Solve the System of Equations 2x+7y=13 -x+14y=-24

$2 x + 7 y = 13$ $- x + 14 y = - 24$

Step 1

Solve for $x$ in $2 x + 7 y = 13$ .

Subtract $7 y$ from both sides of the equation.

$2 x = 13 - 7 y$

$- x + 14 y = - 24$

Divide each term in $2 x = 13 - 7 y$ by $2$ and simplify.

Divide each term in $2 x = 13 - 7 y$ by $2$ .

$\frac{2 x}{2} = \frac{13}{2} + \frac{- 7 y}{2}$

$- x + 14 y = - 24$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{13}{2} + \frac{- 7 y}{2}$

$- x + 14 y = - 24$

Divide $x$ by $1$ .

$x = \frac{13}{2} + \frac{- 7 y}{2}$

$- x + 14 y = - 24$

$x = \frac{13}{2} + \frac{- 7 y}{2}$

$- x + 14 y = - 24$

$x = \frac{13}{2} + \frac{- 7 y}{2}$

$- x + 14 y = - 24$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{13}{2} - \frac{7 y}{2}$

$- x + 14 y = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$- x + 14 y = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$- x + 14 y = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$- x + 14 y = - 24$

Step 2

Replace all occurrences of $x$ with $\frac{13}{2} - \frac{7 y}{2}$ in each equation.

Replace all occurrences of $x$ in $- x + 14 y = - 24$ with $\frac{13}{2} - \frac{7 y}{2}$ .

$- (\frac{13}{2} - \frac{7 y}{2}) + 14 y = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Simplify the left side.

Simplify $- (\frac{13}{2} - \frac{7 y}{2}) + 14 y$ .

Simplify each term.

Apply the distributive property.

$- \frac{13}{2} + \frac{7 y}{2} + 14 y = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Multiply $- - \frac{7 y}{2}$ .

Multiply $- 1$ by $- 1$ .

$- \frac{13}{2} + 1 (\frac{7 y}{2}) + 14 y = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Multiply $\frac{7 y}{2}$ by $1$ .

$- \frac{13}{2} + \frac{7 y}{2} + 14 y = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$- \frac{13}{2} + \frac{7 y}{2} + 14 y = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$- \frac{13}{2} + \frac{7 y}{2} + 14 y = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

To write $14 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{13}{2} + \frac{7 y}{2} + 14 y \cdot \frac{2}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Combine $14 y$ and $\frac{2}{2}$ .

$- \frac{13}{2} + \frac{7 y}{2} + \frac{14 y \cdot 2}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Combine the numerators over the common denominator.

$- \frac{13}{2} + \frac{7 y + 14 y \cdot 2}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Combine the numerators over the common denominator.

$\frac{- 13 + 7 y + 14 y \cdot 2}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Multiply $2$ by $14$ .

$\frac{- 13 + 7 y + 28 y}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Add $7 y$ and $28 y$ .

$\frac{- 13 + 35 y}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Rewrite $- 13$ as $- 1 (13)$ .

$\frac{- 1 \cdot 13 + 35 y}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Factor $- 1$ out of $35 y$ .

$\frac{- 1 \cdot 13 - (- 35 y)}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Factor $- 1$ out of $- 1 (13) - (- 35 y)$ .

$\frac{- 1 (13 - 35 y)}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Move the negative in front of the fraction.

$- \frac{13 - 35 y}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$- \frac{13 - 35 y}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$- \frac{13 - 35 y}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$- \frac{13 - 35 y}{2} = - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Step 3

Solve for $y$ in $- \frac{13 - 35 y}{2} = - 24$ .

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{13 - 35 y}{2}) = - 2 \cdot - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{13 - 35 y}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{13 - 35 y}{2}$ into the numerator.

$- 2 \frac{- (13 - 35 y)}{2} = - 2 \cdot - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- (13 - 35 y)}{2}) = - 2 \cdot - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Cancel the common factor.

$2 \cdot (- 1 \frac{- (13 - 35 y)}{2}) = - 2 \cdot - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Rewrite the expression.

$13 - 35 y = - 2 \cdot - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$13 - 35 y = - 2 \cdot - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (13 - 35 y) = - 2 \cdot - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Multiply $13 - 35 y$ by $1$ .

$13 - 35 y = - 2 \cdot - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$13 - 35 y = - 2 \cdot - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$13 - 35 y = - 2 \cdot - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

$13 - 35 y = - 2 \cdot - 24$

$x = \frac{13}{2} - \frac{7 y}{2}$

Simplify the right side.

Multiply $- 2$ by $- 24$ .

$13 - 35 y = 48$

$x = \frac{13}{2} - \frac{7 y}{2}$

$13 - 35 y = 48$

$x = \frac{13}{2} - \frac{7 y}{2}$

$13 - 35 y = 48$

$x = \frac{13}{2} - \frac{7 y}{2}$

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

Subtract $13$ from both sides of the equation.

$- 35 y = 48 - 13$

$x = \frac{13}{2} - \frac{7 y}{2}$

Subtract $13$ from $48$ .

$- 35 y = 35$

$x = \frac{13}{2} - \frac{7 y}{2}$

$- 35 y = 35$

$x = \frac{13}{2} - \frac{7 y}{2}$

Divide each term in $- 35 y = 35$ by $- 35$ and simplify.

Divide each term in $- 35 y = 35$ by $- 35$ .

$\frac{- 35 y}{- 35} = \frac{35}{- 35}$

$x = \frac{13}{2} - \frac{7 y}{2}$

Simplify the left side.

Cancel the common factor of $- 35$ .

Cancel the common factor.

$\frac{- 35 y}{- 35} = \frac{35}{- 35}$

$x = \frac{13}{2} - \frac{7 y}{2}$

Divide $y$ by $1$ .

$y = \frac{35}{- 35}$

$x = \frac{13}{2} - \frac{7 y}{2}$

$y = \frac{35}{- 35}$

$x = \frac{13}{2} - \frac{7 y}{2}$

$y = \frac{35}{- 35}$

$x = \frac{13}{2} - \frac{7 y}{2}$

Simplify the right side.

Divide $35$ by $- 35$ .

$y = - 1$

$x = \frac{13}{2} - \frac{7 y}{2}$

$y = - 1$

$x = \frac{13}{2} - \frac{7 y}{2}$

$y = - 1$

$x = \frac{13}{2} - \frac{7 y}{2}$

$y = - 1$

$x = \frac{13}{2} - \frac{7 y}{2}$

Step 4

Replace all occurrences of $y$ with $- 1$ in each equation.

Replace all occurrences of $y$ in $x = \frac{13}{2} - \frac{7 y}{2}$ with $- 1$ .

$x = \frac{13}{2} - \frac{7 (- 1)}{2}$

$y = - 1$

Simplify the right side.

Simplify $\frac{13}{2} - \frac{7 (- 1)}{2}$ .

Combine the numerators over the common denominator.

$x = \frac{13 - 7 \cdot - 1}{2}$

$y = - 1$

Simplify the expression.

Multiply $- 7$ by $- 1$ .

$x = \frac{13 + 7}{2}$

$y = - 1$

Add $13$ and $7$ .

$x = \frac{20}{2}$

$y = - 1$

Divide $20$ by $2$ .

$x = 10$

$y = - 1$

$x = 10$

$y = - 1$

$x = 10$

$y = - 1$

$x = 10$

$y = - 1$

$x = 10$

$y = - 1$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(10, - 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(10, - 1)$

Equation Form:

$x = 10, y = - 1$

Step 7

Solve the System of Equations 2x+7y=13 -x+14y=-24

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