Algebra Examples

$2 x^{2} + 3 y^{2} = 4$ $y = 2 x$

Step 1

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

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Step 1.1

Replace all occurrences of $y$ in $2 x^{2} + 3 y^{2} = 4$ with $2 x$ .

$2 x^{2} + 3 {(2 x)}^{2} = 4$

$y = 2 x$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $2 x^{2} + 3 {(2 x)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Apply the product rule to $2 x$ .

$2 x^{2} + 3 (2^{2} x^{2}) = 4$

$y = 2 x$

Step 1.2.1.1.2

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

$2 x^{2} + 3 (4 x^{2}) = 4$

$y = 2 x$

Step 1.2.1.1.3

Multiply $4$ by $3$ .

$2 x^{2} + 12 x^{2} = 4$

$y = 2 x$

$2 x^{2} + 12 x^{2} = 4$

$y = 2 x$

Step 1.2.1.2

Add $2 x^{2}$ and $12 x^{2}$ .

$14 x^{2} = 4$

$y = 2 x$

$14 x^{2} = 4$

$y = 2 x$

$14 x^{2} = 4$

$y = 2 x$

$14 x^{2} = 4$

$y = 2 x$

Step 2

Solve for $x$ in $14 x^{2} = 4$ .

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Step 2.1

Divide each term in $14 x^{2} = 4$ by $14$ and simplify.

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Step 2.1.1

Divide each term in $14 x^{2} = 4$ by $14$ .

$\frac{14 x^{2}}{14} = \frac{4}{14}$

$y = 2 x$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $14$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{14 x^{2}}{14} = \frac{4}{14}$

$y = 2 x$

Step 2.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{4}{14}$

$y = 2 x$

$x^{2} = \frac{4}{14}$

$y = 2 x$

$x^{2} = \frac{4}{14}$

$y = 2 x$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Cancel the common factor of $4$ and $14$ .

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Step 2.1.3.1.1

Factor $2$ out of $4$ .

$x^{2} = \frac{2 (2)}{14}$

$y = 2 x$

Step 2.1.3.1.2

Cancel the common factors.

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Step 2.1.3.1.2.1

Factor $2$ out of $14$ .

$x^{2} = \frac{2 \cdot 2}{2 \cdot 7}$

$y = 2 x$

Step 2.1.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{2 \cdot 2}{2 \cdot 7}$

$y = 2 x$

Step 2.1.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{2}{7}$

$y = 2 x$

$x^{2} = \frac{2}{7}$

$y = 2 x$

$x^{2} = \frac{2}{7}$

$y = 2 x$

$x^{2} = \frac{2}{7}$

$y = 2 x$

$x^{2} = \frac{2}{7}$

$y = 2 x$

Step 2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{2}{7}}$

$y = 2 x$

Step 2.3

Simplify $\pm \sqrt{\frac{2}{7}}$ .

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Step 2.3.1

Rewrite $\sqrt{\frac{2}{7}}$ as $\frac{\sqrt{2}}{\sqrt{7}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{7}}$

$y = 2 x$

Step 2.3.2

Multiply $\frac{\sqrt{2}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

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

$y = 2 x$

Step 2.3.3

Combine and simplify the denominator.

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Step 2.3.3.1

Multiply $\frac{\sqrt{2}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = 2 x$

Step 2.3.3.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = 2 x$

Step 2.3.3.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = 2 x$

Step 2.3.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

$y = 2 x$

Step 2.3.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{2}}$

$y = 2 x$

Step 2.3.3.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

$y = 2 x$

Step 2.3.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

$y = 2 x$

Step 2.3.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

$y = 2 x$

Step 2.3.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

$y = 2 x$

Step 2.3.3.6.4.2

Rewrite the expression.