Finite Math Examples

$3 x + y = - 2$ , $2 x^{2} - y = 0$

Step 1

Subtract $3 x$ from both sides of the equation.

$y = - 2 - 3 x$

$2 x^{2} - y = 0$

Step 2

Replace all occurrences of $y$ with $- 2 - 3 x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $2 x^{2} - y = 0$ with $- 2 - 3 x$ .

$2 x^{2} - (- 2 - 3 x) = 0$

$y = - 2 - 3 x$

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

Apply the distributive property.

$2 x^{2} + 2 - (- 3 x) = 0$

$y = - 2 - 3 x$

Step 2.2.1.2

Multiply $- 1$ by $- 2$ .

$2 x^{2} + 2 - (- 3 x) = 0$

$y = - 2 - 3 x$

Step 2.2.1.3

Multiply $- 3$ by $- 1$ .

$2 x^{2} + 2 + 3 x = 0$

$y = - 2 - 3 x$

$2 x^{2} + 2 + 3 x = 0$

$y = - 2 - 3 x$

$2 x^{2} + 2 + 3 x = 0$

$y = - 2 - 3 x$

$2 x^{2} + 2 + 3 x = 0$

$y = - 2 - 3 x$

Step 3

Solve for $x$ in $2 x^{2} + 2 + 3 x = 0$ .

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Step 3.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$y = - 2 - 3 x$

Step 3.2

Substitute the values $a = 2$ , $b = 3$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (2 \cdot 2)}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot 2}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.3.1.2

Multiply $- 4 \cdot 2 \cdot 2$ .

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Step 3.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot 2}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.3.1.2.2

Multiply $- 8$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 16}}{2 \cdot 2}$

$y = - 2 - 3 x$

$x = \frac{- 3 \pm \sqrt{9 - 16}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.3.1.3

Subtract $16$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.3.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.3.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \sqrt{7}}{2 \cdot 2}$

$y = - 2 - 3 x$

$x = \frac{- 3 \pm i \sqrt{7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.3.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm i \sqrt{7}}{4}$

$y = - 2 - 3 x$

$x = \frac{- 3 \pm i \sqrt{7}}{4}$

$y = - 2 - 3 x$

Step 3.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot 2}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.4.1.2

Multiply $- 4 \cdot 2 \cdot 2$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot 2}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.4.1.2.2

Multiply $- 8$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 16}}{2 \cdot 2}$

$y = - 2 - 3 x$

$x = \frac{- 3 \pm \sqrt{9 - 16}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.4.1.3

Subtract $16$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.4.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.4.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \sqrt{7}}{2 \cdot 2}$

$y = - 2 - 3 x$

$x = \frac{- 3 \pm i \sqrt{7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.4.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm i \sqrt{7}}{4}$

$y = - 2 - 3 x$

Step 3.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + i \sqrt{7}}{4}$

$y = - 2 - 3 x$

Step 3.4.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + i \sqrt{7}}{4}$

$y = - 2 - 3 x$

Step 3.4.5

Factor $- 1$ out of $i \sqrt{7}$ .

$x = \frac{- 1 \cdot 3 - (- i \sqrt{7})}{4}$

$y = - 2 - 3 x$

Step 3.4.6

Factor $- 1$ out of $- 1 (3) - (- i \sqrt{7})$ .

$x = \frac{- 1 (3 - i \sqrt{7})}{4}$

$y = - 2 - 3 x$

Step 3.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - i \sqrt{7}}{4}$

$y = - 2 - 3 x$

$x = - \frac{3 - i \sqrt{7}}{4}$

$y = - 2 - 3 x$

Step 3.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot 2}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.5.1.2

Multiply $- 4 \cdot 2 \cdot 2$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot 2}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.5.1.2.2

Multiply $- 8$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 16}}{2 \cdot 2}$

$y = - 2 - 3 x$

$x = \frac{- 3 \pm \sqrt{9 - 16}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.5.1.3

Subtract $16$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.5.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.5.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \sqrt{7}}{2 \cdot 2}$

$y = - 2 - 3 x$

$x = \frac{- 3 \pm i \sqrt{7}}{2 \cdot 2}$

$y = - 2 - 3 x$

Step 3.5.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm i \sqrt{7}}{4}$

$y = - 2 - 3 x$

Step 3.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - i \sqrt{7}}{4}$

$y = - 2 - 3 x$

Step 3.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - i \sqrt{7}}{4}$

$y = - 2 - 3 x$

Step 3.5.5

Factor $- 1$ out of $- i \sqrt{7}$ .

$x = \frac{- 1 \cdot 3 - (i \sqrt{7})}{4}$

$y = - 2 - 3 x$

Step 3.5.6

Factor $- 1$ out of $- 1 (3) - (i \sqrt{7})$ .

$x = \frac{- 1 (3 + i \sqrt{7})}{4}$

$y = - 2 - 3 x$

Step 3.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + i \sqrt{7}}{4}$

$y = - 2 - 3 x$

$x = - \frac{3 + i \sqrt{7}}{4}$

$y = - 2 - 3 x$

Step 3.6

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{7}}{4}, - \frac{3 + i \sqrt{7}}{4}$

$y = - 2 - 3 x$

$x = - \frac{3 - i \sqrt{7}}{4}, - \frac{3 + i \sqrt{7}}{4}$

$y = - 2 - 3 x$

Step 4

Replace all occurrences of $x$ with $- \frac{3 - i \sqrt{7}}{4}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - 2 - 3 x$ with $- \frac{3 - i \sqrt{7}}{4}$ .

$y = - 2 - 3 (- \frac{3 - i \sqrt{7}}{4})$

$x = - \frac{3 - i \sqrt{7}}{4}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 2 - 3 (- \frac{3 - i \sqrt{7}}{4})$ .

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Step 4.2.1.1

Multiply $- 3 (- \frac{3 - i \sqrt{7}}{4})$ .

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Step 4.2.1.1.1

Multiply $- 1$ by $- 3$ .

$y = - 2 + 3 (\frac{3 - i \sqrt{7}}{4})$

$x = - \frac{3 - i \sqrt{7}}{4}$

Step 4.2.1.1.2

Combine $3$ and $\frac{3 - i \sqrt{7}}{4}$ .

$y = - 2 + \frac{3 (3 - i \sqrt{7})}{4}$

$x = - \frac{3 - i \sqrt{7}}{4}$

$y = - 2 + \frac{3 (3 - i \sqrt{7})}{4}$

$x = - \frac{3 - i \sqrt{7}}{4}$

Step 4.2.1.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - 2 \cdot \frac{4}{4} + \frac{3 (3 - i \sqrt{7})}{4}$

$x = - \frac{3 - i \sqrt{7}}{4}$

Step 4.2.1.3

Combine $- 2$ and $\frac{4}{4}$ .

$y = \frac{- 2 \cdot 4}{4} + \frac{3 (3 - i \sqrt{7})}{4}$

$x = - \frac{3 - i \sqrt{7}}{4}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$y = \frac{1 - 3 i \sqrt{7}}{4}$

$x = - \frac{3 - i \sqrt{7}}{4}$

$y = \frac{1 - 3 i \sqrt{7}}{4}$

$x = - \frac{3 - i \sqrt{7}}{4}$

$y = \frac{1 - 3 i \sqrt{7}}{4}$

$x = - \frac{3 - i \sqrt{7}}{4}$

$y = \frac{1 - 3 i \sqrt{7}}{4}$

$x = - \frac{3 - i \sqrt{7}}{4}$

Step 5

Replace all occurrences of $x$ with $- \frac{3 + i \sqrt{7}}{4}$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = - 2 - 3 x$ with $- \frac{3 + i \sqrt{7}}{4}$ .

$y = - 2 - 3 (- \frac{3 + i \sqrt{7}}{4})$

$x = - \frac{3 + i \sqrt{7}}{4}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 2 - 3 (- \frac{3 + i \sqrt{7}}{4})$ .

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Step 5.2.1.1

Multiply $- 3 (- \frac{3 + i \sqrt{7}}{4})$ .

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Step 5.2.1.1.1

Multiply $- 1$ by $- 3$ .

$y = - 2 + 3 (\frac{3 + i \sqrt{7}}{4})$

$x = - \frac{3 + i \sqrt{7}}{4}$

Step 5.2.1.1.2

Combine $3$ and $\frac{3 + i \sqrt{7}}{4}$ .

$y = - 2 + \frac{3 (3 + i \sqrt{7})}{4}$

$x = - \frac{3 + i \sqrt{7}}{4}$

$y = - 2 + \frac{3 (3 + i \sqrt{7})}{4}$

$x = - \frac{3 + i \sqrt{7}}{4}$

Step 5.2.1.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - 2 \cdot \frac{4}{4} + \frac{3 (3 + i \sqrt{7})}{4}$

$x = - \frac{3 + i \sqrt{7}}{4}$

Step 5.2.1.3

Combine $- 2$ and $\frac{4}{4}$ .

$y = \frac{- 2 \cdot 4}{4} + \frac{3 (3 + i \sqrt{7})}{4}$

$x = - \frac{3 + i \sqrt{7}}{4}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$y = \frac{1 + 3 i \sqrt{7}}{4}$

$x = - \frac{3 + i \sqrt{7}}{4}$

$y = \frac{1 + 3 i \sqrt{7}}{4}$

$x = - \frac{3 + i \sqrt{7}}{4}$

$y = \frac{1 + 3 i \sqrt{7}}{4}$

$x = - \frac{3 + i \sqrt{7}}{4}$

$y = \frac{1 + 3 i \sqrt{7}}{4}$

$x = - \frac{3 + i \sqrt{7}}{4}$

Step 6

List all of the solutions.

$y = \frac{1 - 3 i \sqrt{7}}{4}, x = - \frac{3 - i \sqrt{7}}{4}$

$y = \frac{1 + 3 i \sqrt{7}}{4}, x = - \frac{3 + i \sqrt{7}}{4}$

Step 7