Algebra Examples

$3 x + 6 y - z = 7$ , $x + y + 3 z = 6$

Step 1

Solve the equation for $y$ .

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Step 1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1

Subtract $3 x$ from both sides of the equation.

$6 y - z = 7 - 3 x$

$x + y + 3 z = 6$

Step 1.1.2

Add $z$ to both sides of the equation.

$6 y = 7 - 3 x + z$

$x + y + 3 z = 6$

$6 y = 7 - 3 x + z$

$x + y + 3 z = 6$

Step 1.2

Divide each term in $6 y = 7 - 3 x + z$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 y = 7 - 3 x + z$ by $6$ .

$\frac{6 y}{6} = \frac{7}{6} + \frac{- 3 x}{6} + \frac{z}{6}$

$x + y + 3 z = 6$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{7}{6} + \frac{- 3 x}{6} + \frac{z}{6}$

$x + y + 3 z = 6$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{7}{6} + \frac{- 3 x}{6} + \frac{z}{6}$

$x + y + 3 z = 6$

$y = \frac{7}{6} + \frac{- 3 x}{6} + \frac{z}{6}$

$x + y + 3 z = 6$

$y = \frac{7}{6} + \frac{- 3 x}{6} + \frac{z}{6}$

$x + y + 3 z = 6$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $- 3$ and $6$ .

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Step 1.2.3.1.1.1

Factor $3$ out of $- 3 x$ .

$y = \frac{7}{6} + \frac{3 (- x)}{6} + \frac{z}{6}$

$x + y + 3 z = 6$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $3$ out of $6$ .

$y = \frac{7}{6} + \frac{3 (- x)}{3 (2)} + \frac{z}{6}$

$x + y + 3 z = 6$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{7}{6} + \frac{3 (- x)}{3 \cdot 2} + \frac{z}{6}$

$x + y + 3 z = 6$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{7}{6} + \frac{- x}{2} + \frac{z}{6}$

$x + y + 3 z = 6$

$y = \frac{7}{6} + \frac{- x}{2} + \frac{z}{6}$

$x + y + 3 z = 6$

$y = \frac{7}{6} + \frac{- x}{2} + \frac{z}{6}$

$x + y + 3 z = 6$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$x + y + 3 z = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$x + y + 3 z = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$x + y + 3 z = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$x + y + 3 z = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$x + y + 3 z = 6$

Step 2

Solve the equation for $z$ .

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Step 2.1

Simplify $x + \frac{7}{6} - \frac{x}{2} + \frac{z}{6} + 3 z$ .

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Step 2.1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x \cdot \frac{2}{2} - \frac{x}{2} + \frac{7}{6} + \frac{z}{6} + 3 z = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.2

Combine $x$ and $\frac{2}{2}$ .

$\frac{x \cdot 2}{2} - \frac{x}{2} + \frac{7}{6} + \frac{z}{6} + 3 z = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.3

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - x}{2} + \frac{7}{6} + \frac{z}{6} + 3 z = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.4

Subtract $x$ from $x \cdot 2$ .

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Step 2.1.4.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x - x}{2} + \frac{7}{6} + \frac{z}{6} + 3 z = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.4.2

Subtract $x$ from $2 \cdot x$ .

$\frac{x}{2} + \frac{7}{6} + \frac{z}{6} + 3 z = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$\frac{x}{2} + \frac{7}{6} + \frac{z}{6} + 3 z = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.5

To write $3 z$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{x}{2} + \frac{7}{6} + \frac{z}{6} + 3 z \cdot \frac{6}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.6

Combine $3 z$ and $\frac{6}{6}$ .

$\frac{x}{2} + \frac{7}{6} + \frac{z}{6} + \frac{3 z \cdot 6}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.7

Combine the numerators over the common denominator.

$\frac{x}{2} + \frac{7}{6} + \frac{z + 3 z \cdot 6}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.8

Combine the numerators over the common denominator.

$\frac{x}{2} + \frac{7 + z + 3 z \cdot 6}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.9

Multiply $6$ by $3$ .

$\frac{x}{2} + \frac{7 + z + 18 z}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.10

Add $z$ and $18 z$ .

$\frac{x}{2} + \frac{7 + 19 z}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.11

To write $\frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{2} \cdot \frac{3}{3} + \frac{7 + 19 z}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.12

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.12.1

Multiply $\frac{x}{2}$ by $\frac{3}{3}$ .

$\frac{x \cdot 3}{2 \cdot 3} + \frac{7 + 19 z}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.12.2

Multiply $2$ by $3$ .

$\frac{x \cdot 3}{6} + \frac{7 + 19 z}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$\frac{x \cdot 3}{6} + \frac{7 + 19 z}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.13

Combine the numerators over the common denominator.

$\frac{x \cdot 3 + 7 + 19 z}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.1.14

Move $3$ to the left of $x$ .

$\frac{3 x + 7 + 19 z}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$\frac{3 x + 7 + 19 z}{6} = 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.2

Multiply both sides by $6$ .

$\frac{3 x + 7 + 19 z}{6} \cdot 6 = 6 \cdot 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{3 x + 7 + 19 z}{6} \cdot 6$ .

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Step 2.3.1.1.1

Cancel the common factor of $6$ .

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Step 2.3.1.1.1.1

Cancel the common factor.

$\frac{3 x + 7 + 19 z}{6} \cdot 6 = 6 \cdot 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.3.1.1.1.2

Rewrite the expression.

$3 x + 7 + 19 z = 6 \cdot 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$3 x + 7 + 19 z = 6 \cdot 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.3.1.1.2

Move $7$ .

$3 x + 19 z + 7 = 6 \cdot 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$3 x + 19 z + 7 = 6 \cdot 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$3 x + 19 z + 7 = 6 \cdot 6$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $6$ by $6$ .

$3 x + 19 z + 7 = 36$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$3 x + 19 z + 7 = 36$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$3 x + 19 z + 7 = 36$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.4

Solve for $z$ .

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Step 2.4.1

Move all terms not containing $z$ to the right side of the equation.

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Step 2.4.1.1

Subtract $3 x$ from both sides of the equation.

$19 z + 7 = 36 - 3 x$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.4.1.2

Subtract $7$ from both sides of the equation.

$19 z = 36 - 3 x - 7$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.4.1.3

Subtract $7$ from $36$ .

$19 z = - 3 x + 29$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$19 z = - 3 x + 29$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2

Divide each term in $19 z = - 3 x + 29$ by $19$ and simplify.

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Step 2.4.2.1

Divide each term in $19 z = - 3 x + 29$ by $19$ .

$\frac{19 z}{19} = \frac{- 3 x}{19} + \frac{29}{19}$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $19$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{19 z}{19} = \frac{- 3 x}{19} + \frac{29}{19}$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 3 x}{19} + \frac{29}{19}$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$z = \frac{- 3 x}{19} + \frac{29}{19}$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$z = \frac{- 3 x}{19} + \frac{29}{19}$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Move the negative in front of the fraction.

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{7}{6} - \frac{x}{2} + \frac{z}{6}$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\frac{7}{6} - \frac{x}{2} + \frac{- \frac{3 x}{19} + \frac{29}{19}}{6}$ .

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Step 3.1.1

Combine the numerators over the common denominator.

$y = \frac{7 - \frac{3 x}{19} + \frac{29}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$y = \frac{- \frac{3 x}{19} + 7 \cdot \frac{19}{19} + \frac{29}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.3

Combine $7$ and $\frac{19}{19}$ .

$y = \frac{- \frac{3 x}{19} + \frac{7 \cdot 19}{19} + \frac{29}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.4

Combine the numerators over the common denominator.

$y = \frac{- \frac{3 x}{19} + \frac{7 \cdot 19 + 29}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $7$ by $19$ .

$y = \frac{- \frac{3 x}{19} + \frac{133 + 29}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.5.2

Add $133$ and $29$ .

$y = \frac{- \frac{3 x}{19} + \frac{162}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{- \frac{3 x}{19} + \frac{162}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.6

Simplify each term.

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Step 3.1.6.1

Simplify the numerator.

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Step 3.1.6.1.1

Combine the numerators over the common denominator.

$y = \frac{\frac{- 3 x + 162}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.6.1.2

Factor $3$ out of $- 3 x + 162$ .

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Step 3.1.6.1.2.1

Factor $3$ out of $- 3 x$ .

$y = \frac{\frac{3 (- x) + 162}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.6.1.2.2

Factor $3$ out of $162$ .

$y = \frac{\frac{3 (- x) + 3 (54)}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.6.1.2.3

Factor $3$ out of $3 (- x) + 3 (54)$ .

$y = \frac{\frac{3 (- x + 54)}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{\frac{3 (- x + 54)}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{\frac{3 (- x + 54)}{19}}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.6.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{3 (- x + 54)}{19} \cdot \frac{1}{6} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.6.3

Cancel the common factor of $3$ .

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Step 3.1.6.3.1

Factor $3$ out of $6$ .

$y = \frac{3 (- x + 54)}{19} \cdot \frac{1}{3 (2)} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.6.3.2

Cancel the common factor.

$y = \frac{3 (- x + 54)}{19} \cdot \frac{1}{3 \cdot 2} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.6.3.3

Rewrite the expression.

$y = \frac{- x + 54}{19} \cdot \frac{1}{2} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{- x + 54}{19} \cdot \frac{1}{2} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.6.4

Multiply $\frac{- x + 54}{19}$ by $\frac{1}{2}$ .

$y = \frac{- x + 54}{19 \cdot 2} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.6.5

Multiply $19$ by $2$ .

$y = \frac{- x + 54}{38} + \frac{- x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.6.6

Move the negative in front of the fraction.

$y = \frac{- x + 54}{38} - \frac{x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{- x + 54}{38} - \frac{x}{2}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.7

To write $- \frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$y = \frac{- x + 54}{38} - \frac{x}{2} \cdot \frac{19}{19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.8

Write each expression with a common denominator of $38$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.8.1

Multiply $\frac{x}{2}$ by $\frac{19}{19}$ .

$y = \frac{- x + 54}{38} - \frac{x \cdot 19}{2 \cdot 19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.8.2

Multiply $2$ by $19$ .

$y = \frac{- x + 54}{38} - \frac{x \cdot 19}{38}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{- x + 54}{38} - \frac{x \cdot 19}{38}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.9

Combine the numerators over the common denominator.

$y = \frac{- x + 54 - x \cdot 19}{38}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.10

Simplify the numerator.

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Step 3.1.10.1

Multiply $19$ by $- 1$ .

$y = \frac{- x + 54 - 19 x}{38}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.10.2

Subtract $19 x$ from $- x$ .

$y = \frac{- 20 x + 54}{38}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.10.3

Factor $2$ out of $- 20 x + 54$ .

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Step 3.1.10.3.1

Factor $2$ out of $- 20 x$ .

$y = \frac{2 (- 10 x) + 54}{38}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.10.3.2

Factor $2$ out of $54$ .

$y = \frac{2 (- 10 x) + 2 (27)}{38}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.10.3.3

Factor $2$ out of $2 (- 10 x) + 2 (27)$ .

$y = \frac{2 (- 10 x + 27)}{38}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{2 (- 10 x + 27)}{38}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{2 (- 10 x + 27)}{38}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.11

Cancel the common factors.

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Step 3.1.11.1

Factor $2$ out of $38$ .

$y = \frac{2 (- 10 x + 27)}{2 \cdot 19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.11.2

Cancel the common factor.

$y = \frac{2 (- 10 x + 27)}{2 \cdot 19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.11.3

Rewrite the expression.

$y = \frac{- 10 x + 27}{19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = \frac{- 10 x + 27}{19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.12

Factor $- 1$ out of $- 10 x$ .

$y = \frac{- (10 x) + 27}{19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.13

Rewrite $27$ as $- 1 (- 27)$ .

$y = \frac{- (10 x) - 1 \cdot - 27}{19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.14

Factor $- 1$ out of $- (10 x) - 1 (- 27)$ .

$y = \frac{- (10 x - 27)}{19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.15

Simplify the expression.

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Step 3.1.15.1

Rewrite $- (10 x - 27)$ as $- 1 (10 x - 27)$ .

$y = \frac{- 1 (10 x - 27)}{19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

Step 3.1.15.2

Move the negative in front of the fraction.

$y = - \frac{10 x - 27}{19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = - \frac{10 x - 27}{19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = - \frac{10 x - 27}{19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$

$y = - \frac{10 x - 27}{19}$

$z = - \frac{3 x}{19} + \frac{29}{19}$