Solve by Substitution 3x+2y=23 , 1/2x-4=y

$3 x + 2 y = 23$ , $\frac{1}{2} x - 4 = y$

Step 1

Solve for $x$ in $3 x + 2 y = 23$ .

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

$3 x = 23 - 2 y$

$\frac{1}{2} x - 4 = y$

Divide each term in $3 x = 23 - 2 y$ by $3$ and simplify.

Divide each term in $3 x = 23 - 2 y$ by $3$ .

$\frac{3 x}{3} = \frac{23}{3} + \frac{- 2 y}{3}$

$\frac{1}{2} x - 4 = y$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{23}{3} + \frac{- 2 y}{3}$

$\frac{1}{2} x - 4 = y$

Divide $x$ by $1$ .

$x = \frac{23}{3} + \frac{- 2 y}{3}$

$\frac{1}{2} x - 4 = y$

$x = \frac{23}{3} + \frac{- 2 y}{3}$

$\frac{1}{2} x - 4 = y$

$x = \frac{23}{3} + \frac{- 2 y}{3}$

$\frac{1}{2} x - 4 = y$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{23}{3} - \frac{2 y}{3}$

$\frac{1}{2} x - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

$\frac{1}{2} x - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

$\frac{1}{2} x - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

$\frac{1}{2} x - 4 = y$

Step 2

Replace all occurrences of $x$ with $\frac{23}{3} - \frac{2 y}{3}$ in each equation.

Replace all occurrences of $x$ in $\frac{1}{2} x - 4 = y$ with $\frac{23}{3} - \frac{2 y}{3}$ .

$\frac{1}{2} \cdot (\frac{23}{3} - \frac{2 y}{3}) - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Simplify the left side.

Simplify $\frac{1}{2} \cdot (\frac{23}{3} - \frac{2 y}{3}) - 4$ .

Simplify each term.

Apply the distributive property.

$\frac{1}{2} \cdot \frac{23}{3} + \frac{1}{2} \cdot (- \frac{2 y}{3}) - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Multiply $\frac{1}{2} \cdot \frac{23}{3}$ .

Multiply $\frac{1}{2}$ by $\frac{23}{3}$ .

$\frac{23}{2 \cdot 3} + \frac{1}{2} \cdot (- \frac{2 y}{3}) - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Multiply $2$ by $3$ .

$\frac{23}{6} + \frac{1}{2} \cdot (- \frac{2 y}{3}) - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

$\frac{23}{6} + \frac{1}{2} \cdot (- \frac{2 y}{3}) - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2 y}{3}$ into the numerator.

$\frac{23}{6} + \frac{1}{2} \cdot \frac{- 2 y}{3} - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Factor $2$ out of $- 2 y$ .

$\frac{23}{6} + \frac{1}{2} \cdot \frac{2 (- y)}{3} - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Cancel the common factor.

$\frac{23}{6} + \frac{1}{2} \cdot \frac{2 (- y)}{3} - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Rewrite the expression.

$\frac{23}{6} + \frac{- y}{3} - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

$\frac{23}{6} + \frac{- y}{3} - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Move the negative in front of the fraction.

$\frac{23}{6} - \frac{y}{3} - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

$\frac{23}{6} - \frac{y}{3} - 4 = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{y}{3} + \frac{23}{6} - 4 \cdot \frac{6}{6} = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Combine $- 4$ and $\frac{6}{6}$ .

$- \frac{y}{3} + \frac{23}{6} + \frac{- 4 \cdot 6}{6} = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Combine the numerators over the common denominator.

$- \frac{y}{3} + \frac{23 - 4 \cdot 6}{6} = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Simplify the numerator.

Multiply $- 4$ by $6$ .

$- \frac{y}{3} + \frac{23 - 24}{6} = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Subtract $24$ from $23$ .

$- \frac{y}{3} + \frac{- 1}{6} = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

$- \frac{y}{3} + \frac{- 1}{6} = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Move the negative in front of the fraction.

$- \frac{y}{3} - \frac{1}{6} = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

$- \frac{y}{3} - \frac{1}{6} = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

$- \frac{y}{3} - \frac{1}{6} = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

$- \frac{y}{3} - \frac{1}{6} = y$

$x = \frac{23}{3} - \frac{2 y}{3}$

Step 3

Solve for $y$ in $- \frac{y}{3} - \frac{1}{6} = y$ .

Move all terms containing $y$ to the left side of the equation.

Subtract $y$ from both sides of the equation.

$- \frac{y}{3} - \frac{1}{6} - y = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

To write $- y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{y}{3} - y \cdot \frac{3}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

Combine $- y$ and $\frac{3}{3}$ .

$- \frac{y}{3} + \frac{- y \cdot 3}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

Combine the numerators over the common denominator.

$\frac{- y - y \cdot 3}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

Simplify each term.

Simplify the numerator.

Factor $y$ out of $- y - y \cdot 3$ .

Factor $y$ out of $- y$ .

$\frac{y \cdot - 1 - y \cdot 3}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

Factor $y$ out of $- y \cdot 3$ .

$\frac{y \cdot - 1 + y (- 1 \cdot 3)}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

Factor $y$ out of $y \cdot - 1 + y (- 1 \cdot 3)$ .

$\frac{y (- 1 - 1 \cdot 3)}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

$\frac{y (- 1 - 1 \cdot 3)}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

Multiply $- 1$ by $3$ .

$\frac{y (- 1 - 3)}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

Subtract $3$ from $- 1$ .

$\frac{y \cdot - 4}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

$\frac{y \cdot - 4}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

Move $- 4$ to the left of $y$ .

$\frac{- 4 \cdot y}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

Move the negative in front of the fraction.

$- \frac{4 y}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

$- \frac{4 y}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

$- \frac{4 y}{3} - \frac{1}{6} = 0$

$x = \frac{23}{3} - \frac{2 y}{3}$

Add $\frac{1}{6}$ to both sides of the equation.

$- \frac{4 y}{3} = \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Multiply both sides of the equation by $- \frac{3}{4}$ .

$- \frac{3}{4} \cdot (- \frac{4 y}{3}) = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{3}{4} (- \frac{4 y}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{- 3}{4} \cdot (- \frac{4 y}{3}) = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Move the leading negative in $- \frac{4 y}{3}$ into the numerator.

$\frac{- 3}{4} \cdot \frac{- 4 y}{3} = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{4} \cdot \frac{- 4 y}{3} = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Cancel the common factor.

$\frac{3 \cdot - 1}{4} \cdot \frac{- 4 y}{3} = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Rewrite the expression.

$\frac{- 1}{4} \cdot (- 4 y) = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

$\frac{- 1}{4} \cdot (- 4 y) = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Cancel the common factor of $4$ .

Factor $4$ out of $- 4 y$ .

$\frac{- 1}{4} \cdot (4 (- y)) = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Cancel the common factor.

$\frac{- 1}{4} \cdot (4 (- y)) = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Rewrite the expression.

$y = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

$y = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Multiply $y$ by $1$ .

$y = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

$y = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

$y = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

$y = - \frac{3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Simplify the right side.

Simplify $- \frac{3}{4} \cdot \frac{1}{6}$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$y = \frac{- 3}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Factor $3$ out of $- 3$ .

$y = \frac{3 (- 1)}{4} \cdot \frac{1}{6}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Factor $3$ out of $6$ .

$y = \frac{3 \cdot - 1}{4} \cdot \frac{1}{3 \cdot 2}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Cancel the common factor.

$y = \frac{3 \cdot - 1}{4} \cdot \frac{1}{3 \cdot 2}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Rewrite the expression.

$y = \frac{- 1}{4} \cdot \frac{1}{2}$

$x = \frac{23}{3} - \frac{2 y}{3}$

$y = \frac{- 1}{4} \cdot \frac{1}{2}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Multiply $\frac{- 1}{4}$ by $\frac{1}{2}$ .

$y = \frac{- 1}{4 \cdot 2}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Simplify the expression.

Multiply $4$ by $2$ .

$y = \frac{- 1}{8}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Move the negative in front of the fraction.

$y = - \frac{1}{8}$

$x = \frac{23}{3} - \frac{2 y}{3}$

$y = - \frac{1}{8}$

$x = \frac{23}{3} - \frac{2 y}{3}$

$y = - \frac{1}{8}$

$x = \frac{23}{3} - \frac{2 y}{3}$

$y = - \frac{1}{8}$

$x = \frac{23}{3} - \frac{2 y}{3}$

$y = - \frac{1}{8}$

$x = \frac{23}{3} - \frac{2 y}{3}$

$y = - \frac{1}{8}$

$x = \frac{23}{3} - \frac{2 y}{3}$

Step 4

Replace all occurrences of $y$ with $- \frac{1}{8}$ in each equation.

Replace all occurrences of $y$ in $x = \frac{23}{3} - \frac{2 y}{3}$ with $- \frac{1}{8}$ .

$x = \frac{23}{3} - \frac{2 (- \frac{1}{8})}{3}$

$y = - \frac{1}{8}$

Simplify the right side.

Simplify $\frac{23}{3} - \frac{2 (- \frac{1}{8})}{3}$ .

Combine the numerators over the common denominator.

$x = \frac{23 - 2 (- \frac{1}{8})}{3}$

$y = - \frac{1}{8}$

Simplify each term.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$x = \frac{23 - 2 (\frac{- 1}{8})}{3}$

$y = - \frac{1}{8}$

Factor $2$ out of $- 2$ .

$x = \frac{23 + 2 (- 1) (\frac{- 1}{8})}{3}$

$y = - \frac{1}{8}$

Factor $2$ out of $8$ .

$x = \frac{23 + 2 \cdot (- 1 \frac{- 1}{2 \cdot 4})}{3}$

$y = - \frac{1}{8}$

Cancel the common factor.

$x = \frac{23 + 2 \cdot (- 1 \frac{- 1}{2 \cdot 4})}{3}$

$y = - \frac{1}{8}$

Rewrite the expression.

$x = \frac{23 - 1 (\frac{- 1}{4})}{3}$

$y = - \frac{1}{8}$

$x = \frac{23 - 1 (\frac{- 1}{4})}{3}$

$y = - \frac{1}{8}$

Move the negative in front of the fraction.

$x = \frac{23 - 1 (- \frac{1}{4})}{3}$

$y = - \frac{1}{8}$

Multiply $- 1 (- \frac{1}{4})$ .

Multiply $- 1$ by $- 1$ .

$x = \frac{23 + 1 (\frac{1}{4})}{3}$

$y = - \frac{1}{8}$

Multiply $\frac{1}{4}$ by $1$ .

$x = \frac{23 + \frac{1}{4}}{3}$

$y = - \frac{1}{8}$

$x = \frac{23 + \frac{1}{4}}{3}$

$y = - \frac{1}{8}$

$x = \frac{23 + \frac{1}{4}}{3}$

$y = - \frac{1}{8}$

To write $23$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{23 \cdot \frac{4}{4} + \frac{1}{4}}{3}$

$y = - \frac{1}{8}$

Combine $23$ and $\frac{4}{4}$ .

$x = \frac{\frac{23 \cdot 4}{4} + \frac{1}{4}}{3}$

$y = - \frac{1}{8}$

Combine the numerators over the common denominator.

$x = \frac{\frac{23 \cdot 4 + 1}{4}}{3}$

$y = - \frac{1}{8}$

Simplify the numerator.

Multiply $23$ by $4$ .

$x = \frac{\frac{92 + 1}{4}}{3}$

$y = - \frac{1}{8}$

Add $92$ and $1$ .

$x = \frac{\frac{93}{4}}{3}$

$y = - \frac{1}{8}$

$x = \frac{\frac{93}{4}}{3}$

$y = - \frac{1}{8}$

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{93}{4} \cdot \frac{1}{3}$

$y = - \frac{1}{8}$

Cancel the common factor of $3$ .

Factor $3$ out of $93$ .

$x = \frac{3 (31)}{4} \cdot \frac{1}{3}$

$y = - \frac{1}{8}$

Cancel the common factor.

$x = \frac{3 \cdot 31}{4} \cdot \frac{1}{3}$

$y = - \frac{1}{8}$

Rewrite the expression.

$x = \frac{31}{4}$

$y = - \frac{1}{8}$

$x = \frac{31}{4}$

$y = - \frac{1}{8}$

$x = \frac{31}{4}$

$y = - \frac{1}{8}$

$x = \frac{31}{4}$

$y = - \frac{1}{8}$

$x = \frac{31}{4}$

$y = - \frac{1}{8}$

Step 5

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

$(\frac{31}{4}, - \frac{1}{8})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{31}{4}, - \frac{1}{8})$

Equation Form:

$x = \frac{31}{4}, y = - \frac{1}{8}$

Step 7

Solve by Substitution 3x+2y=23 , 1/2x-4=y

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