Algebra Examples

$x + y + z = 4$ $4 x + \frac{1}{2} y - 3 z = 30$ $- 2 x - 6 y + 5 z = - 7$

Step 1

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

Tap for more steps...

Step 1.1

Subtract $y$ from both sides of the equation.

$x + z = 4 - y$

$4 x + \frac{1}{2} y - 3 z = 30$

$- 2 x - 6 y + 5 z = - 7$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 4 - y - z$

$4 x + \frac{1}{2} y - 3 z = 30$

$- 2 x - 6 y + 5 z = - 7$

$x = 4 - y - z$

$4 x + \frac{1}{2} y - 3 z = 30$

$- 2 x - 6 y + 5 z = - 7$

Step 2

Replace all occurrences of $x$ with $4 - y - z$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $x$ in $4 x + \frac{1}{2} y - 3 z = 30$ with $4 - y - z$ .

$4 (4 - y - z) + \frac{1}{2} y - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify $4 (4 - y - z) + \frac{1}{2} y - 3 z$ .

Tap for more steps...

Step 2.2.1.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.1

Apply the distributive property.

$4 \cdot 4 + 4 (- y) + 4 (- z) + \frac{1}{2} y - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.1.2

Simplify.

Tap for more steps...

Step 2.2.1.1.2.1

Multiply $4$ by $4$ .

$16 + 4 (- y) + 4 (- z) + \frac{1}{2} y - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.1.2.2

Multiply $- 1$ by $4$ .

$16 - 4 y + 4 (- z) + \frac{1}{2} y - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.1.2.3

Multiply $- 1$ by $4$ .

$16 - 4 y - 4 z + \frac{1}{2} y - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

$16 - 4 y - 4 z + \frac{1}{2} y - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.1.3

Combine $\frac{1}{2}$ and $y$ .

$16 - 4 y - 4 z + \frac{y}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

$16 - 4 y - 4 z + \frac{y}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.2

To write $- 4 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$16 - 4 z - 4 y \cdot \frac{2}{2} + \frac{y}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.3

Simplify terms.

Tap for more steps...

Step 2.2.1.3.1

Combine $- 4 y$ and $\frac{2}{2}$ .

$16 - 4 z + \frac{- 4 y \cdot 2}{2} + \frac{y}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$16 - 4 z + \frac{- 4 y \cdot 2 + y}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

$16 - 4 z + \frac{- 4 y \cdot 2 + y}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.4

Simplify each term.

Tap for more steps...

Step 2.2.1.4.1

Simplify the numerator.

Tap for more steps...

Step 2.2.1.4.1.1

Factor $y$ out of $- 4 y \cdot 2 + y$ .

Tap for more steps...

Step 2.2.1.4.1.1.1

Factor $y$ out of $- 4 y \cdot 2$ .

$16 - 4 z + \frac{y (- 4 \cdot 2) + y}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.4.1.1.2

Raise $y$ to the power of $1$ .

$16 - 4 z + \frac{y (- 4 \cdot 2) + y}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.4.1.1.3

Factor $y$ out of $y^{1}$ .

$16 - 4 z + \frac{y (- 4 \cdot 2) + y \cdot 1}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.4.1.1.4

Factor $y$ out of $y (- 4 \cdot 2) + y \cdot 1$ .

$16 - 4 z + \frac{y (- 4 \cdot 2 + 1)}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

$16 - 4 z + \frac{y (- 4 \cdot 2 + 1)}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.4.1.2

Multiply $- 4$ by $2$ .

$16 - 4 z + \frac{y (- 8 + 1)}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.4.1.3

Add $- 8$ and $1$ .

$16 - 4 z + \frac{y \cdot - 7}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

$16 - 4 z + \frac{y \cdot - 7}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.4.2

Move $- 7$ to the left of $y$ .

$16 - 4 z + \frac{- 7 \cdot y}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.4.3

Move the negative in front of the fraction.

$16 - 4 z - \frac{7 y}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

$16 - 4 z - \frac{7 y}{2} - 3 z = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.2.1.5

Subtract $3 z$ from $- 4 z$ .

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$- 2 x - 6 y + 5 z = - 7$

Step 2.3

Replace all occurrences of $x$ in $- 2 x - 6 y + 5 z = - 7$ with $4 - y - z$ .

$- 2 (4 - y - z) - 6 y + 5 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 2.4

Simplify the left side.

Tap for more steps...

Step 2.4.1

Simplify $- 2 (4 - y - z) - 6 y + 5 z$ .

Tap for more steps...

Step 2.4.1.1

Simplify each term.

Tap for more steps...

Step 2.4.1.1.1

Apply the distributive property.

$- 2 \cdot 4 - 2 (- y) - 2 (- z) - 6 y + 5 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 2.4.1.1.2

Simplify.

Tap for more steps...

Step 2.4.1.1.2.1

Multiply $- 2$ by $4$ .

$- 8 - 2 (- y) - 2 (- z) - 6 y + 5 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 2.4.1.1.2.2

Multiply $- 1$ by $- 2$ .

$- 8 + 2 y - 2 (- z) - 6 y + 5 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $- 2$ .

$- 8 + 2 y + 2 z - 6 y + 5 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$- 8 + 2 y + 2 z - 6 y + 5 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$- 8 + 2 y + 2 z - 6 y + 5 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 2.4.1.2

Simplify by adding terms.

Tap for more steps...

Step 2.4.1.2.1

Subtract $6 y$ from $2 y$ .

$- 8 - 4 y + 2 z + 5 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 2.4.1.2.2

Add $2 z$ and $5 z$ .

$- 8 - 4 y + 7 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$- 8 - 4 y + 7 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$- 8 - 4 y + 7 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$- 8 - 4 y + 7 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$- 8 - 4 y + 7 z = - 7$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 3

Solve for $y$ in $- 8 - 4 y + 7 z = - 7$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Add $8$ to both sides of the equation.

$- 4 y + 7 z = - 7 + 8$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 3.1.2

Subtract $7 z$ from both sides of the equation.

$- 4 y = - 7 + 8 - 7 z$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 3.1.3

Add $- 7$ and $8$ .

$- 4 y = 1 - 7 z$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$- 4 y = 1 - 7 z$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 3.2

Divide each term in $- 4 y = 1 - 7 z$ by $- 4$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $- 4 y = 1 - 7 z$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{1}{- 4} + \frac{- 7 z}{- 4}$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{1}{- 4} + \frac{- 7 z}{- 4}$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{- 4} + \frac{- 7 z}{- 4}$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$y = \frac{1}{- 4} + \frac{- 7 z}{- 4}$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$y = \frac{1}{- 4} + \frac{- 7 z}{- 4}$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{1}{4} + \frac{- 7 z}{- 4}$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 3.2.3.1.2

Dividing two negative values results in a positive value.

$y = - \frac{1}{4} + \frac{7 z}{4}$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$16 - 7 z - \frac{7 y}{2} = 30$

$x = 4 - y - z$

Step 4

Replace all occurrences of $y$ with $- \frac{1}{4} + \frac{7 z}{4}$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $y$ in $16 - 7 z - \frac{7 y}{2} = 30$ with $- \frac{1}{4} + \frac{7 z}{4}$ .

$16 - 7 z - \frac{7 (- \frac{1}{4} + \frac{7 z}{4})}{2} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Simplify $16 - 7 z - \frac{7 (- \frac{1}{4} + \frac{7 z}{4})}{2}$ .

Tap for more steps...

Step 4.2.1.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1.1

Combine the numerators over the common denominator.

$16 - 7 z - \frac{7 (\frac{- 1 + 7 z}{4})}{2} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.1.2

Combine $7$ and $\frac{- 1 + 7 z}{4}$ .

$16 - 7 z - \frac{\frac{7 (- 1 + 7 z)}{4}}{2} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$16 - 7 z - (\frac{7 (- 1 + 7 z)}{4} \cdot \frac{1}{2}) = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.1.4

Multiply $\frac{7 (- 1 + 7 z)}{4} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 4.2.1.1.4.1

Multiply $\frac{7 (- 1 + 7 z)}{4}$ by $\frac{1}{2}$ .

$16 - 7 z - \frac{7 (- 1 + 7 z)}{4 \cdot 2} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.1.4.2

Multiply $4$ by $2$ .

$16 - 7 z - \frac{7 (- 1 + 7 z)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

$16 - 7 z - \frac{7 (- 1 + 7 z)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

$16 - 7 z - \frac{7 (- 1 + 7 z)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.2

To write $16$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- 7 z + 16 \cdot \frac{8}{8} - \frac{7 (- 1 + 7 z)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.3

Simplify terms.

Tap for more steps...

Step 4.2.1.3.1

Combine $16$ and $\frac{8}{8}$ .

$- 7 z + \frac{16 \cdot 8}{8} - \frac{7 (- 1 + 7 z)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.3.2

Combine the numerators over the common denominator.

$- 7 z + \frac{16 \cdot 8 - 7 (- 1 + 7 z)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.3.3

Multiply $16$ by $8$ .

$- 7 z + \frac{128 - 7 (- 1 + 7 z)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

$- 7 z + \frac{128 - 7 (- 1 + 7 z)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.4

Simplify the numerator.

Tap for more steps...

Step 4.2.1.4.1

Apply the distributive property.

$- 7 z + \frac{128 - 7 \cdot - 1 - 7 (7 z)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.4.2

Multiply $- 7$ by $- 1$ .

$- 7 z + \frac{128 + 7 - 7 (7 z)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.4.3

Multiply $7$ by $- 7$ .

$- 7 z + \frac{128 + 7 - 49 z}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.4.4

Add $128$ and $7$ .

$- 7 z + \frac{135 - 49 z}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

$- 7 z + \frac{135 - 49 z}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.5

To write $- 7 z$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- 7 z \cdot \frac{8}{8} + \frac{135 - 49 z}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.6

Simplify terms.

Tap for more steps...

Step 4.2.1.6.1

Combine $- 7 z$ and $\frac{8}{8}$ .

$\frac{- 7 z \cdot 8}{8} + \frac{135 - 49 z}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.6.2

Combine the numerators over the common denominator.

$\frac{- 7 z \cdot 8 + 135 - 49 z}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

$\frac{- 7 z \cdot 8 + 135 - 49 z}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.7

Simplify the numerator.

Tap for more steps...

Step 4.2.1.7.1

Multiply $8$ by $- 7$ .

$\frac{- 56 z + 135 - 49 z}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.7.2

Subtract $49 z$ from $- 56 z$ .

$\frac{- 105 z + 135}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.7.3

Factor $15$ out of $- 105 z + 135$ .

Tap for more steps...

Step 4.2.1.7.3.1

Factor $15$ out of $- 105 z$ .

$\frac{15 (- 7 z) + 135}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.7.3.2

Factor $15$ out of $135$ .

$\frac{15 (- 7 z) + 15 (9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.7.3.3

Factor $15$ out of $15 (- 7 z) + 15 (9)$ .

$\frac{15 (- 7 z + 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

$\frac{15 (- 7 z + 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

$\frac{15 (- 7 z + 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.8

Simplify with factoring out.

Tap for more steps...

Step 4.2.1.8.1

Factor $- 1$ out of $- 7 z$ .

$\frac{15 (- (7 z) + 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.8.2

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{15 (- (7 z) - 1 \cdot - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.8.3

Factor $- 1$ out of $- (7 z) - 1 (- 9)$ .

$\frac{15 (- (7 z - 9))}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.8.4

Simplify the expression.

Tap for more steps...

Step 4.2.1.8.4.1

Rewrite $- (7 z - 9)$ as $- 1 (7 z - 9)$ .

$\frac{15 (- 1 (7 z - 9))}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.2.1.8.4.2

Move the negative in front of the fraction.

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 - y - z$

Step 4.3

Replace all occurrences of $y$ in $x = 4 - y - z$ with $- \frac{1}{4} + \frac{7 z}{4}$ .

$x = 4 - (- \frac{1}{4} + \frac{7 z}{4}) - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4

Simplify the right side.

Tap for more steps...

Step 4.4.1

Simplify $4 - (- \frac{1}{4} + \frac{7 z}{4}) - z$ .

Tap for more steps...

Step 4.4.1.1

Simplify each term.

Tap for more steps...

Step 4.4.1.1.1

Apply the distributive property.

$x = 4 + \frac{1}{4} - \frac{7 z}{4} - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.1.2

Multiply $- - \frac{1}{4}$ .

Tap for more steps...

Step 4.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$x = 4 + 1 (\frac{1}{4}) - \frac{7 z}{4} - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.1.2.2

Multiply $\frac{1}{4}$ by $1$ .

$x = 4 + \frac{1}{4} - \frac{7 z}{4} - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 + \frac{1}{4} - \frac{7 z}{4} - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 4 + \frac{1}{4} - \frac{7 z}{4} - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 4 \cdot \frac{4}{4} + \frac{1}{4} - \frac{7 z}{4} - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.3

Combine $4$ and $\frac{4}{4}$ .

$x = \frac{4 \cdot 4}{4} + \frac{1}{4} - \frac{7 z}{4} - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$x = \frac{4 \cdot 4 + 1}{4} - \frac{7 z}{4} - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.5

Simplify the numerator.

Tap for more steps...

Step 4.4.1.5.1

Multiply $4$ by $4$ .

$x = \frac{16 + 1}{4} - \frac{7 z}{4} - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.5.2

Add $16$ and $1$ .

$x = \frac{17}{4} - \frac{7 z}{4} - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = \frac{17}{4} - \frac{7 z}{4} - z$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.6

To write $- z$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{17}{4} - \frac{7 z}{4} - z \cdot \frac{4}{4}$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.7

Combine $- z$ and $\frac{4}{4}$ .

$x = \frac{17}{4} - \frac{7 z}{4} + \frac{- z \cdot 4}{4}$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.8

Combine the numerators over the common denominator.

$x = \frac{17}{4} + \frac{- 7 z - z \cdot 4}{4}$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.9

Combine the numerators over the common denominator.

$x = \frac{17 - 7 z - z \cdot 4}{4}$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.10

Multiply $4$ by $- 1$ .

$x = \frac{17 - 7 z - 4 z}{4}$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 4.4.1.11

Subtract $4 z$ from $- 7 z$ .

$x = \frac{17 - 11 z}{4}$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = \frac{17 - 11 z}{4}$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = \frac{17 - 11 z}{4}$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = \frac{17 - 11 z}{4}$

$- \frac{15 (7 z - 9)}{8} = 30$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5

Solve for $z$ in $- \frac{15 (7 z - 9)}{8} = 30$ .

Tap for more steps...

Step 5.1

Multiply both sides of the equation by $- \frac{8}{15}$ .

$- \frac{8}{15} \cdot (- \frac{15 (7 z - 9)}{8}) = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2

Simplify both sides of the equation.

Tap for more steps...

Step 5.2.1

Simplify the left side.

Tap for more steps...

Step 5.2.1.1

Simplify $- \frac{8}{15} (- \frac{15 (7 z - 9)}{8})$ .

Tap for more steps...

Step 5.2.1.1.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 5.2.1.1.1.1

Move the leading negative in $- \frac{8}{15}$ into the numerator.

$\frac{- 8}{15} \cdot (- \frac{15 (7 z - 9)}{8}) = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.1.1.1.2

Move the leading negative in $- \frac{15 (7 z - 9)}{8}$ into the numerator.

$\frac{- 8}{15} \cdot \frac{- 15 (7 z - 9)}{8} = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.1.1.1.3

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{15} \cdot \frac{- 15 (7 z - 9)}{8} = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.1.1.1.4

Cancel the common factor.

$\frac{8 \cdot - 1}{15} \cdot \frac{- 15 (7 z - 9)}{8} = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.1.1.1.5

Rewrite the expression.

$\frac{- 1}{15} \cdot (- 15 (7 z - 9)) = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$\frac{- 1}{15} \cdot (- 15 (7 z - 9)) = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.1.1.2

Cancel the common factor of $15$ .

Tap for more steps...

Step 5.2.1.1.2.1

Factor $15$ out of $- 15 (7 z - 9)$ .

$\frac{- 1}{15} \cdot (15 (- (7 z - 9))) = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.1.1.2.2

Cancel the common factor.

$\frac{- 1}{15} \cdot (15 (- (7 z - 9))) = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.1.1.2.3

Rewrite the expression.

$7 z - 9 = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$7 z - 9 = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.1.1.3

Multiply.

Tap for more steps...

Step 5.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 (7 z - 9) = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.1.1.3.2

Multiply $7 z - 9$ by $1$ .

$7 z - 9 = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$7 z - 9 = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$7 z - 9 = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$7 z - 9 = - \frac{8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.2

Simplify the right side.

Tap for more steps...

Step 5.2.2.1

Simplify $- \frac{8}{15} \cdot 30$ .

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor of $15$ .

Tap for more steps...

Step 5.2.2.1.1.1

Move the leading negative in $- \frac{8}{15}$ into the numerator.

$7 z - 9 = \frac{- 8}{15} \cdot 30$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.2.1.1.2

Factor $15$ out of $30$ .

$7 z - 9 = \frac{- 8}{15} \cdot (15 (2))$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.2.1.1.3

Cancel the common factor.

$7 z - 9 = \frac{- 8}{15} \cdot (15 \cdot 2)$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.2.1.1.4

Rewrite the expression.

$7 z - 9 = - 8 \cdot 2$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$7 z - 9 = - 8 \cdot 2$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.2.2.1.2

Multiply $- 8$ by $2$ .

$7 z - 9 = - 16$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$7 z - 9 = - 16$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$7 z - 9 = - 16$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$7 z - 9 = - 16$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Add $9$ to both sides of the equation.

$7 z = - 16 + 9$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.3.2

Add $- 16$ and $9$ .

$7 z = - 7$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$7 z = - 7$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.4

Divide each term in $7 z = - 7$ by $7$ and simplify.

Tap for more steps...

Step 5.4.1

Divide each term in $7 z = - 7$ by $7$ .

$\frac{7 z}{7} = \frac{- 7}{7}$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

$\frac{7 z}{7} = \frac{- 7}{7}$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.4.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 7}{7}$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$z = \frac{- 7}{7}$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$z = \frac{- 7}{7}$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Divide $- 7$ by $7$ .

$z = - 1$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$z = - 1$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$z = - 1$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$z = - 1$

$x = \frac{17 - 11 z}{4}$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 6

Replace all occurrences of $z$ with $- 1$ in each equation.

Tap for more steps...

Step 6.1

Replace all occurrences of $z$ in $x = \frac{17 - 11 z}{4}$ with $- 1$ .

$x = \frac{17 - 11 \cdot - 1}{4}$

$z = - 1$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 6.2

Simplify the right side.

Tap for more steps...

Step 6.2.1

Simplify $\frac{17 - 11 \cdot - 1}{4}$ .

Tap for more steps...

Step 6.2.1.1

Simplify the numerator.

Tap for more steps...

Step 6.2.1.1.1

Multiply $- 11$ by $- 1$ .

$x = \frac{17 + 11}{4}$

$z = - 1$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 6.2.1.1.2

Add $17$ and $11$ .

$x = \frac{28}{4}$

$z = - 1$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = \frac{28}{4}$

$z = - 1$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 6.2.1.2

Divide $28$ by $4$ .

$x = 7$

$z = - 1$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 7$

$z = - 1$

$y = - \frac{1}{4} + \frac{7 z}{4}$

$x = 7$

$z = - 1$

$y = - \frac{1}{4} + \frac{7 z}{4}$

Step 6.3

Replace all occurrences of $z$ in $y = - \frac{1}{4} + \frac{7 z}{4}$ with $- 1$ .

$y = - \frac{1}{4} + \frac{7 (- 1)}{4}$

$x = 7$

$z = - 1$

Step 6.4

Simplify the right side.

Tap for more steps...

Step 6.4.1

Simplify $- \frac{1}{4} + \frac{7 (- 1)}{4}$ .

Tap for more steps...

Step 6.4.1.1

Combine the numerators over the common denominator.

$y = \frac{- 1 + 7 (- 1)}{4}$

$x = 7$

$z = - 1$

Step 6.4.1.2

Simplify the expression.

Tap for more steps...

Step 6.4.1.2.1

Multiply $7$ by $- 1$ .

$y = \frac{- 1 - 7}{4}$

$x = 7$

$z = - 1$

Step 6.4.1.2.2

Subtract $7$ from $- 1$ .

$y = \frac{- 8}{4}$

$x = 7$

$z = - 1$

Step 6.4.1.2.3

Divide $- 8$ by $4$ .

$y = - 2$

$x = 7$

$z = - 1$

$y = - 2$

$x = 7$

$z = - 1$

$y = - 2$

$x = 7$

$z = - 1$

$y = - 2$

$x = 7$

$z = - 1$

$y = - 2$

$x = 7$

$z = - 1$

Step 7

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(7, - 2, - 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(7, - 2, - 1)$

Equation Form:

$x = 7, y = - 2, z = - 1$