Algebra Examples

$5 - \frac{3}{10} z + 2 = \frac{6}{5} z - \frac{1}{2} z + 3$

Step 1

Since $z$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{6}{5} z - \frac{1}{2} z + 3 = 5 - \frac{3}{10} z + 2$

Step 2

Simplify $\frac{6}{5} z - \frac{1}{2} z + 3$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $\frac{6}{5}$ and $z$ .

$\frac{6 z}{5} - \frac{1}{2} z + 3 = 5 - \frac{3}{10} z + 2$

Step 2.1.2

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

$\frac{6 z}{5} - \frac{z}{2} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.2

To write $\frac{6 z}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{6 z}{5} \cdot \frac{2}{2} - \frac{z}{2} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.3

To write $- \frac{z}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{6 z}{5} \cdot \frac{2}{2} - \frac{z}{2} \cdot \frac{5}{5} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{6 z}{5}$ by $\frac{2}{2}$ .

$\frac{6 z \cdot 2}{5 \cdot 2} - \frac{z}{2} \cdot \frac{5}{5} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.4.2

Multiply $5$ by $2$ .

$\frac{6 z \cdot 2}{10} - \frac{z}{2} \cdot \frac{5}{5} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.4.3

Multiply $\frac{z}{2}$ by $\frac{5}{5}$ .

$\frac{6 z \cdot 2}{10} - \frac{z \cdot 5}{2 \cdot 5} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.4.4

Multiply $2$ by $5$ .

$\frac{6 z \cdot 2}{10} - \frac{z \cdot 5}{10} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.5

Combine the numerators over the common denominator.

$\frac{6 z \cdot 2 - z \cdot 5}{10} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.6

Simplify each term.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Factor $z$ out of $6 z \cdot 2 - z \cdot 5$ .

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Step 2.6.1.1.1

Factor $z$ out of $6 z \cdot 2$ .

$\frac{z (6 \cdot 2) - z \cdot 5}{10} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.6.1.1.2

Factor $z$ out of $- z \cdot 5$ .

$\frac{z (6 \cdot 2) + z (- 1 \cdot 5)}{10} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.6.1.1.3

Factor $z$ out of $z (6 \cdot 2) + z (- 1 \cdot 5)$ .

$\frac{z (6 \cdot 2 - 1 \cdot 5)}{10} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.6.1.2

Multiply $6$ by $2$ .

$\frac{z (12 - 1 \cdot 5)}{10} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.6.1.3

Multiply $- 1$ by $5$ .

$\frac{z (12 - 5)}{10} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.6.1.4

Subtract $5$ from $12$ .

$\frac{z \cdot 7}{10} + 3 = 5 - \frac{3}{10} z + 2$

Step 2.6.2

Move $7$ to the left of $z$ .

$\frac{7 z}{10} + 3 = 5 - \frac{3}{10} z + 2$

Step 3

Simplify $5 - \frac{3}{10} z + 2$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine $z$ and $\frac{3}{10}$ .

$\frac{7 z}{10} + 3 = 5 - \frac{z \cdot 3}{10} + 2$

Step 3.1.2

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

$\frac{7 z}{10} + 3 = 5 - \frac{3 z}{10} + 2$

Step 3.2

Add $5$ and $2$ .

$\frac{7 z}{10} + 3 = - \frac{3 z}{10} + 7$

Step 4

Move all terms containing $z$ to the left side of the equation.

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Step 4.1

Add $\frac{3 z}{10}$ to both sides of the equation.

$\frac{7 z}{10} + 3 + \frac{3 z}{10} = 7$

Step 4.2

Combine the numerators over the common denominator.

$3 + \frac{7 z + 3 z}{10} = 7$

Step 4.3

Add $7 z$ and $3 z$ .

$3 + \frac{10 z}{10} = 7$

Step 4.4

Cancel the common factor of $10$ .

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Step 4.4.1

Cancel the common factor.

$3 + \frac{10 z}{10} = 7$

Step 4.4.2

Divide $z$ by $1$ .

$3 + z = 7$

Step 5

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

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Step 5.1

Subtract $3$ from both sides of the equation.

$z = 7 - 3$

Step 5.2

Subtract $3$ from $7$ .

$z = 4$