} and \frac{3d}{4} have the same denominator, subtract them by subtracting their numerators.

$$-32=459+54c-27d+4\left(\frac{2c-3d}{4}-\frac{9}{4}\right)\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Since \frac{2c-3d}{4} and \frac{9}{4} have the same denominator, subtract them by subtracting their numerators.

$$-32=459+54c-27d+4\times \frac{2c-3d-9}{4}\left(-3\right)^{2}+4c\left(-3\right)-4d$$

Calculate -3 to the power of 2 and get 9.

$$-32=459+54c-27d+4\times \frac{2c-3d-9}{4}\times 9+4c\left(-3\right)-4d$$

Multiply 4 and 9 to get 36.

$$-32=459+54c-27d+36\times \frac{2c-3d-9}{4}+4c\left(-3\right)-4d$$

Cancel out 4, the greatest common factor in 36 and 4.

$$-32=459+54c-27d+9\left(2c-3d-9\right)+4c\left(-3\right)-4d$$

Use the distributive property to multiply 9 by 2c-3d-9.

$$-32=459+54c-27d+18c-27d-81+4c\left(-3\right)-4d$$

Combine 54c and 18c to get 72c.

$$-32=459+72c-27d-27d-81+4c\left(-3\right)-4d$$

Combine -27d and -27d to get -54d.

$$-32=459+72c-54d-81+4c\left(-3\right)-4d$$

Subtract 81 from 459 to get 378.

$$-32=378+72c-54d+4c\left(-3\right)-4d$$

Multiply 4 and -3 to get -12.

$$-32=378+72c-54d-12c-4d$$

Combine 72c and -12c to get 60c.

$$-32=378+60c-54d-4d$$

Combine -54d and -4d to get -58d.

$$-32=378+60c-58d$$

Swap sides so that all variable terms are on the left hand side.

$$378+60c-58d=-32$$

Subtract 378 from both sides.

$$60c-58d=-32-378$$

Subtract 378 from -32 to get -410.

$$60c-58d=-410$$

Subtract 60c from both sides.

$$-58d=-410-60c$$

The equation is in standard form.

$$-58d=-60c-410$$

Divide both sides by -58.

$$\frac{-58d}{-58}=\frac{-60c-410}{-58}$$

Dividing by -58 undoes the multiplication by -58.

$$d=\frac{-60c-410}{-58}$$

Divide -410-60c by -58.

$$d=\frac{30c+205}{29}$$