Pre-Algebra Examples

$((x - 1 + y - 1) - 1) (x - 1 - y - 1) - 1 \div (x - 2 - y - 2) - 1 = 1$

Step 1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$(((0) - 1 + y - 1) - 1) ((0) - 1 - y - 1) - 1 \div ((0) - 2 - y - 2) - 1 = 1$

Step 2

Solve the equation.

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Step 2.1

Simplify $(((0) - 1 + y - 1) - 1) ((0) - 1 - y - 1) - 1 \div ((0) - 2 - y - 2) - 1$ .

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Step 2.1.1

Combine the opposite terms in $(((0) - 1 + y - 1) - 1) ((0) - 1 - y - 1) - 1 \div ((0) - 2 - y - 2) - 1$ .

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Step 2.1.1.1

Subtract $1$ from $0$ .

$(- 1 + y - 1 - 1) ((0) - 1 - y - 1) - 1 \div ((0) - 2 - y - 2) - 1 = 1$

Step 2.1.1.2

Subtract $1$ from $0$ .

$(- 1 + y - 1 - 1) (- 1 - y - 1) - 1 \div ((0) - 2 - y - 2) - 1 = 1$

Step 2.1.1.3

Subtract $2$ from $0$ .

$(- 1 + y - 1 - 1) (- 1 - y - 1) - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2

Simplify each term.

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Step 2.1.2.1

Subtract $1$ from $- 1$ .

$(y - 2 - 1) (- 1 - y - 1) - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.2

Subtract $1$ from $- 2$ .

$(y - 3) (- 1 - y - 1) - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.3

Subtract $1$ from $- 1$ .

$(y - 3) (- y - 2) - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.4

Expand $(y - 3) (- y - 2)$ using the FOIL Method.

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Step 2.1.2.4.1

Apply the distributive property.

$y (- y - 2) - 3 (- y - 2) - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.4.2

Apply the distributive property.

$y (- y) + y \cdot - 2 - 3 (- y - 2) - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.4.3

Apply the distributive property.

$y (- y) + y \cdot - 2 - 3 (- y) - 3 \cdot - 2 - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.5

Simplify and combine like terms.

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Step 2.1.2.5.1

Simplify each term.

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Step 2.1.2.5.1.1

Rewrite using the commutative property of multiplication.

$- y \cdot y + y \cdot - 2 - 3 (- y) - 3 \cdot - 2 - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.5.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.2.5.1.2.1

Move $y$ .

$- (y \cdot y) + y \cdot - 2 - 3 (- y) - 3 \cdot - 2 - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.5.1.2.2

Multiply $y$ by $y$ .

$- y^{2} + y \cdot - 2 - 3 (- y) - 3 \cdot - 2 - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.5.1.3

Move $- 2$ to the left of $y$ .

$- y^{2} - 2 \cdot y - 3 (- y) - 3 \cdot - 2 - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.5.1.4

Multiply $- 1$ by $- 3$ .

$- y^{2} - 2 y + 3 y - 3 \cdot - 2 - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.5.1.5

Multiply $- 3$ by $- 2$ .

$- y^{2} - 2 y + 3 y + 6 - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.5.2

Add $- 2 y$ and $3 y$ .

$- y^{2} + y + 6 - 1 \div (- 2 - y - 2) - 1 = 1$

Step 2.1.2.6

Rewrite the division as a fraction.

$- y^{2} + y + 6 - \frac{1}{- 2 - y - 2} - 1 = 1$

Step 2.1.2.7

Subtract $2$ from $- 2$ .

$- y^{2} + y + 6 - \frac{1}{- y - 4} - 1 = 1$

Step 2.1.3

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

$y + 6 - y^{2} \cdot \frac{- y - 4}{- y - 4} - \frac{1}{- y - 4} - 1 = 1$

Step 2.1.4

Simplify terms.

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Step 2.1.4.1

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

$y + 6 + \frac{- y^{2} (- y - 4)}{- y - 4} - \frac{1}{- y - 4} - 1 = 1$

Step 2.1.4.2

Combine the numerators over the common denominator.

$y + 6 + \frac{- y^{2} (- y - 4) - 1}{- y - 4} - 1 = 1$

Step 2.1.5

Simplify the numerator.

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Step 2.1.5.1

Apply the distributive property.

$y + 6 + \frac{- y^{2} (- y) - y^{2} \cdot - 4 - 1}{- y - 4} - 1 = 1$

Step 2.1.5.2

Rewrite using the commutative property of multiplication.

$y + 6 + \frac{- 1 \cdot - 1 y^{2} y - y^{2} \cdot - 4 - 1}{- y - 4} - 1 = 1$

Step 2.1.5.3

Multiply $- 4$ by $- 1$ .

$y + 6 + \frac{- 1 \cdot - 1 y^{2} y + 4 y^{2} - 1}{- y - 4} - 1 = 1$

Step 2.1.5.4

Simplify each term.

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Step 2.1.5.4.1

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 2.1.5.4.1.1

Move $y$ .

$y + 6 + \frac{- 1 \cdot - 1 (y \cdot y^{2}) + 4 y^{2} - 1}{- y - 4} - 1 = 1$

Step 2.1.5.4.1.2

Multiply $y$ by $y^{2}$ .

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Step 2.1.5.4.1.2.1

Raise $y$ to the power of $1$ .

$y + 6 + \frac{- 1 \cdot - 1 (y^{1} y^{2}) + 4 y^{2} - 1}{- y - 4} - 1 = 1$

Step 2.1.5.4.1.2.2

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

$y + 6 + \frac{- 1 \cdot - 1 y^{1 + 2} + 4 y^{2} - 1}{- y - 4} - 1 = 1$

Step 2.1.5.4.1.3

Add $1$ and $2$ .

$y + 6 + \frac{- 1 \cdot - 1 y^{3} + 4 y^{2} - 1}{- y - 4} - 1 = 1$

Step 2.1.5.4.2

Multiply $- 1$ by $- 1$ .

$y + 6 + \frac{1 y^{3} + 4 y^{2} - 1}{- y - 4} - 1 = 1$

Step 2.1.5.4.3

Multiply $y^{3}$ by $1$ .

$y + 6 + \frac{y^{3} + 4 y^{2} - 1}{- y - 4} - 1 = 1$

Step 2.1.6

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

$\frac{y (- y - 4)}{- y - 4} + \frac{y^{3} + 4 y^{2} - 1}{- y - 4} + 6 - 1 = 1$

Step 2.1.7

Combine the numerators over the common denominator.

$\frac{y (- y - 4) + y^{3} + 4 y^{2} - 1}{- y - 4} + 6 - 1 = 1$

Step 2.1.8

Find the common denominator.

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Step 2.1.8.1

Write $6$ as a fraction with denominator $1$ .

$\frac{y (- y - 4) + y^{3} + 4 y^{2} - 1}{- y - 4} + \frac{6}{1} - 1 = 1$

Step 2.1.8.2

Multiply $\frac{6}{1}$ by $\frac{- y - 4}{- y - 4}$ .

$\frac{y (- y - 4) + y^{3} + 4 y^{2} - 1}{- y - 4} + \frac{6}{1} \cdot \frac{- y - 4}{- y - 4} - 1 = 1$

Step 2.1.8.3

Multiply $\frac{6}{1}$ by $\frac{- y - 4}{- y - 4}$ .

$\frac{y (- y - 4) + y^{3} + 4 y^{2} - 1}{- y - 4} + \frac{6 (- y - 4)}{- y - 4} - 1 = 1$

Step 2.1.8.4

Write $- 1$ as a fraction with denominator $1$ .

$\frac{y (- y - 4) + y^{3} + 4 y^{2} - 1}{- y - 4} + \frac{6 (- y - 4)}{- y - 4} + \frac{- 1}{1} = 1$

Step 2.1.8.5

Multiply $\frac{- 1}{1}$ by $\frac{- y - 4}{- y - 4}$ .

$\frac{y (- y - 4) + y^{3} + 4 y^{2} - 1}{- y - 4} + \frac{6 (- y - 4)}{- y - 4} + \frac{- 1}{1} \cdot \frac{- y - 4}{- y - 4} = 1$

Step 2.1.8.6

Multiply $\frac{- 1}{1}$ by $\frac{- y - 4}{- y - 4}$ .

$\frac{y (- y - 4) + y^{3} + 4 y^{2} - 1}{- y - 4} + \frac{6 (- y - 4)}{- y - 4} + \frac{- (- y - 4)}{- y - 4} = 1$

Step 2.1.9

Combine the numerators over the common denominator.

$\frac{y (- y - 4) + y^{3} + 4 y^{2} - 1 + 6 (- y - 4) - (- y - 4)}{- y - 4} = 1$

Step 2.1.10

Simplify each term.

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Step 2.1.10.1

Apply the distributive property.

$\frac{y (- y) + y \cdot - 4 + y^{3} + 4 y^{2} - 1 + 6 (- y - 4) - (- y - 4)}{- y - 4} = 1$

Step 2.1.10.2

Rewrite using the commutative property of multiplication.

$\frac{- y \cdot y + y \cdot - 4 + y^{3} + 4 y^{2} - 1 + 6 (- y - 4) - (- y - 4)}{- y - 4} = 1$

Step 2.1.10.3

Move $- 4$ to the left of $y$ .

$\frac{- y \cdot y - 4 \cdot y + y^{3} + 4 y^{2} - 1 + 6 (- y - 4) - (- y - 4)}{- y - 4} = 1$

Step 2.1.10.4

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.10.4.1

Move $y$ .

$\frac{- (y \cdot y) - 4 \cdot y + y^{3} + 4 y^{2} - 1 + 6 (- y - 4) - (- y - 4)}{- y - 4} = 1$

Step 2.1.10.4.2

Multiply $y$ by $y$ .

$\frac{- y^{2} - 4 \cdot y + y^{3} + 4 y^{2} - 1 + 6 (- y - 4) - (- y - 4)}{- y - 4} = 1$

$\frac{- y^{2} - 4 y + y^{3} + 4 y^{2} - 1 + 6 (- y - 4) - (- y - 4)}{- y - 4} = 1$

Step 2.1.10.5

Apply the distributive property.

$\frac{- y^{2} - 4 y + y^{3} + 4 y^{2} - 1 + 6 (- y) + 6 \cdot - 4 - (- y - 4)}{- y - 4} = 1$

Step 2.1.10.6

Multiply $- 1$ by $6$ .

$\frac{- y^{2} - 4 y + y^{3} + 4 y^{2} - 1 - 6 y + 6 \cdot - 4 - (- y - 4)}{- y - 4} = 1$

Step 2.1.10.7

Multiply $6$ by $- 4$ .

$\frac{- y^{2} - 4 y + y^{3} + 4 y^{2} - 1 - 6 y - 24 - (- y - 4)}{- y - 4} = 1$

Step 2.1.10.8

Apply the distributive property.

$\frac{- y^{2} - 4 y + y^{3} + 4 y^{2} - 1 - 6 y - 24 - - y - - 4}{- y - 4} = 1$

Step 2.1.10.9

Multiply $- - y$ .

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Step 2.1.10.9.1

Multiply $- 1$ by $- 1$ .

$\frac{- y^{2} - 4 y + y^{3} + 4 y^{2} - 1 - 6 y - 24 + 1 y - - 4}{- y - 4} = 1$

Step 2.1.10.9.2

Multiply $y$ by $1$ .

$\frac{- y^{2} - 4 y + y^{3} + 4 y^{2} - 1 - 6 y - 24 + y - - 4}{- y - 4} = 1$

Step 2.1.10.10

Multiply $- 1$ by $- 4$ .

$\frac{- y^{2} - 4 y + y^{3} + 4 y^{2} - 1 - 6 y - 24 + y + 4}{- y - 4} = 1$

Step 2.1.11

Simplify terms.

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Step 2.1.11.1

Add $- y^{2}$ and $4 y^{2}$ .

$\frac{3 y^{2} - 4 y + y^{3} - 1 - 6 y - 24 + y + 4}{- y - 4} = 1$

Step 2.1.11.2

Subtract $6 y$ from $- 4 y$ .

$\frac{3 y^{2} - 10 y + y^{3} - 1 - 24 + y + 4}{- y - 4} = 1$

Step 2.1.11.3

Add $- 10 y$ and $y$ .

$\frac{3 y^{2} - 9 y + y^{3} - 1 - 24 + 4}{- y - 4} = 1$

Step 2.1.11.4

Simplify the expression.

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Step 2.1.11.4.1

Subtract $24$ from $- 1$ .

$\frac{3 y^{2} - 9 y + y^{3} - 25 + 4}{- y - 4} = 1$

Step 2.1.11.4.2

Add $- 25$ and $4$ .

$\frac{3 y^{2} - 9 y + y^{3} - 21}{- y - 4} = 1$

Step 2.1.11.4.3

Reorder terms.

$\frac{y^{3} + 3 y^{2} - 9 y - 21}{- y - 4} = 1$

Step 2.1.11.5

Factor $- 1$ out of $- y$ .

$\frac{y^{3} + 3 y^{2} - 9 y - 21}{- (y) - 4} = 1$

Step 2.1.11.6

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{y^{3} + 3 y^{2} - 9 y - 21}{- (y) - 1 (4)} = 1$

Step 2.1.11.7

Factor $- 1$ out of $- (y) - 1 (4)$ .

$\frac{y^{3} + 3 y^{2} - 9 y - 21}{- (y + 4)} = 1$

Step 2.1.11.8

Rewrite negatives.

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Step 2.1.11.8.1

Rewrite $- (y + 4)$ as $- 1 (y + 4)$ .

$\frac{y^{3} + 3 y^{2} - 9 y - 21}{- 1 (y + 4)} = 1$

Step 2.1.11.8.2

Move the negative in front of the fraction.

$- \frac{y^{3} + 3 y^{2} - 9 y - 21}{y + 4} = 1$

Step 2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$y \approx - 3.93515155, - 1.66283827, 2.59798982$

Step 3

y-intercept(s) in point form.

y-intercept(s): $(0, - 3.93515155), (0, - 1.66283827), (0, 2.59798982)$

Step 4