Solve by Substitution 5x+2y=21 , -2x+6y=-34

$5 x + 2 y = 21$ , $- 2 x + 6 y = - 34$

Step 1

Solve for $x$ in $5 x + 2 y = 21$ .

Subtract $2 y$ from both sides of the equation.

$5 x = 21 - 2 y$

$- 2 x + 6 y = - 34$

Divide each term in $5 x = 21 - 2 y$ by $5$ and simplify.

Divide each term in $5 x = 21 - 2 y$ by $5$ .

$\frac{5 x}{5} = \frac{21}{5} + \frac{- 2 y}{5}$

$- 2 x + 6 y = - 34$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{21}{5} + \frac{- 2 y}{5}$

$- 2 x + 6 y = - 34$

Divide $x$ by $1$ .

$x = \frac{21}{5} + \frac{- 2 y}{5}$

$- 2 x + 6 y = - 34$

$x = \frac{21}{5} + \frac{- 2 y}{5}$

$- 2 x + 6 y = - 34$

$x = \frac{21}{5} + \frac{- 2 y}{5}$

$- 2 x + 6 y = - 34$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{21}{5} - \frac{2 y}{5}$

$- 2 x + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$- 2 x + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$- 2 x + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$- 2 x + 6 y = - 34$

Step 2

Replace all occurrences of $x$ with $\frac{21}{5} - \frac{2 y}{5}$ in each equation.

Replace all occurrences of $x$ in $- 2 x + 6 y = - 34$ with $\frac{21}{5} - \frac{2 y}{5}$ .

$- 2 (\frac{21}{5} - \frac{2 y}{5}) + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Simplify the left side.

Simplify $- 2 (\frac{21}{5} - \frac{2 y}{5}) + 6 y$ .

Simplify each term.

Apply the distributive property.

$- 2 (\frac{21}{5}) - 2 (- \frac{2 y}{5}) + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Multiply $- 2 (\frac{21}{5})$ .

Combine $- 2$ and $\frac{21}{5}$ .

$\frac{- 2 \cdot 21}{5} - 2 (- \frac{2 y}{5}) + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Multiply $- 2$ by $21$ .

$\frac{- 42}{5} - 2 (- \frac{2 y}{5}) + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$\frac{- 42}{5} - 2 (- \frac{2 y}{5}) + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Multiply $- 2 (- \frac{2 y}{5})$ .

Multiply $- 1$ by $- 2$ .

$\frac{- 42}{5} + 2 (\frac{2 y}{5}) + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Combine $2$ and $\frac{2 y}{5}$ .

$\frac{- 42}{5} + \frac{2 (2 y)}{5} + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Multiply $2$ by $2$ .

$\frac{- 42}{5} + \frac{4 y}{5} + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$\frac{- 42}{5} + \frac{4 y}{5} + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Move the negative in front of the fraction.

$- \frac{42}{5} + \frac{4 y}{5} + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$- \frac{42}{5} + \frac{4 y}{5} + 6 y = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

To write $6 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{42}{5} + \frac{4 y}{5} + 6 y \cdot \frac{5}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

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

$- \frac{42}{5} + \frac{4 y}{5} + \frac{6 y \cdot 5}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Combine the numerators over the common denominator.

$- \frac{42}{5} + \frac{4 y + 6 y \cdot 5}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Combine the numerators over the common denominator.

$\frac{- 42 + 4 y + 6 y \cdot 5}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Multiply $5$ by $6$ .

$\frac{- 42 + 4 y + 30 y}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Add $4 y$ and $30 y$ .

$\frac{- 42 + 34 y}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Factor $2$ out of $- 42 + 34 y$ .

Factor $2$ out of $- 42$ .

$\frac{2 (- 21) + 34 y}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Factor $2$ out of $34 y$ .

$\frac{2 (- 21) + 2 (17 y)}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Factor $2$ out of $2 (- 21) + 2 (17 y)$ .

$\frac{2 (- 21 + 17 y)}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$\frac{2 (- 21 + 17 y)}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Rewrite $- 21$ as $- 1 (21)$ .

$\frac{2 (- 1 \cdot 21 + 17 y)}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Factor $- 1$ out of $17 y$ .

$\frac{2 (- 1 \cdot 21 - (- 17 y))}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Factor $- 1$ out of $- 1 (21) - (- 17 y)$ .

$\frac{2 (- 1 (21 - 17 y))}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Move the negative in front of the fraction.

$- \frac{2 (21 - 17 y)}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$- \frac{2 (21 - 17 y)}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$- \frac{2 (21 - 17 y)}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$- \frac{2 (21 - 17 y)}{5} = - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Step 3

Solve for $y$ in $- \frac{2 (21 - 17 y)}{5} = - 34$ .

Multiply both sides of the equation by $- \frac{5}{2}$ .

$- \frac{5}{2} \cdot (- \frac{2 (21 - 17 y)}{5}) = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{5}{2} (- \frac{2 (21 - 17 y)}{5})$ .

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$\frac{- 5}{2} \cdot (- \frac{2 (21 - 17 y)}{5}) = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Move the leading negative in $- \frac{2 (21 - 17 y)}{5}$ into the numerator.

$\frac{- 5}{2} \cdot \frac{- 2 (21 - 17 y)}{5} = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{2} \cdot \frac{- 2 (21 - 17 y)}{5} = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Cancel the common factor.

$\frac{5 \cdot - 1}{2} \cdot \frac{- 2 (21 - 17 y)}{5} = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Rewrite the expression.

$\frac{- 1}{2} \cdot (- 2 (21 - 17 y)) = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$\frac{- 1}{2} \cdot (- 2 (21 - 17 y)) = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2 (21 - 17 y)$ .

$\frac{- 1}{2} \cdot (2 (- (21 - 17 y))) = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Cancel the common factor.

$\frac{- 1}{2} \cdot (2 (- (21 - 17 y))) = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Rewrite the expression.

$21 - 17 y = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$21 - 17 y = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (21 - 17 y) = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Multiply $21 - 17 y$ by $1$ .

$21 - 17 y = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$21 - 17 y = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$21 - 17 y = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

$21 - 17 y = - \frac{5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Simplify the right side.

Simplify $- \frac{5}{2} \cdot - 34$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$21 - 17 y = \frac{- 5}{2} \cdot - 34$

$x = \frac{21}{5} - \frac{2 y}{5}$

Factor $2$ out of $- 34$ .

$21 - 17 y = \frac{- 5}{2} \cdot (2 (- 17))$

$x = \frac{21}{5} - \frac{2 y}{5}$

Cancel the common factor.

$21 - 17 y = \frac{- 5}{2} \cdot (2 \cdot - 17)$

$x = \frac{21}{5} - \frac{2 y}{5}$

Rewrite the expression.

$21 - 17 y = - 5 \cdot - 17$

$x = \frac{21}{5} - \frac{2 y}{5}$

$21 - 17 y = - 5 \cdot - 17$

$x = \frac{21}{5} - \frac{2 y}{5}$

Multiply $- 5$ by $- 17$ .

$21 - 17 y = 85$

$x = \frac{21}{5} - \frac{2 y}{5}$

$21 - 17 y = 85$

$x = \frac{21}{5} - \frac{2 y}{5}$

$21 - 17 y = 85$

$x = \frac{21}{5} - \frac{2 y}{5}$

$21 - 17 y = 85$

$x = \frac{21}{5} - \frac{2 y}{5}$

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

Subtract $21$ from both sides of the equation.

$- 17 y = 85 - 21$

$x = \frac{21}{5} - \frac{2 y}{5}$

Subtract $21$ from $85$ .

$- 17 y = 64$

$x = \frac{21}{5} - \frac{2 y}{5}$

$- 17 y = 64$

$x = \frac{21}{5} - \frac{2 y}{5}$

Divide each term in $- 17 y = 64$ by $- 17$ and simplify.

Divide each term in $- 17 y = 64$ by $- 17$ .

$\frac{- 17 y}{- 17} = \frac{64}{- 17}$

$x = \frac{21}{5} - \frac{2 y}{5}$

Simplify the left side.

Cancel the common factor of $- 17$ .

Cancel the common factor.

$\frac{- 17 y}{- 17} = \frac{64}{- 17}$

$x = \frac{21}{5} - \frac{2 y}{5}$

Divide $y$ by $1$ .

$y = \frac{64}{- 17}$

$x = \frac{21}{5} - \frac{2 y}{5}$

$y = \frac{64}{- 17}$

$x = \frac{21}{5} - \frac{2 y}{5}$

$y = \frac{64}{- 17}$

$x = \frac{21}{5} - \frac{2 y}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{64}{17}$

$x = \frac{21}{5} - \frac{2 y}{5}$

$y = - \frac{64}{17}$

$x = \frac{21}{5} - \frac{2 y}{5}$

$y = - \frac{64}{17}$

$x = \frac{21}{5} - \frac{2 y}{5}$

$y = - \frac{64}{17}$

$x = \frac{21}{5} - \frac{2 y}{5}$

Step 4

Replace all occurrences of $y$ with $- \frac{64}{17}$ in each equation.

Replace all occurrences of $y$ in $x = \frac{21}{5} - \frac{2 y}{5}$ with $- \frac{64}{17}$ .

$x = \frac{21}{5} - \frac{2 (- \frac{64}{17})}{5}$

$y = - \frac{64}{17}$

Simplify the right side.

Simplify $\frac{21}{5} - \frac{2 (- \frac{64}{17})}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{21 - 2 (- \frac{64}{17})}{5}$

$y = - \frac{64}{17}$

Multiply $- 2 (- \frac{64}{17})$ .

Multiply $- 1$ by $- 2$ .

$x = \frac{21 + 2 (\frac{64}{17})}{5}$

$y = - \frac{64}{17}$

Combine $2$ and $\frac{64}{17}$ .

$x = \frac{21 + \frac{2 \cdot 64}{17}}{5}$

$y = - \frac{64}{17}$

Multiply $2$ by $64$ .

$x = \frac{21 + \frac{128}{17}}{5}$

$y = - \frac{64}{17}$

$x = \frac{21 + \frac{128}{17}}{5}$

$y = - \frac{64}{17}$

To write $21$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$x = \frac{21 \cdot \frac{17}{17} + \frac{128}{17}}{5}$

$y = - \frac{64}{17}$

Combine $21$ and $\frac{17}{17}$ .

$x = \frac{\frac{21 \cdot 17}{17} + \frac{128}{17}}{5}$

$y = - \frac{64}{17}$

Combine the numerators over the common denominator.

$x = \frac{\frac{21 \cdot 17 + 128}{17}}{5}$

$y = - \frac{64}{17}$

Simplify the numerator.

Multiply $21$ by $17$ .

$x = \frac{\frac{357 + 128}{17}}{5}$

$y = - \frac{64}{17}$

Add $357$ and $128$ .

$x = \frac{\frac{485}{17}}{5}$

$y = - \frac{64}{17}$

$x = \frac{\frac{485}{17}}{5}$

$y = - \frac{64}{17}$

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{485}{17} \cdot \frac{1}{5}$

$y = - \frac{64}{17}$

Cancel the common factor of $5$ .

Factor $5$ out of $485$ .

$x = \frac{5 (97)}{17} \cdot \frac{1}{5}$

$y = - \frac{64}{17}$

Cancel the common factor.

$x = \frac{5 \cdot 97}{17} \cdot \frac{1}{5}$

$y = - \frac{64}{17}$

Rewrite the expression.

$x = \frac{97}{17}$

$y = - \frac{64}{17}$

$x = \frac{97}{17}$

$y = - \frac{64}{17}$

$x = \frac{97}{17}$

$y = - \frac{64}{17}$

$x = \frac{97}{17}$

$y = - \frac{64}{17}$

$x = \frac{97}{17}$

$y = - \frac{64}{17}$

Step 5

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

$(\frac{97}{17}, - \frac{64}{17})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{97}{17}, - \frac{64}{17})$

Equation Form:

$x = \frac{97}{17}, y = - \frac{64}{17}$

Step 7

Solve by Substitution 5x+2y=21 , -2x+6y=-34

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