Algebra Examples

$2 a + 3 b = 6$ $5 a + 2 b - 4 = 0$

Step 1

Add $4$ to both sides of the equation.

$2 a + 3 b = 6, 5 a + 2 b = 4$

Step 2

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

$(- 2) \cdot (2 a + 3 b) = (- 2) (6)$

$(3) \cdot (5 a + 2 b) = (3) (4)$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify the left side.

Tap for more steps...

Step 3.1.1

Simplify $(- 2) \cdot (2 a + 3 b)$ .

Tap for more steps...

Step 3.1.1.1

Apply the distributive property.

$- 2 (2 a) - 2 (3 b) = (- 2) (6)$

$(3) \cdot (5 a + 2 b) = (3) (4)$

Step 3.1.1.2

Multiply.

Tap for more steps...

Step 3.1.1.2.1

Multiply $2$ by $- 2$ .

$- 4 a - 2 (3 b) = (- 2) (6)$

$(3) \cdot (5 a + 2 b) = (3) (4)$

Step 3.1.1.2.2

Multiply $3$ by $- 2$ .

$- 4 a - 6 b = (- 2) (6)$

$(3) \cdot (5 a + 2 b) = (3) (4)$

$- 4 a - 6 b = (- 2) (6)$

$(3) \cdot (5 a + 2 b) = (3) (4)$

$- 4 a - 6 b = (- 2) (6)$

$(3) \cdot (5 a + 2 b) = (3) (4)$

$- 4 a - 6 b = (- 2) (6)$

$(3) \cdot (5 a + 2 b) = (3) (4)$

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

Multiply $- 2$ by $6$ .

$- 4 a - 6 b = - 12$

$(3) \cdot (5 a + 2 b) = (3) (4)$

$- 4 a - 6 b = - 12$

$(3) \cdot (5 a + 2 b) = (3) (4)$

Step 3.3

Simplify the left side.

Tap for more steps...

Step 3.3.1

Simplify $(3) \cdot (5 a + 2 b)$ .

Tap for more steps...

Step 3.3.1.1

Apply the distributive property.

$- 4 a - 6 b = - 12$

$3 (5 a) + 3 (2 b) = (3) (4)$

Step 3.3.1.2

Multiply.

Tap for more steps...

Step 3.3.1.2.1

Multiply $5$ by $3$ .

$- 4 a - 6 b = - 12$

$15 a + 3 (2 b) = (3) (4)$

Step 3.3.1.2.2

Multiply $2$ by $3$ .

$- 4 a - 6 b = - 12$

$15 a + 6 b = (3) (4)$

$- 4 a - 6 b = - 12$

$15 a + 6 b = (3) (4)$

$- 4 a - 6 b = - 12$

$15 a + 6 b = (3) (4)$

$- 4 a - 6 b = - 12$

$15 a + 6 b = (3) (4)$

Step 3.4

Simplify the right side.

Tap for more steps...

Step 3.4.1

Multiply $3$ by $4$ .

$- 4 a - 6 b = - 12$

$15 a + 6 b = 12$

$- 4 a - 6 b = - 12$

$15 a + 6 b = 12$

$- 4 a - 6 b = - 12$

$15 a + 6 b = 12$

Step 4

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

	$-$	$4$	$a$	$-$	$6$	$b$	$=$	$-$	$1$	$2$
$+$	$1$	$5$	$a$	$+$	$6$	$b$	$=$		$1$	$2$
	$1$	$1$	$a$				$=$			$0$

Step 5

Divide each term in $11 a = 0$ by $11$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $11 a = 0$ by $11$ .

$\frac{11 a}{11} = \frac{0}{11}$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $11$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

$\frac{11 a}{11} = \frac{0}{11}$

Step 5.2.1.2

Divide $a$ by $1$ .

$a = \frac{0}{11}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Divide $0$ by $11$ .

$a = 0$

Step 6

Substitute the value found for $a$ into one of the original equations, then solve for $b$ .

Tap for more steps...

Step 6.1

Substitute the value found for $a$ into one of the original equations to solve for $b$ .

$- 4 \cdot 0 - 6 b = - 12$

Step 6.2

Simplify $- 4 \cdot 0 - 6 b$ .

Tap for more steps...

Step 6.2.1

Multiply $- 4$ by $0$ .

$0 - 6 b = - 12$

Step 6.2.2

Subtract $6 b$ from $0$ .

$- 6 b = - 12$

Step 6.3

Divide each term in $- 6 b = - 12$ by $- 6$ and simplify.

Tap for more steps...

Step 6.3.1

Divide each term in $- 6 b = - 12$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{- 12}{- 6}$

Step 6.3.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.1

Cancel the common factor of $- 6$ .

Tap for more steps...

Step 6.3.2.1.1

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{- 12}{- 6}$

Step 6.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 12}{- 6}$

Step 6.3.3

Simplify the right side.

Tap for more steps...

Step 6.3.3.1

Divide $- 12$ by $- 6$ .

$b = 2$

Step 7

This is the final solution to the independent system of equations.

$a = 0$

$b = 2$

	$-$	$4$	$a$	$-$	$6$	$b$	$=$	$-$	$1$	$2$
$+$	$1$	$5$	$a$	$+$	$6$	$b$	$=$		$1$	$2$
	$1$	$1$	$a$				$=$			$0$

	$-$	$4$	$a$	$-$	$6$	$b$	$=$	$-$	$1$	$2$
$+$	$1$	$5$	$a$	$+$	$6$	$b$	$=$		$1$	$2$
	$1$	$1$	$a$				$=$			$0$

	$-$	$4$	$a$	$-$	$6$	$b$	$=$	$-$	$1$	$2$
$+$	$1$	$5$	$a$	$+$	$6$	$b$	$=$		$1$	$2$
	$1$	$1$	$a$				$=$			$0$