Solve the System of Equations -7x+5y=14 and 6x-7y=7

$- 7 x + 5 y = 14$ and $6 x - 7 y = 7$

Step 1

Solve for $x$ in $- 7 x + 5 y = 14$ .

Subtract $5 y$ from both sides of the equation.

$- 7 x = 14 - 5 y$

$6 x - 7 y = 7$

Divide each term in $- 7 x = 14 - 5 y$ by $- 7$ and simplify.

Divide each term in $- 7 x = 14 - 5 y$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{14}{- 7} + \frac{- 5 y}{- 7}$

$6 x - 7 y = 7$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{14}{- 7} + \frac{- 5 y}{- 7}$

$6 x - 7 y = 7$

Divide $x$ by $1$ .

$x = \frac{14}{- 7} + \frac{- 5 y}{- 7}$

$6 x - 7 y = 7$

$x = \frac{14}{- 7} + \frac{- 5 y}{- 7}$

$6 x - 7 y = 7$

$x = \frac{14}{- 7} + \frac{- 5 y}{- 7}$

$6 x - 7 y = 7$

Simplify the right side.

Simplify each term.

Divide $14$ by $- 7$ .

$x = - 2 + \frac{- 5 y}{- 7}$

$6 x - 7 y = 7$

Dividing two negative values results in a positive value.

$x = - 2 + \frac{5 y}{7}$

$6 x - 7 y = 7$

$x = - 2 + \frac{5 y}{7}$

$6 x - 7 y = 7$

$x = - 2 + \frac{5 y}{7}$

$6 x - 7 y = 7$

$x = - 2 + \frac{5 y}{7}$

$6 x - 7 y = 7$

$x = - 2 + \frac{5 y}{7}$

$6 x - 7 y = 7$

Step 2

Replace all occurrences of $x$ with $- 2 + \frac{5 y}{7}$ in each equation.

Replace all occurrences of $x$ in $6 x - 7 y = 7$ with $- 2 + \frac{5 y}{7}$ .

$6 (- 2 + \frac{5 y}{7}) - 7 y = 7$

$x = - 2 + \frac{5 y}{7}$

Simplify the left side.

Simplify $6 (- 2 + \frac{5 y}{7}) - 7 y$ .

Simplify each term.

Apply the distributive property.

$6 \cdot - 2 + 6 (\frac{5 y}{7}) - 7 y = 7$

$x = - 2 + \frac{5 y}{7}$

Multiply $6$ by $- 2$ .

$- 12 + 6 (\frac{5 y}{7}) - 7 y = 7$

$x = - 2 + \frac{5 y}{7}$

Multiply $6 \frac{5 y}{7}$ .

Combine $6$ and $\frac{5 y}{7}$ .

$- 12 + \frac{6 (5 y)}{7} - 7 y = 7$

$x = - 2 + \frac{5 y}{7}$

Multiply $5$ by $6$ .

$- 12 + \frac{30 y}{7} - 7 y = 7$

$x = - 2 + \frac{5 y}{7}$

$- 12 + \frac{30 y}{7} - 7 y = 7$

$x = - 2 + \frac{5 y}{7}$

$- 12 + \frac{30 y}{7} - 7 y = 7$

$x = - 2 + \frac{5 y}{7}$

To write $- 7 y$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- 12 + \frac{30 y}{7} - 7 y \cdot \frac{7}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

Simplify terms.

Combine $- 7 y$ and $\frac{7}{7}$ .

$- 12 + \frac{30 y}{7} + \frac{- 7 y \cdot 7}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

Combine the numerators over the common denominator.

$- 12 + \frac{30 y - 7 y \cdot 7}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

$- 12 + \frac{30 y - 7 y \cdot 7}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

Simplify each term.

Simplify the numerator.

Factor $y$ out of $30 y - 7 y \cdot 7$ .

Factor $y$ out of $30 y$ .

$- 12 + \frac{y \cdot 30 - 7 y \cdot 7}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

Factor $y$ out of $- 7 y \cdot 7$ .

$- 12 + \frac{y \cdot 30 + y (- 7 \cdot 7)}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

Factor $y$ out of $y \cdot 30 + y (- 7 \cdot 7)$ .

$- 12 + \frac{y (30 - 7 \cdot 7)}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

$- 12 + \frac{y (30 - 7 \cdot 7)}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

Multiply $- 7$ by $7$ .

$- 12 + \frac{y (30 - 49)}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

Subtract $49$ from $30$ .

$- 12 + \frac{y \cdot - 19}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

$- 12 + \frac{y \cdot - 19}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

Move $- 19$ to the left of $y$ .

$- 12 + \frac{- 19 \cdot y}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

Move the negative in front of the fraction.

$- 12 - \frac{19 y}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

$- 12 - \frac{19 y}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

$- 12 - \frac{19 y}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

$- 12 - \frac{19 y}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

$- 12 - \frac{19 y}{7} = 7$

$x = - 2 + \frac{5 y}{7}$

Step 3

Solve for $y$ in $- 12 - \frac{19 y}{7} = 7$ .

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

Add $12$ to both sides of the equation.

$- \frac{19 y}{7} = 7 + 12$

$x = - 2 + \frac{5 y}{7}$

Add $7$ and $12$ .

$- \frac{19 y}{7} = 19$

$x = - 2 + \frac{5 y}{7}$

$- \frac{19 y}{7} = 19$

$x = - 2 + \frac{5 y}{7}$

Multiply both sides of the equation by $- \frac{7}{19}$ .

$- \frac{7}{19} \cdot (- \frac{19 y}{7}) = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{7}{19} (- \frac{19 y}{7})$ .

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{7}{19}$ into the numerator.

$\frac{- 7}{19} \cdot (- \frac{19 y}{7}) = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Move the leading negative in $- \frac{19 y}{7}$ into the numerator.

$\frac{- 7}{19} \cdot \frac{- 19 y}{7} = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Factor $7$ out of $- 7$ .

$\frac{7 (- 1)}{19} \cdot \frac{- 19 y}{7} = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Cancel the common factor.

$\frac{7 \cdot - 1}{19} \cdot \frac{- 19 y}{7} = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Rewrite the expression.

$\frac{- 1}{19} \cdot (- 19 y) = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

$\frac{- 1}{19} \cdot (- 19 y) = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Cancel the common factor of $19$ .

Factor $19$ out of $- 19 y$ .

$\frac{- 1}{19} \cdot (19 (- y)) = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Cancel the common factor.

$\frac{- 1}{19} \cdot (19 (- y)) = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Rewrite the expression.

$y = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

$y = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Multiply $y$ by $1$ .

$y = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

$y = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

$y = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

$y = - \frac{7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Simplify the right side.

Cancel the common factor of $19$ .

Move the leading negative in $- \frac{7}{19}$ into the numerator.

$y = \frac{- 7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Cancel the common factor.

$y = \frac{- 7}{19} \cdot 19$

$x = - 2 + \frac{5 y}{7}$

Rewrite the expression.

$y = - 7$

$x = - 2 + \frac{5 y}{7}$

$y = - 7$

$x = - 2 + \frac{5 y}{7}$

$y = - 7$

$x = - 2 + \frac{5 y}{7}$

$y = - 7$

$x = - 2 + \frac{5 y}{7}$

$y = - 7$

$x = - 2 + \frac{5 y}{7}$

Step 4

Replace all occurrences of $y$ with $- 7$ in each equation.

Replace all occurrences of $y$ in $x = - 2 + \frac{5 y}{7}$ with $- 7$ .

$x = - 2 + \frac{5 (- 7)}{7}$

$y = - 7$

Simplify the right side.

Simplify $- 2 + \frac{5 (- 7)}{7}$ .

Multiply $5$ by $- 7$ .

$x = - 2 + \frac{- 35}{7}$

$y = - 7$

Divide $- 35$ by $7$ .

$x = - 2 - 5$

$y = - 7$

Subtract $5$ from $- 2$ .

$x = - 7$

$y = - 7$

$x = - 7$

$y = - 7$

$x = - 7$

$y = - 7$

$x = - 7$

$y = - 7$

Step 5

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

$(- 7, - 7)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 7, - 7)$

Equation Form:

$x = - 7, y = - 7$

Step 7

Solve the System of Equations -7x+5y=14 and 6x-7y=7

Related Questions

Support LCMGCF.com by disabling your adblocker.