Algebra Examples

3 y = 9 x - 6

$3 y = 9 x - 6$

2 y + 6 x = 4

$2 y + 6 x = 4$

Step 1

Solve the system of equations.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Reorder

2 y

$2 y$ and

6 x

$6 x$ .

6 x + 2 y = 4

$6 x + 2 y = 4$

3 y = 9 x - 6

$3 y = 9 x - 6$

6 x + 2 y = 4

$6 x + 2 y = 4$

3 y = 9 x - 6

$3 y = 9 x - 6$

Step 1.2

Subtract

9 x

$9 x$ from both sides of the equation.

3 y - 9 x = - 6, 6 x + 2 y = 4

$3 y - 9 x = - 6, 6 x + 2 y = 4$

Step 1.3

Reorder the polynomial.

- 9 x + 3 y = - 6

$- 9 x + 3 y = - 6$

6 x + 2 y = 4

$6 x + 2 y = 4$

Step 1.4

Multiply each equation by the value that makes the coefficients of

y

$y$ opposite.

(- 2) \cdot (- 9 x + 3 y) = (- 2) (- 6)

$(- 2) \cdot (- 9 x + 3 y) = (- 2) (- 6)$

(3) \cdot (6 x + 2 y) = (3) (4)

$(3) \cdot (6 x + 2 y) = (3) (4)$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the left side.

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Step 1.5.1.1

Simplify

(- 2) \cdot (- 9 x + 3 y)

$(- 2) \cdot (- 9 x + 3 y)$ .

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Step 1.5.1.1.1

Apply the distributive property.

- 2 (- 9 x) - 2 (3 y) = (- 2) (- 6)

$- 2 (- 9 x) - 2 (3 y) = (- 2) (- 6)$

(3) \cdot (6 x + 2 y) = (3) (4)

$(3) \cdot (6 x + 2 y) = (3) (4)$

Step 1.5.1.1.2

Multiply.

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Step 1.5.1.1.2.1

Multiply

- 9

$- 9$ by

- 2

$- 2$ .

18 x - 2 (3 y) = (- 2) (- 6)

$18 x - 2 (3 y) = (- 2) (- 6)$

(3) \cdot (6 x + 2 y) = (3) (4)

$(3) \cdot (6 x + 2 y) = (3) (4)$

Step 1.5.1.1.2.2

Multiply

3

$3$ by

- 2

$- 2$ .

18 x - 6 y = (- 2) (- 6)

$18 x - 6 y = (- 2) (- 6)$

(3) \cdot (6 x + 2 y) = (3) (4)

$(3) \cdot (6 x + 2 y) = (3) (4)$

18 x - 6 y = (- 2) (- 6)

$18 x - 6 y = (- 2) (- 6)$

(3) \cdot (6 x + 2 y) = (3) (4)

$(3) \cdot (6 x + 2 y) = (3) (4)$

18 x - 6 y = (- 2) (- 6)

$18 x - 6 y = (- 2) (- 6)$

(3) \cdot (6 x + 2 y) = (3) (4)

$(3) \cdot (6 x + 2 y) = (3) (4)$

18 x - 6 y = (- 2) (- 6)

$18 x - 6 y = (- 2) (- 6)$

(3) \cdot (6 x + 2 y) = (3) (4)

$(3) \cdot (6 x + 2 y) = (3) (4)$

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Multiply

- 2

$- 2$ by

- 6

$- 6$ .

18 x - 6 y = 12

$18 x - 6 y = 12$

(3) \cdot (6 x + 2 y) = (3) (4)

$(3) \cdot (6 x + 2 y) = (3) (4)$

18 x - 6 y = 12

$18 x - 6 y = 12$

(3) \cdot (6 x + 2 y) = (3) (4)

$(3) \cdot (6 x + 2 y) = (3) (4)$

Step 1.5.3

Simplify the left side.

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Step 1.5.3.1

Simplify

(3) \cdot (6 x + 2 y)

$(3) \cdot (6 x + 2 y)$ .

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Step 1.5.3.1.1

Apply the distributive property.

18 x - 6 y = 12

$18 x - 6 y = 12$

3 (6 x) + 3 (2 y) = (3) (4)

$3 (6 x) + 3 (2 y) = (3) (4)$

Step 1.5.3.1.2

Multiply.

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Step 1.5.3.1.2.1

Multiply

6

$6$ by

3

$3$ .

18 x - 6 y = 12

$18 x - 6 y = 12$

18 x + 3 (2 y) = (3) (4)

$18 x + 3 (2 y) = (3) (4)$

Step 1.5.3.1.2.2

Multiply

2

$2$ by

3

$3$ .

18 x - 6 y = 12

$18 x - 6 y = 12$

18 x + 6 y = (3) (4)

$18 x + 6 y = (3) (4)$

18 x - 6 y = 12

$18 x - 6 y = 12$

18 x + 6 y = (3) (4)

$18 x + 6 y = (3) (4)$

18 x - 6 y = 12

$18 x - 6 y = 12$

18 x + 6 y = (3) (4)

$18 x + 6 y = (3) (4)$

18 x - 6 y = 12

$18 x - 6 y = 12$

18 x + 6 y = (3) (4)

$18 x + 6 y = (3) (4)$

Step 1.5.4

Simplify the right side.

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Step 1.5.4.1

Multiply

3

$3$ by

4

$4$ .

18 x - 6 y = 12

$18 x - 6 y = 12$

18 x + 6 y = 12

$18 x + 6 y = 12$

18 x - 6 y = 12

$18 x - 6 y = 12$

18 x + 6 y = 12

$18 x + 6 y = 12$

18 x - 6 y = 12

$18 x - 6 y = 12$

18 x + 6 y = 12

$18 x + 6 y = 12$

Step 1.6

Add the two equations together to eliminate

y

$y$ from the system.

	$1$ $1$	$8$ $8$	$x$ $x$	$-$ $-$	$6$ $6$	$y$ $y$	$=$ $=$	$1$ $1$	$2$ $2$
$+$ $+$	$1$ $1$	$8$ $8$	$x$ $x$	$+$ $+$	$6$ $6$	$y$ $y$	$=$ $=$	$1$ $1$	$2$ $2$
	$3$ $3$	$6$ $6$	$x$ $x$				$=$ $=$	$2$ $2$	$4$ $4$

Step 1.7

Divide each term in

36 x = 24

$36 x = 24$ by

36

$36$ and simplify.

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Step 1.7.1

Divide each term in

36 x = 24

$36 x = 24$ by

36

$36$ .

\frac{36 x}{36} = \frac{24}{36}

$\frac{36 x}{36} = \frac{24}{36}$

Step 1.7.2

Simplify the left side.

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Step 1.7.2.1

Cancel the common factor of

36

$36$ .

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Step 1.7.2.1.1

Cancel the common factor.

$\frac{36 x}{36} = \frac{24}{36}$

Step 1.7.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{24}{36}$

Step 1.7.3

Simplify the right side.

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Step 1.7.3.1

Cancel the common factor of

$24$ and

$36$ .

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Step 1.7.3.1.1

Factor

$12$ out of

$24$ .

$x = \frac{12 (2)}{36}$

Step 1.7.3.1.2

Cancel the common factors.

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Step 1.7.3.1.2.1

Factor

$12$ out of

$36$ .

$x = \frac{12 \cdot 2}{12 \cdot 3}$

Step 1.7.3.1.2.2

Cancel the common factor.

$x = \frac{12 \cdot 2}{12 \cdot 3}$

Step 1.7.3.1.2.3

Rewrite the expression.

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

Step 1.8

Substitute the value found for

$x$ into one of the original equations, then solve for

$y$ .

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Step 1.8.1

Substitute the value found for

$x$ into one of the original equations to solve for

$y$ .

$18 (\frac{2}{3}) - 6 y = 12$

Step 1.8.2

Simplify each term.

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Step 1.8.2.1

Cancel the common factor of

$3$ .

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Step 1.8.2.1.1

Factor

$3$ out of

$18$ .

$3 (6) \frac{2}{3} - 6 y = 12$

Step 1.8.2.1.2

Cancel the common factor.

$3 \cdot 6 \frac{2}{3} - 6 y = 12$

Step 1.8.2.1.3

Rewrite the expression.

$6 \cdot 2 - 6 y = 12$

Step 1.8.2.2

Multiply

$6$ by

$2$ .

$12 - 6 y = 12$

Step 1.8.3

Move all terms not containing

$y$ to the right side of the equation.

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Step 1.8.3.1

Subtract

$12$ from both sides of the equation.

$- 6 y = 12 - 12$

Step 1.8.3.2

Subtract

$12$ from

$12$ .

$- 6 y = 0$

Step 1.8.4

Divide each term in

$- 6 y = 0$ by

$- 6$ and simplify.

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Step 1.8.4.1

Divide each term in

$- 6 y = 0$ by

$- 6$ .

$\frac{- 6 y}{- 6} = \frac{0}{- 6}$

Step 1.8.4.2

Simplify the left side.

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Step 1.8.4.2.1

Cancel the common factor of

$- 6$ .

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Step 1.8.4.2.1.1

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{0}{- 6}$

Step 1.8.4.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{0}{- 6}$

Step 1.8.4.3

Simplify the right side.

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Step 1.8.4.3.1

Divide

$0$ by

$- 6$ .

$y = 0$

Step 1.9

The solution to the independent system of equations can be represented as a point.

$(\frac{2}{3}, 0)$

Step 2

Since the system has a point of intersection, the system is independent.

Independent

Step 3