Algebra Examples

$k x - 3 y = 4$ $4 x - 5 y = 7$

Step 1

Solve the equation for $x$ .

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Step 1.1

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

$k x = 4 + 3 y$

$4 x - 5 y = 7$

Step 1.2

Divide each term in $k x = 4 + 3 y$ by $k$ and simplify.

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Step 1.2.1

Divide each term in $k x = 4 + 3 y$ by $k$ .

$\frac{k x}{k} = \frac{4}{k} + \frac{3 y}{k}$

$4 x - 5 y = 7$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $k$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{k x}{k} = \frac{4}{k} + \frac{3 y}{k}$

$4 x - 5 y = 7$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{k} + \frac{3 y}{k}$

$4 x - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

$4 x - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

$4 x - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

$4 x - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

$4 x - 5 y = 7$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$4 (\frac{4}{k}) + 4 (\frac{3 y}{k}) - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.1.2

Multiply $4 \frac{4}{k}$ .

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Step 2.1.2.1

Combine $4$ and $\frac{4}{k}$ .

$\frac{4 \cdot 4}{k} + 4 (\frac{3 y}{k}) - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.1.2.2

Multiply $4$ by $4$ .

$\frac{16}{k} + 4 (\frac{3 y}{k}) - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

$\frac{16}{k} + 4 (\frac{3 y}{k}) - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.1.3

Multiply $4 \frac{3 y}{k}$ .

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Step 2.1.3.1

Combine $4$ and $\frac{3 y}{k}$ .

$\frac{16}{k} + \frac{4 (3 y)}{k} - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.1.3.2

Multiply $3$ by $4$ .

$\frac{16}{k} + \frac{12 y}{k} - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

$\frac{16}{k} + \frac{12 y}{k} - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

$\frac{16}{k} + \frac{12 y}{k} - 5 y = 7$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$k, k, 1, 1$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.2.2

Since $k, k, 1, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part $k^{1}, k^{1}$ .

Since $k, k, 1, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part k,k.

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

$1$ List the prime factors of each number.

$2$ Multiply each factor the greatest number of times it occurs in either number.

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.2.5

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.2.6

The factor for $k^{1}$ is $k$ itself.

$k = k$

k occurs $1$ time.

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.2.7

The LCM of $k^{1}, k^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.3

Multiply each term in $\frac{16}{k} + \frac{12 y}{k} - 5 y = 7$ by $k$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{16}{k} + \frac{12 y}{k} - 5 y = 7$ by $k$ .

$\frac{16}{k} \cdot k + \frac{12 y}{k} \cdot k - 5 y k = 7 k$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Cancel the common factor of $k$ .

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Step 2.3.2.1.1.1

Cancel the common factor.

$\frac{16}{k} \cdot k + \frac{12 y}{k} \cdot k - 5 y k = 7 k$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.3.2.1.1.2

Rewrite the expression.

$16 + \frac{12 y}{k} \cdot k - 5 y k = 7 k$

$x = \frac{4}{k} + \frac{3 y}{k}$

$16 + \frac{12 y}{k} \cdot k - 5 y k = 7 k$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.3.2.1.2

Cancel the common factor of $k$ .

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Step 2.3.2.1.2.1

Cancel the common factor.

$16 + \frac{12 y}{k} \cdot k - 5 y k = 7 k$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.3.2.1.2.2

Rewrite the expression.

$16 + 12 y - 5 y k = 7 k$

$x = \frac{4}{k} + \frac{3 y}{k}$

$16 + 12 y - 5 y k = 7 k$

$x = \frac{4}{k} + \frac{3 y}{k}$

$16 + 12 y - 5 y k = 7 k$

$x = \frac{4}{k} + \frac{3 y}{k}$

$16 + 12 y - 5 y k = 7 k$

$x = \frac{4}{k} + \frac{3 y}{k}$

$16 + 12 y - 5 y k = 7 k$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.4

Solve the equation.

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Step 2.4.1

Subtract $16$ from both sides of the equation.

$12 y - 5 y k = 7 k - 16$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.4.2

Factor $y$ out of $12 y - 5 y k$ .

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Step 2.4.2.1

Factor $y$ out of $12 y$ .

$y \cdot 12 - 5 y k = 7 k - 16$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.4.2.2

Factor $y$ out of $- 5 y k$ .

$y \cdot 12 + y (- 5 k) = 7 k - 16$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.4.2.3

Factor $y$ out of $y \cdot 12 + y (- 5 k)$ .

$y \cdot (12 - 5 k) = 7 k - 16$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.4.2.4

Multiply $y$ by $12 - 5 k$ .

$y (12 - 5 k) = 7 k - 16$

$x = \frac{4}{k} + \frac{3 y}{k}$

$y (12 - 5 k) = 7 k - 16$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.4.3

Divide each term in $y (12 - 5 k) = 7 k - 16$ by $12 - 5 k$ and simplify.

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Step 2.4.3.1

Divide each term in $y (12 - 5 k) = 7 k - 16$ by $12 - 5 k$ .

$\frac{y (12 - 5 k)}{12 - 5 k} = \frac{7 k}{12 - 5 k} + \frac{- 16}{12 - 5 k}$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $12 - 5 k$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{y (12 - 5 k)}{12 - 5 k} = \frac{7 k}{12 - 5 k} + \frac{- 16}{12 - 5 k}$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{7 k}{12 - 5 k} + \frac{- 16}{12 - 5 k}$

$x = \frac{4}{k} + \frac{3 y}{k}$

$y = \frac{7 k}{12 - 5 k} + \frac{- 16}{12 - 5 k}$

$x = \frac{4}{k} + \frac{3 y}{k}$

$y = \frac{7 k}{12 - 5 k} + \frac{- 16}{12 - 5 k}$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Combine the numerators over the common denominator.

$y = \frac{7 k - 16}{12 - 5 k}$

$x = \frac{4}{k} + \frac{3 y}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

$x = \frac{4}{k} + \frac{3 y}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

$x = \frac{4}{k} + \frac{3 y}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

$x = \frac{4}{k} + \frac{3 y}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

$x = \frac{4}{k} + \frac{3 y}{k}$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\frac{4}{k} + \frac{3 (\frac{7 k - 16}{12 - 5 k})}{k}$ .

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Step 3.1.1

Combine the numerators over the common denominator.

$x = \frac{4 + 3 (\frac{7 k - 16}{12 - 5 k})}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.2

Combine $3$ and $\frac{7 k - 16}{12 - 5 k}$ .

$x = \frac{4 + \frac{3 (7 k - 16)}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{12 - 5 k}{12 - 5 k}$ .

$x = \frac{4 \cdot \frac{12 - 5 k}{12 - 5 k} + \frac{3 (7 k - 16)}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.4

Simplify terms.

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Step 3.1.4.1

Combine $4$ and $\frac{12 - 5 k}{12 - 5 k}$ .

$x = \frac{\frac{4 (12 - 5 k)}{12 - 5 k} + \frac{3 (7 k - 16)}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.4.2

Combine the numerators over the common denominator.

$x = \frac{\frac{4 (12 - 5 k) + 3 (7 k - 16)}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

$x = \frac{\frac{4 (12 - 5 k) + 3 (7 k - 16)}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Apply the distributive property.

$x = \frac{\frac{4 \cdot 12 + 4 (- 5 k) + 3 (7 k - 16)}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.5.2

Multiply $4$ by $12$ .

$x = \frac{\frac{48 + 4 (- 5 k) + 3 (7 k - 16)}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.5.3

Multiply $- 5$ by $4$ .

$x = \frac{\frac{48 - 20 k + 3 (7 k - 16)}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.5.4

Apply the distributive property.

$x = \frac{\frac{48 - 20 k + 3 (7 k) + 3 \cdot - 16}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.5.5

Multiply $7$ by $3$ .

$x = \frac{\frac{48 - 20 k + 21 k + 3 \cdot - 16}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.5.6

Multiply $3$ by $- 16$ .

$x = \frac{\frac{48 - 20 k + 21 k - 48}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.5.7

Subtract $48$ from $48$ .

$x = \frac{\frac{- 20 k + 21 k + 0}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.5.8

Add $- 20 k + 21 k$ and $0$ .

$x = \frac{\frac{- 20 k + 21 k}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.5.9

Add $- 20 k$ and $21 k$ .

$x = \frac{\frac{k}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

$x = \frac{\frac{k}{12 - 5 k}}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.6

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{k}{12 - 5 k} \cdot \frac{1}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.7

Cancel the common factor of $k$ .

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Step 3.1.7.1

Cancel the common factor.

$x = \frac{k}{12 - 5 k} \cdot \frac{1}{k}$

$y = \frac{7 k - 16}{12 - 5 k}$

Step 3.1.7.2

Rewrite the expression.

$x = \frac{1}{12 - 5 k}$

$y = \frac{7 k - 16}{12 - 5 k}$

$x = \frac{1}{12 - 5 k}$

$y = \frac{7 k - 16}{12 - 5 k}$

$x = \frac{1}{12 - 5 k}$

$y = \frac{7 k - 16}{12 - 5 k}$

$x = \frac{1}{12 - 5 k}$

$y = \frac{7 k - 16}{12 - 5 k}$