Algebra Examples

$y = - 3 x + 5$ $3 x + y = 5$

Step 1

Solve the system of equations.

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Step 1.1

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

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

Step 1.2

Reorder the polynomial.

$3 x + y = 5$

Step 1.3

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

$3 x + y = 5$

$(- 1) \cdot (3 x + y) = (- 1) (5)$

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 $(- 1) \cdot (3 x + y)$ .

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Step 1.4.1.1.1

Apply the distributive property.

$3 x + y = 5$

$- 1 (3 x) - 1 y = (- 1) (5)$

Step 1.4.1.1.2

Simplify the expression.

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Step 1.4.1.1.2.1

Multiply $3$ by $- 1$ .

$3 x + y = 5$

$- 3 x - 1 y = (- 1) (5)$

Step 1.4.1.1.2.2

Rewrite $- 1 y$ as $- y$ .

$3 x + y = 5$

$- 3 x - y = (- 1) (5)$

$3 x + y = 5$

$- 3 x - y = (- 1) (5)$

$3 x + y = 5$

$- 3 x - y = (- 1) (5)$

$3 x + y = 5$

$- 3 x - y = (- 1) (5)$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Multiply $- 1$ by $5$ .

$3 x + y = 5$

$- 3 x - y = - 5$

$3 x + y = 5$

$- 3 x - y = - 5$

$3 x + y = 5$

$- 3 x - y = - 5$

Step 1.5

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

		$3$	$x$	$+$	$y$	$=$		$5$
$+$	$-$	$3$	$x$	$-$	$y$	$=$	$-$	$5$
					$0$	$=$		$0$

Step 1.6

Since $0 = 0$ , the equations intersect at an infinite number of points.

Infinite number of solutions

Step 1.7

Subtract $3 x$ from both sides of the equation.

$y = 5 - 3 x$

Step 1.8

The solution is the set of ordered pairs that make $y = 5 - 3 x$ true.

$(x, 5 - 3 x)$

Step 2

Since the system is always true, the equations are equal and the graphs are the same line. Thus, the system is dependent.

Dependent

Step 3