Solve by Addition/Elimination 4x-14y=16 , 6x-21y=24

$4 x - 14 y = 16$ , $6 x - 21 y = 24$

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

$(- 3) \cdot (4 x - 14 y) = (- 3) (16)$

$(2) \cdot (6 x - 21 y) = (2) (24)$

Simplify.

Simplify the left side.

Simplify $(- 3) \cdot (4 x - 14 y)$ .

Apply the distributive property.

$- 3 (4 x) - 3 (- 14 y) = (- 3) (16)$

$(2) \cdot (6 x - 21 y) = (2) (24)$

Multiply.

Multiply $4$ by $- 3$ .

$- 12 x - 3 (- 14 y) = (- 3) (16)$

$(2) \cdot (6 x - 21 y) = (2) (24)$

Multiply $- 14$ by $- 3$ .

$- 12 x + 42 y = (- 3) (16)$

$(2) \cdot (6 x - 21 y) = (2) (24)$

$- 12 x + 42 y = (- 3) (16)$

$(2) \cdot (6 x - 21 y) = (2) (24)$

$- 12 x + 42 y = (- 3) (16)$

$(2) \cdot (6 x - 21 y) = (2) (24)$

$- 12 x + 42 y = (- 3) (16)$

$(2) \cdot (6 x - 21 y) = (2) (24)$

Simplify the right side.

Multiply $- 3$ by $16$ .

$- 12 x + 42 y = - 48$

$(2) \cdot (6 x - 21 y) = (2) (24)$

$- 12 x + 42 y = - 48$

$(2) \cdot (6 x - 21 y) = (2) (24)$

Simplify the left side.

Simplify $(2) \cdot (6 x - 21 y)$ .

Apply the distributive property.

$- 12 x + 42 y = - 48$

$2 (6 x) + 2 (- 21 y) = (2) (24)$

Multiply.

Multiply $6$ by $2$ .

$- 12 x + 42 y = - 48$

$12 x + 2 (- 21 y) = (2) (24)$

Multiply $- 21$ by $2$ .

$- 12 x + 42 y = - 48$

$12 x - 42 y = (2) (24)$

$- 12 x + 42 y = - 48$

$12 x - 42 y = (2) (24)$

$- 12 x + 42 y = - 48$

$12 x - 42 y = (2) (24)$

$- 12 x + 42 y = - 48$

$12 x - 42 y = (2) (24)$

Simplify the right side.

Multiply $2$ by $24$ .

$- 12 x + 42 y = - 48$

$12 x - 42 y = 48$

$- 12 x + 42 y = - 48$

$12 x - 42 y = 48$

$- 12 x + 42 y = - 48$

$12 x - 42 y = 48$

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

	$-$	$1$	$2$	$x$	$+$	$4$	$2$	$y$	$=$	$-$	$4$	$8$
$+$		$1$	$2$	$x$	$-$	$4$	$2$	$y$	$=$		$4$	$8$
								$0$	$=$			$0$

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

Infinite number of solutions

Solve one of the equations for $y$ .

Add $12 x$ to both sides of the equation.

$42 y = - 48 + 12 x$

Divide each term in $42 y = - 48 + 12 x$ by $42$ and simplify.

Divide each term in $42 y = - 48 + 12 x$ by $42$ .

$\frac{42 y}{42} = \frac{- 48}{42} + \frac{12 x}{42}$

Simplify the left side.

Cancel the common factor of $42$ .

Cancel the common factor.

$\frac{42 y}{42} = \frac{- 48}{42} + \frac{12 x}{42}$

Divide $y$ by $1$ .

$y = \frac{- 48}{42} + \frac{12 x}{42}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 48$ and $42$ .

Factor $6$ out of $- 48$ .

$y = \frac{6 (- 8)}{42} + \frac{12 x}{42}$

Cancel the common factors.

Factor $6$ out of $42$ .

$y = \frac{6 \cdot - 8}{6 \cdot 7} + \frac{12 x}{42}$

Cancel the common factor.

$y = \frac{6 \cdot - 8}{6 \cdot 7} + \frac{12 x}{42}$

Rewrite the expression.

$y = \frac{- 8}{7} + \frac{12 x}{42}$

Move the negative in front of the fraction.

$y = - \frac{8}{7} + \frac{12 x}{42}$

Cancel the common factor of $12$ and $42$ .

Factor $6$ out of $12 x$ .

$y = - \frac{8}{7} + \frac{6 (2 x)}{42}$

Cancel the common factors.

Factor $6$ out of $42$ .

$y = - \frac{8}{7} + \frac{6 (2 x)}{6 (7)}$

Cancel the common factor.

$y = - \frac{8}{7} + \frac{6 (2 x)}{6 \cdot 7}$

Rewrite the expression.

$y = - \frac{8}{7} + \frac{2 x}{7}$

The solution is the set of ordered pairs that make $y = - \frac{8}{7} + \frac{2 x}{7}$ true.

$(x, - \frac{8}{7} + \frac{2 x}{7})$

The result can be shown in multiple forms.

Point Form:

$(x, - \frac{8}{7} + \frac{2 x}{7})$

Equation Form:

$x = x, y = - \frac{8}{7} + \frac{2 x}{7}$

Solve by Addition/Elimination 4x-14y=16 , 6x-21y=24

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