Algebra Examples

$a + 5 b - c = - 20$ $4 a - 5 b + 4 c = 19$ $- a - 5 b - 5 c = 2$

Step 1

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

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Step 1.1

Subtract $5 b$ from both sides of the equation.

$a - c = - 20 - 5 b$

$4 a - 5 b + 4 c = 19$

$- a - 5 b - 5 c = 2$

Step 1.2

Add $c$ to both sides of the equation.

$a = - 20 - 5 b + c$

$4 a - 5 b + 4 c = 19$

$- a - 5 b - 5 c = 2$

$a = - 20 - 5 b + c$

$4 a - 5 b + 4 c = 19$

$- a - 5 b - 5 c = 2$

Step 2

Replace all occurrences of $a$ with $- 20 - 5 b + c$ in each equation.

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Step 2.1

Replace all occurrences of $a$ in $4 a - 5 b + 4 c = 19$ with $- 20 - 5 b + c$ .

$4 (- 20 - 5 b + c) - 5 b + 4 c = 19$

$a = - 20 - 5 b + c$

$- a - 5 b - 5 c = 2$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 (- 20 - 5 b + c) - 5 b + 4 c$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$4 \cdot - 20 + 4 (- 5 b) + 4 c - 5 b + 4 c = 19$

$a = - 20 - 5 b + c$

$- a - 5 b - 5 c = 2$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $4$ by $- 20$ .

$- 80 + 4 (- 5 b) + 4 c - 5 b + 4 c = 19$

$a = - 20 - 5 b + c$

$- a - 5 b - 5 c = 2$

Step 2.2.1.1.2.2

Multiply $- 5$ by $4$ .

$- 80 - 20 b + 4 c - 5 b + 4 c = 19$

$a = - 20 - 5 b + c$

$- a - 5 b - 5 c = 2$

$- 80 - 20 b + 4 c - 5 b + 4 c = 19$

$a = - 20 - 5 b + c$

$- a - 5 b - 5 c = 2$

$- 80 - 20 b + 4 c - 5 b + 4 c = 19$

$a = - 20 - 5 b + c$

$- a - 5 b - 5 c = 2$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $5 b$ from $- 20 b$ .

$- 80 - 25 b + 4 c + 4 c = 19$

$a = - 20 - 5 b + c$

$- a - 5 b - 5 c = 2$

Step 2.2.1.2.2

Add $4 c$ and $4 c$ .

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$- a - 5 b - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$- a - 5 b - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$- a - 5 b - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$- a - 5 b - 5 c = 2$

Step 2.3

Replace all occurrences of $a$ in $- a - 5 b - 5 c = 2$ with $- 20 - 5 b + c$ .

$- (- 20 - 5 b + c) - 5 b - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- (- 20 - 5 b + c) - 5 b - 5 c$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$20 - (- 5 b) - c - 5 b - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 1$ by $- 20$ .

$20 - (- 5 b) - c - 5 b - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 2.4.1.1.2.2

Multiply $- 5$ by $- 1$ .

$20 + 5 b - c - 5 b - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$20 + 5 b - c - 5 b - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$20 + 5 b - c - 5 b - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Combine the opposite terms in $20 + 5 b - c - 5 b - 5 c$ .

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Step 2.4.1.2.1.1

Subtract $5 b$ from $5 b$ .

$20 - c + 0 - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 2.4.1.2.1.2

Add $20 - c$ and $0$ .

$20 - c - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$20 - c - 5 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 2.4.1.2.2

Subtract $5 c$ from $- c$ .

$20 - 6 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$20 - 6 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$20 - 6 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$20 - 6 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$20 - 6 c = 2$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 3

Solve for $c$ in $20 - 6 c = 2$ .

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Step 3.1

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

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Step 3.1.1

Subtract $20$ from both sides of the equation.

$- 6 c = 2 - 20$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 3.1.2

Subtract $20$ from $2$ .

$- 6 c = - 18$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$- 6 c = - 18$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 3.2

Divide each term in $- 6 c = - 18$ by $- 6$ and simplify.

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Step 3.2.1

Divide each term in $- 6 c = - 18$ by $- 6$ .

$\frac{- 6 c}{- 6} = \frac{- 18}{- 6}$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 6 c}{- 6} = \frac{- 18}{- 6}$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 3.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{- 18}{- 6}$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$c = \frac{- 18}{- 6}$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$c = \frac{- 18}{- 6}$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 18$ by $- 6$ .

$c = 3$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$c = 3$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$c = 3$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

$c = 3$

$- 80 - 25 b + 8 c = 19$

$a = - 20 - 5 b + c$

Step 4

Replace all occurrences of $c$ with $3$ in each equation.

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Step 4.1

Replace all occurrences of $c$ in $- 80 - 25 b + 8 c = 19$ with $3$ .

$- 80 - 25 b + 8 (3) = 19$

$c = 3$

$a = - 20 - 5 b + c$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 80 - 25 b + 8 (3)$ .

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Step 4.2.1.1

Multiply $8$ by $3$ .

$- 80 - 25 b + 24 = 19$

$c = 3$

$a = - 20 - 5 b + c$

Step 4.2.1.2

Add $- 80$ and $24$ .

$- 25 b - 56 = 19$

$c = 3$

$a = - 20 - 5 b + c$

$- 25 b - 56 = 19$

$c = 3$

$a = - 20 - 5 b + c$

$- 25 b - 56 = 19$

$c = 3$

$a = - 20 - 5 b + c$

Step 4.3

Replace all occurrences of $c$ in $a = - 20 - 5 b + c$ with $3$ .

$a = - 20 - 5 b + 3$

$- 25 b - 56 = 19$

$c = 3$

Step 4.4

Simplify $a = - 20 - 5 b + 3$ .

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Remove parentheses.

$a = - 20 - 5 b + 3$

$- 25 b - 56 = 19$

$c = 3$

$a = - 20 - 5 b + 3$

$- 25 b - 56 = 19$

$c = 3$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Add $- 20$ and $3$ .

$a = - 5 b - 17$

$- 25 b - 56 = 19$

$c = 3$

$a = - 5 b - 17$

$- 25 b - 56 = 19$

$c = 3$

$a = - 5 b - 17$

$- 25 b - 56 = 19$

$c = 3$

$a = - 5 b - 17$

$- 25 b - 56 = 19$

$c = 3$

Step 5

Solve for $b$ in $- 25 b - 56 = 19$ .

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Step 5.1

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

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Step 5.1.1

Add $56$ to both sides of the equation.

$- 25 b = 19 + 56$

$a = - 5 b - 17$

$c = 3$

Step 5.1.2

Add $19$ and $56$ .

$- 25 b = 75$

$a = - 5 b - 17$

$c = 3$

$- 25 b = 75$

$a = - 5 b - 17$

$c = 3$

Step 5.2

Divide each term in $- 25 b = 75$ by $- 25$ and simplify.

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Step 5.2.1

Divide each term in $- 25 b = 75$ by $- 25$ .

$\frac{- 25 b}{- 25} = \frac{75}{- 25}$

$a = - 5 b - 17$

$c = 3$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 25$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 25 b}{- 25} = \frac{75}{- 25}$

$a = - 5 b - 17$

$c = 3$

Step 5.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{75}{- 25}$

$a = - 5 b - 17$

$c = 3$

$b = \frac{75}{- 25}$

$a = - 5 b - 17$

$c = 3$

$b = \frac{75}{- 25}$

$a = - 5 b - 17$

$c = 3$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $75$ by $- 25$ .

$b = - 3$

$a = - 5 b - 17$

$c = 3$

$b = - 3$

$a = - 5 b - 17$

$c = 3$

$b = - 3$

$a = - 5 b - 17$

$c = 3$

$b = - 3$

$a = - 5 b - 17$

$c = 3$

Step 6

Replace all occurrences of $b$ with $- 3$ in each equation.

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Step 6.1

Replace all occurrences of $b$ in $a = - 5 b - 17$ with $- 3$ .

$a = - 5 \cdot - 3 - 17$

$b = - 3$

$c = 3$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 5 \cdot - 3 - 17$ .

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Step 6.2.1.1

Multiply $- 5$ by $- 3$ .

$a = 15 - 17$

$b = - 3$

$c = 3$

Step 6.2.1.2

Subtract $17$ from $15$ .

$a = - 2$

$b = - 3$

$c = 3$

$a = - 2$

$b = - 3$

$c = 3$

$a = - 2$

$b = - 3$

$c = 3$

$a = - 2$

$b = - 3$

$c = 3$

Step 7

List all of the solutions.

$a = - 2, b = - 3, c = 3$