Algebra Examples

$2 a - 3 b - 4 c = 5$ $5 a - 4 b - 2 c = 4$ $6 a - 9 b - 12 c = 5$

Step 1

Solve for $a$ in $2 a - 3 b - 4 c = 5$ .

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Step 1.1

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

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Step 1.1.1

Add $3 b$ to both sides of the equation.

$2 a - 4 c = 5 + 3 b$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

Step 1.1.2

Add $4 c$ to both sides of the equation.

$2 a = 5 + 3 b + 4 c$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

$2 a = 5 + 3 b + 4 c$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

Step 1.2

Divide each term in $2 a = 5 + 3 b + 4 c$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 a = 5 + 3 b + 4 c$ by $2$ .

$\frac{2 a}{2} = \frac{5}{2} + \frac{3 b}{2} + \frac{4 c}{2}$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{5}{2} + \frac{3 b}{2} + \frac{4 c}{2}$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

Step 1.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{5}{2} + \frac{3 b}{2} + \frac{4 c}{2}$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

$a = \frac{5}{2} + \frac{3 b}{2} + \frac{4 c}{2}$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

$a = \frac{5}{2} + \frac{3 b}{2} + \frac{4 c}{2}$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $4$ and $2$ .

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Step 1.2.3.1.1

Factor $2$ out of $4 c$ .

$a = \frac{5}{2} + \frac{3 b}{2} + \frac{2 (2 c)}{2}$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $2$ out of $2$ .

$a = \frac{5}{2} + \frac{3 b}{2} + \frac{2 (2 c)}{2 (1)}$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

Step 1.2.3.1.2.2

Cancel the common factor.

$a = \frac{5}{2} + \frac{3 b}{2} + \frac{2 (2 c)}{2 \cdot 1}$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

Step 1.2.3.1.2.3

Rewrite the expression.

$a = \frac{5}{2} + \frac{3 b}{2} + \frac{2 c}{1}$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

Step 1.2.3.1.2.4

Divide $2 c$ by $1$ .

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$5 a - 4 b - 2 c = 4$

$6 a - 9 b - 12 c = 5$

Step 2

Replace all occurrences of $a$ with $\frac{5}{2} + \frac{3 b}{2} + 2 c$ in each equation.

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Step 2.1

Replace all occurrences of $a$ in $5 a - 4 b - 2 c = 4$ with $\frac{5}{2} + \frac{3 b}{2} + 2 c$ .

$5 (\frac{5}{2} + \frac{3 b}{2} + 2 c) - 4 b - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (\frac{5}{2} + \frac{3 b}{2} + 2 c) - 4 b - 2 c$ .

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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.

$5 (\frac{5}{2}) + 5 (\frac{3 b}{2}) + 5 (2 c) - 4 b - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $5 (\frac{5}{2})$ .

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Step 2.2.1.1.2.1.1

Combine $5$ and $\frac{5}{2}$ .

$\frac{5 \cdot 5}{2} + 5 (\frac{3 b}{2}) + 5 (2 c) - 4 b - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.1.2.1.2

Multiply $5$ by $5$ .

$\frac{25}{2} + 5 (\frac{3 b}{2}) + 5 (2 c) - 4 b - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

$\frac{25}{2} + 5 (\frac{3 b}{2}) + 5 (2 c) - 4 b - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.1.2.2

Multiply $5 \frac{3 b}{2}$ .

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Step 2.2.1.1.2.2.1

Combine $5$ and $\frac{3 b}{2}$ .

$\frac{25}{2} + \frac{5 (3 b)}{2} + 5 (2 c) - 4 b - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.1.2.2.2

Multiply $3$ by $5$ .

$\frac{25}{2} + \frac{15 b}{2} + 5 (2 c) - 4 b - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

$\frac{25}{2} + \frac{15 b}{2} + 5 (2 c) - 4 b - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.1.2.3

Multiply $2$ by $5$ .

$\frac{25}{2} + \frac{15 b}{2} + 10 c - 4 b - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

$\frac{25}{2} + \frac{15 b}{2} + 10 c - 4 b - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

$\frac{25}{2} + \frac{15 b}{2} + 10 c - 4 b - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.2

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

$\frac{25}{2} + 10 c + \frac{15 b}{2} - 4 b \cdot \frac{2}{2} - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.3

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

$\frac{25}{2} + 10 c + \frac{15 b}{2} + \frac{- 4 b \cdot 2}{2} - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{25}{2} + 10 c + \frac{15 b - 4 b \cdot 2}{2} - 2 c = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.5

Combine the numerators over the common denominator.

$10 c - 2 c + \frac{25 + 15 b - 4 b \cdot 2}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.6

Multiply $2$ by $- 4$ .

$10 c - 2 c + \frac{25 + 15 b - 8 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.7

Subtract $8 b$ from $15 b$ .

$10 c - 2 c + \frac{25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.8

Subtract $2 c$ from $10 c$ .

$8 c + \frac{25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.9

To write $8 c$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$8 c \cdot \frac{2}{2} + \frac{25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.10

Combine $8 c$ and $\frac{2}{2}$ .

$\frac{8 c \cdot 2}{2} + \frac{25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.11

Combine the numerators over the common denominator.

$\frac{8 c \cdot 2 + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.2.1.12

Multiply $2$ by $8$ .

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$6 a - 9 b - 12 c = 5$

Step 2.3

Replace all occurrences of $a$ in $6 a - 9 b - 12 c = 5$ with $\frac{5}{2} + \frac{3 b}{2} + 2 c$ .

$6 (\frac{5}{2} + \frac{3 b}{2} + 2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $6 (\frac{5}{2} + \frac{3 b}{2} + 2 c) - 9 b - 12 c$ .

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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.

$6 (\frac{5}{2}) + 6 (\frac{3 b}{2}) + 6 (2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.1.1

Factor $2$ out of $6$ .

$2 (3) (\frac{5}{2}) + 6 (\frac{3 b}{2}) + 6 (2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.1.2.1.2

Cancel the common factor.

$2 \cdot (3 (\frac{5}{2})) + 6 (\frac{3 b}{2}) + 6 (2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.1.2.1.3

Rewrite the expression.

$3 \cdot 5 + 6 (\frac{3 b}{2}) + 6 (2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$3 \cdot 5 + 6 (\frac{3 b}{2}) + 6 (2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.1.2.2

Multiply $3$ by $5$ .

$15 + 6 (\frac{3 b}{2}) + 6 (2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.1.2.3

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.3.1

Factor $2$ out of $6$ .

$15 + 2 (3) (\frac{3 b}{2}) + 6 (2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.1.2.3.2

Cancel the common factor.

$15 + 2 \cdot (3 (\frac{3 b}{2})) + 6 (2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.1.2.3.3

Rewrite the expression.

$15 + 3 (3 b) + 6 (2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$15 + 3 (3 b) + 6 (2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.1.2.4

Multiply $3$ by $3$ .

$15 + 9 b + 6 (2 c) - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.1.2.5

Multiply $2$ by $6$ .

$15 + 9 b + 12 c - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$15 + 9 b + 12 c - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$15 + 9 b + 12 c - 9 b - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.2

Combine the opposite terms in $15 + 9 b + 12 c - 9 b - 12 c$ .

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Step 2.4.1.2.1

Subtract $9 b$ from $9 b$ .

$15 + 12 c + 0 - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.2.2

Add $15 + 12 c$ and $0$ .

$15 + 12 c - 12 c = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.2.3

Subtract $12 c$ from $12 c$ .

$15 + 0 = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 2.4.1.2.4

Add $15$ and $0$ .

$15 = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$15 = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$15 = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$15 = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

$15 = 5$

$\frac{16 c + 25 + 7 b}{2} = 4$

$a = \frac{5}{2} + \frac{3 b}{2} + 2 c$

Step 3

Since $15 = 5$ is not true, there is no solution.

No solution