Precalculus Examples

$y = \frac{1}{7} x + 3$ , $x - 7 y = - 21$

Step 1

Solve the system of equations.

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Step 1.1

Multiply each term in $y = \frac{1}{7} x + 3$ by $7$ to eliminate the fractions.

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Step 1.1.1

Multiply each term in $y = \frac{1}{7} x + 3$ by $7$ .

$y \cdot 7 = \frac{1}{7} x \cdot 7 + 3 \cdot 7$

$x - 7 y = - 21$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Move $7$ to the left of $y$ .

$7 y = \frac{1}{7} x \cdot 7 + 3 \cdot 7$

$x - 7 y = - 21$

$7 y = \frac{1}{7} x \cdot 7 + 3 \cdot 7$

$x - 7 y = - 21$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Combine $\frac{1}{7}$ and $x$ .

$7 y = \frac{x}{7} \cdot 7 + 3 \cdot 7$

$x - 7 y = - 21$

Step 1.1.3.1.2

Cancel the common factor of $7$ .

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Step 1.1.3.1.2.1

Cancel the common factor.

$7 y = \frac{x}{7} \cdot 7 + 3 \cdot 7$

$x - 7 y = - 21$

Step 1.1.3.1.2.2

Rewrite the expression.

$7 y = x + 3 \cdot 7$

$x - 7 y = - 21$

$7 y = x + 3 \cdot 7$

$x - 7 y = - 21$

Step 1.1.3.1.3

Multiply $3$ by $7$ .

$7 y = x + 21$

$x - 7 y = - 21$

$7 y = x + 21$

$x - 7 y = - 21$

$7 y = x + 21$

$x - 7 y = - 21$

$7 y = x + 21$

$x - 7 y = - 21$

Step 1.2

Subtract $x$ from both sides of the equation.

$7 y - x = 21, x - 7 y = - 21$

Step 1.3

Reorder the polynomial.

$- x + 7 y = 21$

$x - 7 y = - 21$

Step 1.4

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

	$-$	$x$	$+$	$7$	$y$	$=$		$2$	$1$
$+$		$x$	$-$	$7$	$y$	$=$	$-$	$2$	$1$
					$0$	$=$			$0$

Step 1.5

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

Infinite number of solutions

Step 1.6

Solve one of the equations for $y$ .

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Step 1.6.1

Add $x$ to both sides of the equation.

$7 y = 21 + x$

Step 1.6.2

Divide each term in $7 y = 21 + x$ by $7$ and simplify.

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Step 1.6.2.1

Divide each term in $7 y = 21 + x$ by $7$ .

$\frac{7 y}{7} = \frac{21}{7} + \frac{x}{7}$

Step 1.6.2.2

Simplify the left side.

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Step 1.6.2.2.1

Cancel the common factor of $7$ .

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Step 1.6.2.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{21}{7} + \frac{x}{7}$

Step 1.6.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{21}{7} + \frac{x}{7}$

Step 1.6.2.3

Simplify the right side.

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Step 1.6.2.3.1

Divide $21$ by $7$ .

$y = 3 + \frac{x}{7}$

Step 1.7

The solution is the set of ordered pairs that make $y = 3 + \frac{x}{7}$ true.

$(x, 3 + \frac{x}{7})$

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

	$-$	$x$	$+$	$7$	$y$	$=$		$2$	$1$
$+$		$x$	$-$	$7$	$y$	$=$	$-$	$2$	$1$
					$0$	$=$			$0$

	$-$	$x$	$+$	$7$	$y$	$=$		$2$	$1$
$+$		$x$	$-$	$7$	$y$	$=$	$-$	$2$	$1$
					$0$	$=$			$0$

	$-$	$x$	$+$	$7$	$y$	$=$		$2$	$1$
$+$		$x$	$-$	$7$	$y$	$=$	$-$	$2$	$1$
					$0$	$=$			$0$