Algebra Examples

$y = x + 4$ , $3 x + 5 y = 10$

Step 1

Solve the system of equations.

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Step 1.1

Subtract $x$ from both sides of the equation.

$y - x = 4, 3 x + 5 y = 10$

Step 1.2

Reorder the polynomial.

$- x + y = 4$

$3 x + 5 y = 10$

Step 1.3

Multiply each equation by the value that makes the coefficients of $x$ opposite.

$(3) \cdot (- x + y) = (3) (4)$

$3 x + 5 y = 10$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Simplify $(3) \cdot (- x + y)$ .

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Step 1.4.1.1.1

Apply the distributive property.

$3 (- x) + 3 y = (3) (4)$

$3 x + 5 y = 10$

Step 1.4.1.1.2

Multiply $- 1$ by $3$ .

$- 3 x + 3 y = (3) (4)$

$3 x + 5 y = 10$

$- 3 x + 3 y = (3) (4)$

$3 x + 5 y = 10$

$- 3 x + 3 y = (3) (4)$

$3 x + 5 y = 10$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Multiply $3$ by $4$ .

$- 3 x + 3 y = 12$

$3 x + 5 y = 10$

$- 3 x + 3 y = 12$

$3 x + 5 y = 10$

$- 3 x + 3 y = 12$

$3 x + 5 y = 10$

Step 1.5

Add the two equations together to eliminate $x$ from the system.

	$-$	$3$	$x$	$+$	$3$	$y$	$=$	$1$	$2$
$+$		$3$	$x$	$+$	$5$	$y$	$=$	$1$	$0$
					$8$	$y$	$=$	$2$	$2$

Step 1.6

Divide each term in $8 y = 22$ by $8$ and simplify.

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Step 1.6.1

Divide each term in $8 y = 22$ by $8$ .

$\frac{8 y}{8} = \frac{22}{8}$

Step 1.6.2

Simplify the left side.

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Step 1.6.2.1

Cancel the common factor of $8$ .

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Step 1.6.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{22}{8}$

Step 1.6.2.1.2

Divide $y$ by $1$ .

$y = \frac{22}{8}$

Step 1.6.3

Simplify the right side.

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Step 1.6.3.1

Cancel the common factor of $22$ and $8$ .

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Step 1.6.3.1.1

Factor $2$ out of $22$ .

$y = \frac{2 (11)}{8}$

Step 1.6.3.1.2

Cancel the common factors.

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Step 1.6.3.1.2.1

Factor $2$ out of $8$ .

$y = \frac{2 \cdot 11}{2 \cdot 4}$

Step 1.6.3.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 11}{2 \cdot 4}$

Step 1.6.3.1.2.3

Rewrite the expression.

$y = \frac{11}{4}$

Step 1.7

Substitute the value found for $y$ into one of the original equations, then solve for $x$ .

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Step 1.7.1

Substitute the value found for $y$ into one of the original equations to solve for $x$ .

$- 3 x + 3 (\frac{11}{4}) = 12$

Step 1.7.2

Multiply $3 (\frac{11}{4})$ .

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Step 1.7.2.1

Combine $3$ and $\frac{11}{4}$ .

$- 3 x + \frac{3 \cdot 11}{4} = 12$

Step 1.7.2.2

Multiply $3$ by $11$ .

$- 3 x + \frac{33}{4} = 12$

Step 1.7.3

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

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Step 1.7.3.1

Subtract $\frac{33}{4}$ from both sides of the equation.

$- 3 x = 12 - \frac{33}{4}$

Step 1.7.3.2

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

$- 3 x = 12 \cdot \frac{4}{4} - \frac{33}{4}$

Step 1.7.3.3

Combine $12$ and $\frac{4}{4}$ .

$- 3 x = \frac{12 \cdot 4}{4} - \frac{33}{4}$

Step 1.7.3.4

Combine the numerators over the common denominator.

$- 3 x = \frac{12 \cdot 4 - 33}{4}$

Step 1.7.3.5

Simplify the numerator.

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Step 1.7.3.5.1

Multiply $12$ by $4$ .

$- 3 x = \frac{48 - 33}{4}$

Step 1.7.3.5.2

Subtract $33$ from $48$ .

$- 3 x = \frac{15}{4}$

Step 1.7.4

Divide each term in $- 3 x = \frac{15}{4}$ by $- 3$ and simplify.

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Step 1.7.4.1

Divide each term in $- 3 x = \frac{15}{4}$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{\frac{15}{4}}{- 3}$

Step 1.7.4.2

Simplify the left side.

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Step 1.7.4.2.1

Cancel the common factor of $- 3$ .

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Step 1.7.4.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{\frac{15}{4}}{- 3}$

Step 1.7.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{15}{4}}{- 3}$

Step 1.7.4.3

Simplify the right side.

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Step 1.7.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{15}{4} \cdot \frac{1}{- 3}$

Step 1.7.4.3.2

Cancel the common factor of $3$ .

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Step 1.7.4.3.2.1

Factor $3$ out of $15$ .

$x = \frac{3 (5)}{4} \cdot \frac{1}{- 3}$

Step 1.7.4.3.2.2

Factor $3$ out of $- 3$ .

$x = \frac{3 \cdot 5}{4} \cdot \frac{1}{3 \cdot - 1}$

Step 1.7.4.3.2.3

Cancel the common factor.

$x = \frac{3 \cdot 5}{4} \cdot \frac{1}{3 \cdot - 1}$

Step 1.7.4.3.2.4

Rewrite the expression.

$x = \frac{5}{4} \cdot \frac{1}{- 1}$

Step 1.7.4.3.3

Multiply $\frac{5}{4}$ by $\frac{1}{- 1}$ .

$x = \frac{5}{4 \cdot - 1}$

Step 1.7.4.3.4

Simplify the expression.

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Step 1.7.4.3.4.1

Multiply $4$ by $- 1$ .

$x = \frac{5}{- 4}$

Step 1.7.4.3.4.2

Move the negative in front of the fraction.

$x = - \frac{5}{4}$

Step 1.8

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

$(- \frac{5}{4}, \frac{11}{4})$

Step 2

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

Independent

Step 3

	$-$	$3$	$x$	$+$	$3$	$y$	$=$	$1$	$2$
$+$		$3$	$x$	$+$	$5$	$y$	$=$	$1$	$0$
					$8$	$y$	$=$	$2$	$2$

	$-$	$3$	$x$	$+$	$3$	$y$	$=$	$1$	$2$
$+$		$3$	$x$	$+$	$5$	$y$	$=$	$1$	$0$
					$8$	$y$	$=$	$2$	$2$

	$-$	$3$	$x$	$+$	$3$	$y$	$=$	$1$	$2$
$+$		$3$	$x$	$+$	$5$	$y$	$=$	$1$	$0$
					$8$	$y$	$=$	$2$	$2$