Algebra Examples

$A + B = 1$ $A + C = 0$ $A - B - C = 0$

Step 1

Subtract $B$ from both sides of the equation.

$A = 1 - B$

$A + C = 0$

$A - B - C = 0$

Step 2

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

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Step 2.1

Replace all occurrences of $A$ in $A + C = 0$ with $1 - B$ .

$(1 - B) + C = 0$

$A = 1 - B$

$A - B - C = 0$

Step 2.2

Simplify the left side.

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Step 2.2.1

Remove parentheses.

$1 - B + C = 0$

$A = 1 - B$

$A - B - C = 0$

$1 - B + C = 0$

$A = 1 - B$

$A - B - C = 0$

Step 2.3

Replace all occurrences of $A$ in $A - B - C = 0$ with $1 - B$ .

$(1 - B) - B - C = 0$

$1 - B + C = 0$

$A = 1 - B$

Step 2.4

Simplify the left side.

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Step 2.4.1

Subtract $B$ from $- B$ .

$1 - 2 B - C = 0$

$1 - B + C = 0$

$A = 1 - B$

$1 - 2 B - C = 0$

$1 - B + C = 0$

$A = 1 - B$

$1 - 2 B - C = 0$

$1 - B + C = 0$

$A = 1 - B$

Step 3

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

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Step 3.1

Subtract $1$ from both sides of the equation.

$- B + C = - 1$

$1 - 2 B - C = 0$

$A = 1 - B$

Step 3.2

Add $B$ to both sides of the equation.

$C = - 1 + B$

$1 - 2 B - C = 0$

$A = 1 - B$

$C = - 1 + B$

$1 - 2 B - C = 0$

$A = 1 - B$

Step 4

Replace all occurrences of $C$ with $- 1 + B$ in each equation.

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Step 4.1

Replace all occurrences of $C$ in $1 - 2 B - C = 0$ with $- 1 + B$ .

$1 - 2 B - (- 1 + B) = 0$

$C = - 1 + B$

$A = 1 - B$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $1 - 2 B - (- 1 + B)$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$1 - 2 B + 1 - B = 0$

$C = - 1 + B$

$A = 1 - B$

Step 4.2.1.1.2

Multiply $- 1$ by $- 1$ .

$1 - 2 B + 1 - B = 0$

$C = - 1 + B$

$A = 1 - B$

$1 - 2 B + 1 - B = 0$

$C = - 1 + B$

$A = 1 - B$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $1$ and $1$ .

$- 2 B + 2 - B = 0$

$C = - 1 + B$

$A = 1 - B$

Step 4.2.1.2.2

Subtract $B$ from $- 2 B$ .

$- 3 B + 2 = 0$

$C = - 1 + B$

$A = 1 - B$

$- 3 B + 2 = 0$

$C = - 1 + B$

$A = 1 - B$

$- 3 B + 2 = 0$

$C = - 1 + B$

$A = 1 - B$

$- 3 B + 2 = 0$

$C = - 1 + B$

$A = 1 - B$

$- 3 B + 2 = 0$

$C = - 1 + B$

$A = 1 - B$

Step 5

Solve for $B$ in $- 3 B + 2 = 0$ .

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Step 5.1

Subtract $2$ from both sides of the equation.

$- 3 B = - 2$

$C = - 1 + B$

$A = 1 - B$

Step 5.2

Divide each term in $- 3 B = - 2$ by $- 3$ and simplify.

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Step 5.2.1

Divide each term in $- 3 B = - 2$ by $- 3$ .

$\frac{- 3 B}{- 3} = \frac{- 2}{- 3}$

$C = - 1 + B$

$A = 1 - B$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 3 B}{- 3} = \frac{- 2}{- 3}$

$C = - 1 + B$

$A = 1 - B$

Step 5.2.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 2}{- 3}$

$C = - 1 + B$

$A = 1 - B$

$B = \frac{- 2}{- 3}$

$C = - 1 + B$

$A = 1 - B$

$B = \frac{- 2}{- 3}$

$C = - 1 + B$

$A = 1 - B$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Dividing two negative values results in a positive value.

$B = \frac{2}{3}$

$C = - 1 + B$

$A = 1 - B$

$B = \frac{2}{3}$

$C = - 1 + B$

$A = 1 - B$

$B = \frac{2}{3}$

$C = - 1 + B$

$A = 1 - B$

$B = \frac{2}{3}$

$C = - 1 + B$

$A = 1 - B$

Step 6

Replace all occurrences of $B$ with $\frac{2}{3}$ in each equation.

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Step 6.1

Replace all occurrences of $B$ in $C = - 1 + B$ with $\frac{2}{3}$ .

$C = - 1 + \frac{2}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

Step 6.2

Simplify $C = - 1 + \frac{2}{3}$ .

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Remove parentheses.

$C = - 1 + \frac{2}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

$C = - 1 + \frac{2}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$C = - 1 \cdot \frac{3}{3} + \frac{2}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

Step 6.2.2.1.2

Combine $- 1$ and $\frac{3}{3}$ .

$C = \frac{- 1 \cdot 3}{3} + \frac{2}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

Step 6.2.2.1.3

Combine the numerators over the common denominator.

$C = \frac{- 1 \cdot 3 + 2}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

Step 6.2.2.1.4

Simplify the numerator.

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Step 6.2.2.1.4.1

Multiply $- 1$ by $3$ .

$C = \frac{- 3 + 2}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

Step 6.2.2.1.4.2

Add $- 3$ and $2$ .

$C = \frac{- 1}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

$C = \frac{- 1}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

Step 6.2.2.1.5

Move the negative in front of the fraction.

$C = - \frac{1}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

$C = - \frac{1}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

$C = - \frac{1}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

$C = - \frac{1}{3}$

$B = \frac{2}{3}$

$A = 1 - B$

Step 6.3

Replace all occurrences of $B$ in $A = 1 - B$ with $\frac{2}{3}$ .

$A = 1 - (\frac{2}{3})$

$C = - \frac{1}{3}$

$B = \frac{2}{3}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $1 - (\frac{2}{3})$ .

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Step 6.4.1.1

Write $1$ as a fraction with a common denominator.

$A = \frac{3}{3} - \frac{2}{3}$

$C = - \frac{1}{3}$

$B = \frac{2}{3}$

Step 6.4.1.2

Combine the numerators over the common denominator.

$A = \frac{3 - 2}{3}$

$C = - \frac{1}{3}$

$B = \frac{2}{3}$

Step 6.4.1.3

Subtract $2$ from $3$ .

$A = \frac{1}{3}$

$C = - \frac{1}{3}$

$B = \frac{2}{3}$

$A = \frac{1}{3}$

$C = - \frac{1}{3}$

$B = \frac{2}{3}$

$A = \frac{1}{3}$

$C = - \frac{1}{3}$

$B = \frac{2}{3}$

$A = \frac{1}{3}$

$C = - \frac{1}{3}$

$B = \frac{2}{3}$

Step 7

List all of the solutions.

$A = \frac{1}{3}, C = - \frac{1}{3}, B = \frac{2}{3}$