Algebra Examples

$p - 6 q + 4 r = 2$ $2 p + 4 q - 8 r = 16$ $p - 2 q = 5$

Step 1

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

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Step 1.1

Add $6 q$ to both sides of the equation.

$p + 4 r = 2 + 6 q$

$2 p + 4 q - 8 r = 16$

$p - 2 q = 5$

Step 1.2

Subtract $4 r$ from both sides of the equation.

$p = 2 + 6 q - 4 r$

$2 p + 4 q - 8 r = 16$

$p - 2 q = 5$

$p = 2 + 6 q - 4 r$

$2 p + 4 q - 8 r = 16$

$p - 2 q = 5$

Step 2

Replace all occurrences of $p$ with $2 + 6 q - 4 r$ in each equation.

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Step 2.1

Replace all occurrences of $p$ in $2 p + 4 q - 8 r = 16$ with $2 + 6 q - 4 r$ .

$2 (2 + 6 q - 4 r) + 4 q - 8 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (2 + 6 q - 4 r) + 4 q - 8 r$ .

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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.

$2 \cdot 2 + 2 (6 q) + 2 (- 4 r) + 4 q - 8 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $2$ by $2$ .

$4 + 2 (6 q) + 2 (- 4 r) + 4 q - 8 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

Step 2.2.1.1.2.2

Multiply $6$ by $2$ .

$4 + 12 q + 2 (- 4 r) + 4 q - 8 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

Step 2.2.1.1.2.3

Multiply $- 4$ by $2$ .

$4 + 12 q - 8 r + 4 q - 8 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

$4 + 12 q - 8 r + 4 q - 8 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

$4 + 12 q - 8 r + 4 q - 8 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $12 q$ and $4 q$ .

$4 + 16 q - 8 r - 8 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

Step 2.2.1.2.2

Subtract $8 r$ from $- 8 r$ .

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$p - 2 q = 5$

Step 2.3

Replace all occurrences of $p$ in $p - 2 q = 5$ with $2 + 6 q - 4 r$ .

$(2 + 6 q - 4 r) - 2 q = 5$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

Step 2.4

Simplify the left side.

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Step 2.4.1

Subtract $2 q$ from $6 q$ .

$2 + 4 q - 4 r = 5$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$2 + 4 q - 4 r = 5$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$2 + 4 q - 4 r = 5$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

Step 3

Solve for $q$ in $2 + 4 q - 4 r = 5$ .

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Step 3.1

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

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Step 3.1.1

Subtract $2$ from both sides of the equation.

$4 q - 4 r = 5 - 2$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

Step 3.1.2

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

$4 q = 5 - 2 + 4 r$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

Step 3.1.3

Subtract $2$ from $5$ .

$4 q = 3 + 4 r$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$4 q = 3 + 4 r$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

Step 3.2

Divide each term in $4 q = 3 + 4 r$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 q = 3 + 4 r$ by $4$ .

$\frac{4 q}{4} = \frac{3}{4} + \frac{4 r}{4}$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 q}{4} = \frac{3}{4} + \frac{4 r}{4}$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

Step 3.2.2.1.2

Divide $q$ by $1$ .

$q = \frac{3}{4} + \frac{4 r}{4}$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$q = \frac{3}{4} + \frac{4 r}{4}$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$q = \frac{3}{4} + \frac{4 r}{4}$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $4$ .

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Step 3.2.3.1.1

Cancel the common factor.

$q = \frac{3}{4} + \frac{4 r}{4}$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

Step 3.2.3.1.2

Divide $r$ by $1$ .

$q = \frac{3}{4} + r$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$q = \frac{3}{4} + r$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$q = \frac{3}{4} + r$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$q = \frac{3}{4} + r$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

$q = \frac{3}{4} + r$

$4 + 16 q - 16 r = 16$

$p = 2 + 6 q - 4 r$

Step 4

Replace all occurrences of $q$ with $\frac{3}{4} + r$ in each equation.

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Step 4.1

Replace all occurrences of $q$ in $4 + 16 q - 16 r = 16$ with $\frac{3}{4} + r$ .

$4 + 16 (\frac{3}{4} + r) - 16 r = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $4 + 16 (\frac{3}{4} + r) - 16 r$ .

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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.

$4 + 16 (\frac{3}{4}) + 16 r - 16 r = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

Step 4.2.1.1.2

Cancel the common factor of $4$ .

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Step 4.2.1.1.2.1

Factor $4$ out of $16$ .

$4 + 4 (4) (\frac{3}{4}) + 16 r - 16 r = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

Step 4.2.1.1.2.2

Cancel the common factor.

$4 + 4 \cdot (4 (\frac{3}{4})) + 16 r - 16 r = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

Step 4.2.1.1.2.3

Rewrite the expression.

$4 + 4 \cdot 3 + 16 r - 16 r = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

$4 + 4 \cdot 3 + 16 r - 16 r = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

Step 4.2.1.1.3

Multiply $4$ by $3$ .

$4 + 12 + 16 r - 16 r = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

$4 + 12 + 16 r - 16 r = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $4 + 12 + 16 r - 16 r$ .

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Step 4.2.1.2.1.1

Subtract $16 r$ from $16 r$ .

$4 + 12 + 0 = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

Step 4.2.1.2.1.2

Add $4 + 12$ and $0$ .

$4 + 12 = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

$4 + 12 = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

Step 4.2.1.2.2

Add $4$ and $12$ .

$16 = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

$p = 2 + 6 q - 4 r$

Step 4.3

Replace all occurrences of $q$ in $p = 2 + 6 q - 4 r$ with $\frac{3}{4} + r$ .

$p = 2 + 6 (\frac{3}{4} + r) - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $2 + 6 (\frac{3}{4} + r) - 4 r$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$p = 2 + 6 (\frac{3}{4}) + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.1.1.2.1

Factor $2$ out of $6$ .

$p = 2 + 2 (3) (\frac{3}{4}) + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.1.2.2

Factor $2$ out of $4$ .

$p = 2 + 2 \cdot (3 (\frac{3}{2 \cdot 2})) + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.1.2.3

Cancel the common factor.

$p = 2 + 2 \cdot (3 (\frac{3}{2 \cdot 2})) + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.1.2.4

Rewrite the expression.

$p = 2 + 3 (\frac{3}{2}) + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

$p = 2 + 3 (\frac{3}{2}) + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.1.3

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

$p = 2 + \frac{3 \cdot 3}{2} + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.1.4

Multiply $3$ by $3$ .

$p = 2 + \frac{9}{2} + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

$p = 2 + \frac{9}{2} + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.2

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

$p = 2 \cdot \frac{2}{2} + \frac{9}{2} + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.3

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

$p = \frac{2 \cdot 2}{2} + \frac{9}{2} + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.4

Combine the numerators over the common denominator.

$p = \frac{2 \cdot 2 + 9}{2} + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $2$ by $2$ .

$p = \frac{4 + 9}{2} + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.5.2

Add $4$ and $9$ .

$p = \frac{13}{2} + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

$p = \frac{13}{2} + 6 r - 4 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 4.4.1.6

Subtract $4 r$ from $6 r$ .

$p = \frac{13}{2} + 2 r$

$16 = 16$

$q = \frac{3}{4} + r$

$p = \frac{13}{2} + 2 r$

$16 = 16$

$q = \frac{3}{4} + r$

$p = \frac{13}{2} + 2 r$

$16 = 16$

$q = \frac{3}{4} + r$

$p = \frac{13}{2} + 2 r$

$16 = 16$

$q = \frac{3}{4} + r$

Step 5

Remove any equations from the system that are always true.

$p = \frac{13}{2} + 2 r$

$q = \frac{3}{4} + r$