Algebra Examples

$y = \sqrt{9 - x}$ $x + 2 y = 6$

Step 1

Replace all occurrences of $y$ with $\sqrt{9 - x}$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $x + 2 y = 6$ with $\sqrt{9 - x}$ .

$x + 2 (\sqrt{9 - x}) = 6$

$y = \sqrt{9 - x}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Multiply $2$ by $\sqrt{9 - x}$ .

$x + 2 \sqrt{9 - x} = 6$

$y = \sqrt{9 - x}$

$x + 2 \sqrt{9 - x} = 6$

$y = \sqrt{9 - x}$

$x + 2 \sqrt{9 - x} = 6$

$y = \sqrt{9 - x}$

Step 2

Solve for $x$ in $x + 2 \sqrt{9 - x} = 6$ .

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Step 2.1

Subtract $x$ from both sides of the equation.

$2 \sqrt{9 - x} = 6 - x$

$y = \sqrt{9 - x}$

Step 2.2

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

${(2 \sqrt{9 - x})}^{2} = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - x}$ as ${(9 - x)}^{\frac{1}{2}}$ .

${(2 {(9 - x)}^{\frac{1}{2}})}^{2} = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify ${(2 {(9 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.1

Apply the product rule to $2 {(9 - x)}^{\frac{1}{2}}$ .

$2^{2} {({(9 - x)}^{\frac{1}{2}})}^{2} = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

Step 2.3.2.1.2

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

$4 {({(9 - x)}^{\frac{1}{2}})}^{2} = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

Step 2.3.2.1.3

Multiply the exponents in ${({(9 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.3.1

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

$4 {(9 - x)}^{\frac{1}{2} \cdot 2} = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

Step 2.3.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1.3.2.1

Cancel the common factor.

$4 {(9 - x)}^{\frac{1}{2} \cdot 2} = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

Step 2.3.2.1.3.2.2

Rewrite the expression.

$4 (9 - x) = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

$4 (9 - x) = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

$4 (9 - x) = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

Step 2.3.2.1.4

Simplify.

$4 (9 - x) = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

Step 2.3.2.1.5

Apply the distributive property.

$4 \cdot 9 + 4 (- x) = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

Step 2.3.2.1.6

Multiply.

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Step 2.3.2.1.6.1

Multiply $4$ by $9$ .

$36 + 4 (- x) = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

Step 2.3.2.1.6.2

Multiply $- 1$ by $4$ .

$36 - 4 x = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

$36 - 4 x = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

$36 - 4 x = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

$36 - 4 x = {(6 - x)}^{2}$

$y = \sqrt{9 - x}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify ${(6 - x)}^{2}$ .

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Step 2.3.3.1.1

Rewrite ${(6 - x)}^{2}$ as $(6 - x) (6 - x)$ .

$36 - 4 x = (6 - x) (6 - x)$

$y = \sqrt{9 - x}$

Step 2.3.3.1.2

Expand $(6 - x) (6 - x)$ using the FOIL Method.

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Step 2.3.3.1.2.1

Apply the distributive property.

$36 - 4 x = 6 (6 - x) - x (6 - x)$

$y = \sqrt{9 - x}$

Step 2.3.3.1.2.2

Apply the distributive property.

$36 - 4 x = 6 \cdot 6 + 6 (- x) - x (6 - x)$

$y = \sqrt{9 - x}$

Step 2.3.3.1.2.3

Apply the distributive property.

$36 - 4 x = 6 \cdot 6 + 6 (- x) - x \cdot 6 - x (- x)$

$y = \sqrt{9 - x}$

$36 - 4 x = 6 \cdot 6 + 6 (- x) - x \cdot 6 - x (- x)$

$y = \sqrt{9 - x}$

Step 2.3.3.1.3

Simplify and combine like terms.

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Step 2.3.3.1.3.1

Simplify each term.

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Step 2.3.3.1.3.1.1

Multiply $6$ by $6$ .

$36 - 4 x = 36 + 6 (- x) - x \cdot 6 - x (- x)$

$y = \sqrt{9 - x}$

Step 2.3.3.1.3.1.2

Multiply $- 1$ by $6$ .

$36 - 4 x = 36 - 6 x - x \cdot 6 - x (- x)$

$y = \sqrt{9 - x}$

Step 2.3.3.1.3.1.3

Multiply $6$ by $- 1$ .

$36 - 4 x = 36 - 6 x - 6 x - x (- x)$

$y = \sqrt{9 - x}$

Step 2.3.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$36 - 4 x = 36 - 6 x - 6 x - 1 \cdot (- 1 x \cdot x)$

$y = \sqrt{9 - x}$

Step 2.3.3.1.3.1.5

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

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Step 2.3.3.1.3.1.5.1

Move $x$ .

$36 - 4 x = 36 - 6 x - 6 x - 1 \cdot (- 1 (x \cdot x))$

$y = \sqrt{9 - x}$

Step 2.3.3.1.3.1.5.2

Multiply $x$ by $x$ .

$36 - 4 x = 36 - 6 x - 6 x - 1 \cdot (- 1 x^{2})$

$y = \sqrt{9 - x}$

$36 - 4 x = 36 - 6 x - 6 x - 1 \cdot (- 1 x^{2})$

$y = \sqrt{9 - x}$

Step 2.3.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$36 - 4 x = 36 - 6 x - 6 x + 1 x^{2}$

$y = \sqrt{9 - x}$

Step 2.3.3.1.3.1.7

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

$36 - 4 x = 36 - 6 x - 6 x + x^{2}$

$y = \sqrt{9 - x}$

$36 - 4 x = 36 - 6 x - 6 x + x^{2}$

$y = \sqrt{9 - x}$

Step 2.3.3.1.3.2

Subtract $6 x$ from $- 6 x$ .