Finite Math Examples

$- x + 4 y = 17$ , $10 x + 3 y = - 41$ , $- 4 + 9 y = - 55$ , $2 x + y = 11$

Step 1

Solve for $y$ in $- 4 + 9 y = - 55$ .

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Step 1.1

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

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Step 1.1.1

Add $4$ to both sides of the equation.

$9 y = - 55 + 4$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 1.1.2

Add $- 55$ and $4$ .

$9 y = - 51$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

$9 y = - 51$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 1.2

Divide each term in $9 y = - 51$ by $9$ and simplify.

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Step 1.2.1

Divide each term in $9 y = - 51$ by $9$ .

$\frac{9 y}{9} = \frac{- 51}{9}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{9 y}{9} = \frac{- 51}{9}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 51}{9}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

$y = \frac{- 51}{9}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

$y = \frac{- 51}{9}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $- 51$ and $9$ .

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Step 1.2.3.1.1

Factor $3$ out of $- 51$ .

$y = \frac{3 (- 17)}{9}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $3$ out of $9$ .

$y = \frac{3 \cdot - 17}{3 \cdot 3}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 1.2.3.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot - 17}{3 \cdot 3}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 1.2.3.1.2.3

Rewrite the expression.

$y = \frac{- 17}{3}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

$y = \frac{- 17}{3}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

$y = \frac{- 17}{3}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 1.2.3.2

Move the negative in front of the fraction.

$y = - \frac{17}{3}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

$y = - \frac{17}{3}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

$y = - \frac{17}{3}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

$y = - \frac{17}{3}$

$- x + 4 y = 17$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 2

Replace all occurrences of $y$ with $- \frac{17}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $- x + 4 y = 17$ with $- \frac{17}{3}$ .

$- x + 4 (- \frac{17}{3}) = 17$

$y = - \frac{17}{3}$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $4 (- \frac{17}{3})$ .

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Step 2.2.1.1.1

Multiply $- 1$ by $4$ .

$- x - 4 (\frac{17}{3}) = 17$

$y = - \frac{17}{3}$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 2.2.1.1.2

Combine $- 4$ and $\frac{17}{3}$ .

$- x + \frac{- 4 \cdot 17}{3} = 17$

$y = - \frac{17}{3}$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 2.2.1.1.3

Multiply $- 4$ by $17$ .

$- x + \frac{- 68}{3} = 17$

$y = - \frac{17}{3}$

$10 x + 3 y = - 41$

$2 x + y = 11$

$- x + \frac{- 68}{3} = 17$

$y = - \frac{17}{3}$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 2.2.1.2

Move the negative in front of the fraction.

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$10 x + 3 y = - 41$

$2 x + y = 11$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$10 x + 3 y = - 41$

$2 x + y = 11$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$10 x + 3 y = - 41$

$2 x + y = 11$

Step 2.3

Replace all occurrences of $y$ in $10 x + 3 y = - 41$ with $- \frac{17}{3}$ .

$10 x + 3 (- \frac{17}{3}) = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$2 x + y = 11$

Step 2.4

Simplify the left side.

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Step 2.4.1

Cancel the common factor of $3$ .

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Step 2.4.1.1

Move the leading negative in $- \frac{17}{3}$ into the numerator.

$10 x + 3 (\frac{- 17}{3}) = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$2 x + y = 11$

Step 2.4.1.2

Cancel the common factor.

$10 x + 3 (\frac{- 17}{3}) = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$2 x + y = 11$

Step 2.4.1.3

Rewrite the expression.

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$2 x + y = 11$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$2 x + y = 11$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$2 x + y = 11$

Step 2.5

Replace all occurrences of $y$ in $2 x + y = 11$ with $- \frac{17}{3}$ .

$2 x - \frac{17}{3} = 11$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 2.6

Simplify the left side.

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Step 2.6.1

Remove parentheses.

$2 x - \frac{17}{3} = 11$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$2 x - \frac{17}{3} = 11$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$2 x - \frac{17}{3} = 11$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3

Solve for $x$ in $2 x - \frac{17}{3} = 11$ .

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Step 3.1

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

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Step 3.1.1

Add $\frac{17}{3}$ to both sides of the equation.

$2 x = 11 + \frac{17}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.1.2

To write $11$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 x = 11 \cdot \frac{3}{3} + \frac{17}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.1.3

Combine $11$ and $\frac{3}{3}$ .

$2 x = \frac{11 \cdot 3}{3} + \frac{17}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.1.4

Combine the numerators over the common denominator.

$2 x = \frac{11 \cdot 3 + 17}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $11$ by $3$ .

$2 x = \frac{33 + 17}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.1.5.2

Add $33$ and $17$ .

$2 x = \frac{50}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$2 x = \frac{50}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$2 x = \frac{50}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.2

Divide each term in $2 x = \frac{50}{3}$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = \frac{50}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{50}{3}}{2}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{50}{3}}{2}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{50}{3}}{2}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$x = \frac{\frac{50}{3}}{2}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$x = \frac{\frac{50}{3}}{2}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{50}{3} \cdot \frac{1}{2}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.2.3.2

Cancel the common factor of $2$ .

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Step 3.2.3.2.1

Factor $2$ out of $50$ .

$x = \frac{2 (25)}{3} \cdot \frac{1}{2}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.2.3.2.2

Cancel the common factor.

$x = \frac{2 \cdot 25}{3} \cdot \frac{1}{2}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 3.2.3.2.3

Rewrite the expression.

$x = \frac{25}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$x = \frac{25}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$x = \frac{25}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$x = \frac{25}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$x = \frac{25}{3}$

$10 x - 17 = - 41$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 4

Replace all occurrences of $x$ with $\frac{25}{3}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $10 x - 17 = - 41$ with $\frac{25}{3}$ .

$10 (\frac{25}{3}) - 17 = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $10 (\frac{25}{3}) - 17$ .

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Step 4.2.1.1

Multiply $10 (\frac{25}{3})$ .

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Step 4.2.1.1.1

Combine $10$ and $\frac{25}{3}$ .

$\frac{10 \cdot 25}{3} - 17 = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 4.2.1.1.2

Multiply $10$ by $25$ .

$\frac{250}{3} - 17 = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$\frac{250}{3} - 17 = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 4.2.1.2

To write $- 17$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{250}{3} - 17 \cdot \frac{3}{3} = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 4.2.1.3

Combine $- 17$ and $\frac{3}{3}$ .

$\frac{250}{3} + \frac{- 17 \cdot 3}{3} = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{250 - 17 \cdot 3}{3} = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $- 17$ by $3$ .

$\frac{250 - 51}{3} = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 4.2.1.5.2

Subtract $51$ from $250$ .

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$- x - \frac{68}{3} = 17$

$y = - \frac{17}{3}$

Step 4.3

Replace all occurrences of $x$ in $- x - \frac{68}{3} = 17$ with $\frac{25}{3}$ .

$- (\frac{25}{3}) - \frac{68}{3} = 17$

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$y = - \frac{17}{3}$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify $- (\frac{25}{3}) - \frac{68}{3}$ .

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Step 4.4.1.1

Combine the numerators over the common denominator.

$\frac{- 25 - 68}{3} = 17$

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$y = - \frac{17}{3}$

Step 4.4.1.2

Simplify the expression.

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Step 4.4.1.2.1

Subtract $68$ from $- 25$ .

$\frac{- 93}{3} = 17$

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$y = - \frac{17}{3}$

Step 4.4.1.2.2

Divide $- 93$ by $3$ .

$- 31 = 17$

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$y = - \frac{17}{3}$

$- 31 = 17$

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$y = - \frac{17}{3}$

$- 31 = 17$

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$y = - \frac{17}{3}$

$- 31 = 17$

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$y = - \frac{17}{3}$

$- 31 = 17$

$\frac{199}{3} = - 41$

$x = \frac{25}{3}$

$y = - \frac{17}{3}$

Step 5

Since $- 31 = 17$ is not true, there is no solution.

No solution

Step 6