Algebra Examples

$4 x^{2} + y = 3$ $- x - y = 11$

Step 1

Subtract $4 x^{2}$ from both sides of the equation.

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

$- x - y = 11$

Step 2

Replace all occurrences of $y$ with $3 - 4 x^{2}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $- x - y = 11$ with $3 - 4 x^{2}$ .

$- x - (3 - 4 x^{2}) = 11$

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

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.

$- x - 1 \cdot 3 - (- 4 x^{2}) = 11$

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

Step 2.2.1.2

Multiply $- 1$ by $3$ .

$- x - 3 - (- 4 x^{2}) = 11$

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

Step 2.2.1.3

Multiply $- 4$ by $- 1$ .

$- x - 3 + 4 x^{2} = 11$

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

$- x - 3 + 4 x^{2} = 11$

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

$- x - 3 + 4 x^{2} = 11$

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

$- x - 3 + 4 x^{2} = 11$

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

Step 3

Solve for $x$ in $- x - 3 + 4 x^{2} = 11$ .

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Step 3.1

Subtract $11$ from both sides of the equation.

$- x - 3 + 4 x^{2} - 11 = 0$

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

Step 3.2

Subtract $11$ from $- 3$ .

$- x + 4 x^{2} - 14 = 0$

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

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$- u + 4 u^{2} - 14 = 0$

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

Step 3.3.2

Factor by grouping.

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Step 3.3.2.1

Reorder terms.

$4 u^{2} - u - 14 = 0$

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

Step 3.3.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 14 = - 56$ and whose sum is $b = - 1$ .

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Step 3.3.2.2.1

Factor $- 1$ out of $- u$ .

$4 u^{2} - (u) - 14 = 0$

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

Step 3.3.2.2.2

Rewrite $- 1$ as $7$ plus $- 8$

$4 u^{2} + (7 - 8) u - 14 = 0$

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

Step 3.3.2.2.3

Apply the distributive property.

$4 u^{2} + 7 u - 8 u - 14 = 0$

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

$4 u^{2} + 7 u - 8 u - 14 = 0$

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

Step 3.3.2.3

Factor out the greatest common factor from each group.

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Step 3.3.2.3.1

Group the first two terms and the last two terms.

$(4 u^{2} + 7 u) - 8 u - 14 = 0$

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

Step 3.3.2.3.2

Factor out the greatest common factor (GCF) from each group.

$u (4 u + 7) - 2 (4 u + 7) = 0$

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

$u (4 u + 7) - 2 (4 u + 7) = 0$

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

Step 3.3.2.4

Factor the polynomial by factoring out the greatest common factor, $4 u + 7$ .

$(4 u + 7) (u - 2) = 0$

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

$(4 u + 7) (u - 2) = 0$

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

Step 3.3.3

Replace all occurrences of $u$ with $x$ .

$(4 x + 7) (x - 2) = 0$

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

$(4 x + 7) (x - 2) = 0$

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

Step 3.4

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

$4 x + 7 = 0$

$x - 2 = 0$

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

Step 3.5

Set $4 x + 7$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $4 x + 7$ equal to $0$ .

$4 x + 7 = 0$

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

Step 3.5.2

Solve $4 x + 7 = 0$ for $x$ .

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Step 3.5.2.1

Subtract $7$ from both sides of the equation.

$4 x = - 7$

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

Step 3.5.2.2

Divide each term in $4 x = - 7$ by $4$ and simplify.

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Step 3.5.2.2.1

Divide each term in $4 x = - 7$ by $4$ .

$\frac{4 x}{4} = \frac{- 7}{4}$

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

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.5.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 7}{4}$

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

Step 3.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{4}$

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

$x = \frac{- 7}{4}$

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

$x = \frac{- 7}{4}$

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

Step 3.5.2.2.3

Simplify the right side.

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Step 3.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{7}{4}$

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

$x = - \frac{7}{4}$

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

$x = - \frac{7}{4}$

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

$x = - \frac{7}{4}$

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

$x = - \frac{7}{4}$

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

Step 3.6

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

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

Step 3.6.2

Add $2$ to both sides of the equation.

$x = 2$

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

$x = 2$

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

Step 3.7

The final solution is all the values that make $(4 x + 7) (x - 2) = 0$ true.

$x = - \frac{7}{4}, 2$

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

$x = - \frac{7}{4}, 2$

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

Step 4

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

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Step 4.1

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

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

$x = - \frac{7}{4}$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.1.1.1.1

Apply the product rule to $- \frac{7}{4}$ .

$y = 3 - 4 ({(- 1)}^{2} {(\frac{7}{4})}^{2})$

$x = - \frac{7}{4}$

Step 4.2.1.1.1.2

Apply the product rule to $\frac{7}{4}$ .

$y = 3 - 4 ({(- 1)}^{2} (\frac{7^{2}}{4^{2}}))$

$x = - \frac{7}{4}$

$y = 3 - 4 ({(- 1)}^{2} (\frac{7^{2}}{4^{2}}))$

$x = - \frac{7}{4}$

Step 4.2.1.1.2

Raise $- 1$ to the power of $2$ .

$y = 3 - 4 (1 (\frac{7^{2}}{4^{2}}))$

$x = - \frac{7}{4}$

Step 4.2.1.1.3

Multiply $\frac{7^{2}}{4^{2}}$ by $1$ .

$y = 3 - 4 \frac{7^{2}}{4^{2}}$

$x = - \frac{7}{4}$

Step 4.2.1.1.4

Raise $7$ to the power of $2$ .

$y = 3 - 4 \frac{49}{4^{2}}$

$x = - \frac{7}{4}$

Step 4.2.1.1.5

Raise $4$ to the power of $2$ .

$y = 3 - 4 (\frac{49}{16})$

$x = - \frac{7}{4}$

Step 4.2.1.1.6

Cancel the common factor of $4$ .

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Step 4.2.1.1.6.1

Factor $4$ out of $- 4$ .

$y = 3 + 4 (- 1) (\frac{49}{16})$

$x = - \frac{7}{4}$

Step 4.2.1.1.6.2

Factor $4$ out of $16$ .

$y = 3 + 4 \cdot (- 1 \frac{49}{4 \cdot 4})$

$x = - \frac{7}{4}$

Step 4.2.1.1.6.3

Cancel the common factor.

$y = 3 + 4 \cdot (- 1 \frac{49}{4 \cdot 4})$

$x = - \frac{7}{4}$

Step 4.2.1.1.6.4

Rewrite the expression.

$y = 3 - 1 (\frac{49}{4})$

$x = - \frac{7}{4}$

$y = 3 - 1 (\frac{49}{4})$

$x = - \frac{7}{4}$

Step 4.2.1.1.7

Rewrite $- 1 (\frac{49}{4})$ as $- (\frac{49}{4})$ .

$y = 3 - \frac{49}{4}$

$x = - \frac{7}{4}$

$y = 3 - \frac{49}{4}$

$x = - \frac{7}{4}$

Step 4.2.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = 3 \cdot \frac{4}{4} - \frac{49}{4}$

$x = - \frac{7}{4}$

Step 4.2.1.3

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

$y = \frac{3 \cdot 4}{4} - \frac{49}{4}$

$x = - \frac{7}{4}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$y = \frac{3 \cdot 4 - 49}{4}$

$x = - \frac{7}{4}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $3$ by $4$ .

$y = \frac{12 - 49}{4}$

$x = - \frac{7}{4}$

Step 4.2.1.5.2

Subtract $49$ from $12$ .

$y = \frac{- 37}{4}$

$x = - \frac{7}{4}$

$y = \frac{- 37}{4}$

$x = - \frac{7}{4}$

Step 4.2.1.6

Move the negative in front of the fraction.

$y = - \frac{37}{4}$

$x = - \frac{7}{4}$

$y = - \frac{37}{4}$

$x = - \frac{7}{4}$

$y = - \frac{37}{4}$

$x = - \frac{7}{4}$

$y = - \frac{37}{4}$

$x = - \frac{7}{4}$

Step 5

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

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Step 5.1

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

$y = 3 - 4 {(2)}^{2}$

$x = 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $3 - 4 {(2)}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

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

$y = 3 - 4 \cdot 4$

$x = 2$

Step 5.2.1.1.2

Multiply $- 4$ by $4$ .

$y = 3 - 16$

$x = 2$

$y = 3 - 16$

$x = 2$

Step 5.2.1.2

Subtract $16$ from $3$ .

$y = - 13$

$x = 2$

$y = - 13$

$x = 2$

$y = - 13$

$x = 2$

$y = - 13$

$x = 2$

Step 6

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

$(- \frac{7}{4}, - \frac{37}{4})$

$(2, - 13)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- \frac{7}{4}, - \frac{37}{4}), (2, - 13)$

Equation Form:

$x = - \frac{7}{4}, y = - \frac{37}{4}$

$x = 2, y = - 13$

Step 8