Algebra Examples

$y = \sqrt{1 - x^{2}}$ , $y = 2 x - 1$

Step 1

Eliminate the equal sides of each equation and combine.

$\sqrt{1 - x^{2}} = 2 x - 1$

Step 2

Solve $\sqrt{1 - x^{2}} = 2 x - 1$ for $x$ .

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Step 2.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{1 - x^{2}}}^{2} = {(2 x - 1)}^{2}$

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

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

${({(1 - x^{2})}^{\frac{1}{2}})}^{2} = {(2 x - 1)}^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${({(1 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${({(1 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1.1

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

${(1 - x^{2})}^{\frac{1}{2} \cdot 2} = {(2 x - 1)}^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(1 - x^{2})}^{\frac{1}{2} \cdot 2} = {(2 x - 1)}^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(1 - x^{2})}^{1} = {(2 x - 1)}^{2}$

Step 2.2.2.1.2

Simplify.

$1 - x^{2} = {(2 x - 1)}^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify ${(2 x - 1)}^{2}$ .

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Step 2.2.3.1.1

Rewrite ${(2 x - 1)}^{2}$ as $(2 x - 1) (2 x - 1)$ .

$1 - x^{2} = (2 x - 1) (2 x - 1)$

Step 2.2.3.1.2

Expand $(2 x - 1) (2 x - 1)$ using the FOIL Method.

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Step 2.2.3.1.2.1

Apply the distributive property.

$1 - x^{2} = 2 x (2 x - 1) - 1 (2 x - 1)$

Step 2.2.3.1.2.2

Apply the distributive property.

$1 - x^{2} = 2 x (2 x) + 2 x \cdot - 1 - 1 (2 x - 1)$

Step 2.2.3.1.2.3

Apply the distributive property.

$1 - x^{2} = 2 x (2 x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Step 2.2.3.1.3

Simplify and combine like terms.

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Step 2.2.3.1.3.1

Simplify each term.

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Step 2.2.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$1 - x^{2} = 2 \cdot 2 x \cdot x + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Step 2.2.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.3.1.3.1.2.1

Move $x$ .

$1 - x^{2} = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Step 2.2.3.1.3.1.2.2

Multiply $x$ by $x$ .

$1 - x^{2} = 2 \cdot 2 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Step 2.2.3.1.3.1.3

Multiply $2$ by $2$ .

$1 - x^{2} = 4 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Step 2.2.3.1.3.1.4

Multiply $- 1$ by $2$ .

$1 - x^{2} = 4 x^{2} - 2 x - 1 (2 x) - 1 \cdot - 1$

Step 2.2.3.1.3.1.5

Multiply $2$ by $- 1$ .

$1 - x^{2} = 4 x^{2} - 2 x - 2 x - 1 \cdot - 1$

Step 2.2.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$1 - x^{2} = 4 x^{2} - 2 x - 2 x + 1$

Step 2.2.3.1.3.2

Subtract $2 x$ from $- 2 x$ .

$1 - x^{2} = 4 x^{2} - 4 x + 1$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x^{2} - 4 x + 1 = 1 - x^{2}$

Step 2.3.2

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

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Step 2.3.2.1

Add $x^{2}$ to both sides of the equation.

$4 x^{2} - 4 x + 1 + x^{2} = 1$

Step 2.3.2.2

Add $4 x^{2}$ and $x^{2}$ .

$5 x^{2} - 4 x + 1 = 1$

Step 2.3.3

Subtract $1$ from both sides of the equation.

$5 x^{2} - 4 x + 1 - 1 = 0$

Step 2.3.4

Combine the opposite terms in $5 x^{2} - 4 x + 1 - 1$ .

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Step 2.3.4.1

Subtract $1$ from $1$ .

$5 x^{2} - 4 x + 0 = 0$

Step 2.3.4.2

Add $5 x^{2} - 4 x$ and $0$ .

$5 x^{2} - 4 x = 0$

Step 2.3.5

Factor $x$ out of $5 x^{2} - 4 x$ .

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Step 2.3.5.1

Factor $x$ out of $5 x^{2}$ .

$x (5 x) - 4 x = 0$

Step 2.3.5.2

Factor $x$ out of $- 4 x$ .

$x (5 x) + x \cdot - 4 = 0$

Step 2.3.5.3

Factor $x$ out of $x (5 x) + x \cdot - 4$ .

$x (5 x - 4) = 0$

Step 2.3.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$5 x - 4 = 0 + y = 2 x - 1$

Step 2.3.7

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.8

Set $5 x - 4$ equal to $0$ and solve for $x$ .

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Step 2.3.8.1

Set $5 x - 4$ equal to $0$ .

$5 x - 4 = 0$

Step 2.3.8.2

Solve $5 x - 4 = 0$ for $x$ .

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Step 2.3.8.2.1

Add $4$ to both sides of the equation.

$5 x = 4$

Step 2.3.8.2.2

Divide each term in $5 x = 4$ by $5$ and simplify.

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Step 2.3.8.2.2.1

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

Step 2.3.8.2.2.2

Simplify the left side.

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Step 2.3.8.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.3.8.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{4}{5}$

Step 2.3.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{5}$

Step 2.3.9

The final solution is all the values that make $x (5 x - 4) = 0$ true.

$x = 0, \frac{4}{5}$

Step 2.4

Exclude the solutions that do not make $\sqrt{1 - x^{2}} = 2 x - 1$ true.

$x = \frac{4}{5}$

Step 3

Evaluate $y$ when $x = \frac{4}{5}$ .

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Step 3.1

Substitute $\frac{4}{5}$ for $x$ .

$y = 2 (\frac{4}{5}) - 1$

Step 3.2

Simplify $2 (\frac{4}{5}) - 1$ .

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Step 3.2.1

Multiply $2 (\frac{4}{5})$ .

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Step 3.2.1.1

Combine $2$ and $\frac{4}{5}$ .

$y = \frac{2 \cdot 4}{5} - 1$

Step 3.2.1.2

Multiply $2$ by $4$ .

$y = \frac{8}{5} - 1$

Step 3.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{8}{5} - 1 \cdot \frac{5}{5}$

Step 3.2.3

Combine $- 1$ and $\frac{5}{5}$ .

$y = \frac{8}{5} + \frac{- 1 \cdot 5}{5}$

Step 3.2.4

Combine the numerators over the common denominator.

$y = \frac{8 - 1 \cdot 5}{5}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 1$ by $5$ .

$y = \frac{8 - 5}{5}$

Step 3.2.5.2

Subtract $5$ from $8$ .

$y = \frac{3}{5}$

Step 4

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

$(\frac{4}{5}, \frac{3}{5})$

Step 5

The result can be shown in multiple forms.

Point Form:

$(\frac{4}{5}, \frac{3}{5})$

Equation Form:

$x = \frac{4}{5}, y = \frac{3}{5}$

Step 6