Solve {l}{22*7*13*6y+0z=408*24}{x+13*6y+22*68z=362*88}{9*0*7x+13*6y+22*68z=317*52}{a=13*6}{text{Solvefor}btext{where}}{b=a}

$$\left. \begin{array} { l } { 22 \cdot 7 \cdot 13 \cdot 6 y + 0 z = 408 \cdot 24 }\\ { x + 13 \cdot 6 y + 22 \cdot 68 z = 362 \cdot 88 }\\ { 9 \cdot 0 \cdot 7 x + 13 \cdot 6 y + 22 \cdot 68 z = 317 \cdot 52 }\\ { a = 13 \cdot 6 }\\ { \text{Solve for } b \text{ where} } \\ { b = a } \end{array} \right.$$

Solve for y, z, x, a, b

$$b=78$$

Solution Steps

Consider the fourth equation. Multiply 13 and 6 to get 78.

$$a=78$$

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

$$b=78$$

Consider the first equation. Multiply 22 and 7 to get 154.

$$154\times 13\times 6y+0z=408\times 24$$

Multiply 154 and 13 to get 2002.

$$2002\times 6y+0z=408\times 24$$

Multiply 2002 and 6 to get 12012.

$$12012y+0z=408\times 24$$

Anything times zero gives zero.

$$12012y+0=408\times 24$$

Anything plus zero gives itself.

$$12012y=408\times 24$$

Multiply 408 and 24 to get 9792.

$$12012y=9792$$

Divide both sides by 12012.

$$y=\frac{9792}{12012}$$

Reduce the fraction \frac{9792}{12012} to lowest terms by extracting and canceling out 12.

$$y=\frac{816}{1001}$$

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

$$x+13\times 6\times \frac{816}{1001}+22\times 68z=362\times 88$$

Do the multiplications.

$$x+78\times \frac{816}{1001}+1496z=362\times 88$$

Multiply 78 and \frac{816}{1001} to get \frac{4896}{77}.

$$x+\frac{4896}{77}+1496z=362\times 88$$

Multiply 362 and 88 to get 31856.

$$x+\frac{4896}{77}+1496z=31856$$

Subtract \frac{4896}{77} from both sides.

$$x+1496z=31856-\frac{4896}{77}$$

Subtract \frac{4896}{77} from 31856 to get \frac{2448016}{77}.

$$x+1496z=\frac{2448016}{77}$$

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

$$9\times 0\times 7x+13\times 6\times \frac{816}{1001}+22\times 68z=317\times 52$$

Do the multiplications.

$$0\times 7x+78\times \frac{816}{1001}+1496z=317\times 52$$

Multiply 0 and 7 to get 0.

$$0x+78\times \frac{816}{1001}+1496z=317\times 52$$

Anything times zero gives zero.

$$0+78\times \frac{816}{1001}+1496z=317\times 52$$

Multiply 78 and \frac{816}{1001} to get \frac{4896}{77}.

$$0+\frac{4896}{77}+1496z=317\times 52$$

Add 0 and \frac{4896}{77} to get \frac{4896}{77}.

$$\frac{4896}{77}+1496z=317\times 52$$

Multiply 317 and 52 to get 16484.

$$\frac{4896}{77}+1496z=16484$$

Subtract \frac{4896}{77} from both sides.

$$1496z=16484-\frac{4896}{77}$$

Subtract \frac{4896}{77} from 16484 to get \frac{1264372}{77}.

$$1496z=\frac{1264372}{77}$$

Divide both sides by 1496.

$$z=\frac{\frac{1264372}{77}}{1496}$$

Express \frac{\frac{1264372}{77}}{1496} as a single fraction.

$$z=\frac{1264372}{77\times 1496}$$

Multiply 77 and 1496 to get 115192.

$$z=\frac{1264372}{115192}$$

Reduce the fraction \frac{1264372}{115192} to lowest terms by extracting and canceling out 4.

$$z=\frac{316093}{28798}$$

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

$$x+1496\times \frac{316093}{28798}=\frac{2448016}{77}$$

Multiply 1496 and \frac{316093}{28798} to get \frac{1264372}{77}.

$$x+\frac{1264372}{77}=\frac{2448016}{77}$$

Subtract \frac{1264372}{77} from both sides.

$$x=\frac{2448016}{77}-\frac{1264372}{77}$$

Subtract \frac{1264372}{77} from \frac{2448016}{77} to get 15372.

$$x=15372$$

The system is now solved.

$$y=\frac{816}{1001} z=\frac{316093}{28798} x=15372 a=78 b=78$$