Precalculus Examples

$x + 3 y = 5$ , $x^{2} + y^{2} = 25$

Step 1

Subtract $3 y$ from both sides of the equation.

$x = 5 - 3 y$

$x^{2} + y^{2} = 25$

Step 2

Replace all occurrences of $x$ with $5 - 3 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 25$ with $5 - 3 y$ .

${(5 - 3 y)}^{2} + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(5 - 3 y)}^{2} + y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(5 - 3 y)}^{2}$ as $(5 - 3 y) (5 - 3 y)$ .

$(5 - 3 y) (5 - 3 y) + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.1.2

Expand $(5 - 3 y) (5 - 3 y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$5 (5 - 3 y) - 3 y (5 - 3 y) + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.1.2.2

Apply the distributive property.

$5 \cdot 5 + 5 (- 3 y) - 3 y (5 - 3 y) + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.1.2.3

Apply the distributive property.

$5 \cdot 5 + 5 (- 3 y) - 3 y \cdot 5 - 3 y (- 3 y) + y^{2} = 25$

$x = 5 - 3 y$

$5 \cdot 5 + 5 (- 3 y) - 3 y \cdot 5 - 3 y (- 3 y) + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.1.3

Simplify and combine like terms.

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Step 2.2.1.1.3.1

Simplify each term.

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Step 2.2.1.1.3.1.1

Multiply $5$ by $5$ .

$25 + 5 (- 3 y) - 3 y \cdot 5 - 3 y (- 3 y) + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.1.3.1.2

Multiply $- 3$ by $5$ .

$25 - 15 y - 3 y \cdot 5 - 3 y (- 3 y) + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.1.3.1.3

Multiply $5$ by $- 3$ .

$25 - 15 y - 15 y - 3 y (- 3 y) + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$25 - 15 y - 15 y - 3 \cdot (- 3 y \cdot y) + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.1.3.1.5

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

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Step 2.2.1.1.3.1.5.1

Move $y$ .

$25 - 15 y - 15 y - 3 \cdot (- 3 (y \cdot y)) + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$25 - 15 y - 15 y - 3 \cdot (- 3 y^{2}) + y^{2} = 25$

$x = 5 - 3 y$

$25 - 15 y - 15 y - 3 \cdot (- 3 y^{2}) + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.1.3.1.6

Multiply $- 3$ by $- 3$ .

$25 - 15 y - 15 y + 9 y^{2} + y^{2} = 25$

$x = 5 - 3 y$

$25 - 15 y - 15 y + 9 y^{2} + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.1.3.2

Subtract $15 y$ from $- 15 y$ .

$25 - 30 y + 9 y^{2} + y^{2} = 25$

$x = 5 - 3 y$

$25 - 30 y + 9 y^{2} + y^{2} = 25$

$x = 5 - 3 y$

$25 - 30 y + 9 y^{2} + y^{2} = 25$

$x = 5 - 3 y$

Step 2.2.1.2

Add $9 y^{2}$ and $y^{2}$ .

$25 - 30 y + 10 y^{2} = 25$

$x = 5 - 3 y$

$25 - 30 y + 10 y^{2} = 25$

$x = 5 - 3 y$

$25 - 30 y + 10 y^{2} = 25$

$x = 5 - 3 y$

$25 - 30 y + 10 y^{2} = 25$

$x = 5 - 3 y$

Step 3

Solve for $y$ in $25 - 30 y + 10 y^{2} = 25$ .

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Step 3.1

Subtract $25$ from both sides of the equation.

$25 - 30 y + 10 y^{2} - 25 = 0$

$x = 5 - 3 y$

Step 3.2

Combine the opposite terms in $25 - 30 y + 10 y^{2} - 25$ .

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Step 3.2.1

Subtract $25$ from $25$ .

$- 30 y + 10 y^{2} + 0 = 0$

$x = 5 - 3 y$

Step 3.2.2

Add $- 30 y + 10 y^{2}$ and $0$ .

$- 30 y + 10 y^{2} = 0$

$x = 5 - 3 y$

$- 30 y + 10 y^{2} = 0$

$x = 5 - 3 y$

Step 3.3

Factor $- 10 y$ out of $- 30 y + 10 y^{2}$ .

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Step 3.3.1

Reorder $- 30 y$ and $10 y^{2}$ .

$10 y^{2} - 30 y = 0$

$x = 5 - 3 y$

Step 3.3.2

Factor $- 10 y$ out of $10 y^{2}$ .

$- 10 y (- y) - 30 y = 0$

$x = 5 - 3 y$

Step 3.3.3

Factor $- 10 y$ out of $- 30 y$ .

$- 10 y (- y) - 10 y \cdot 3 = 0$

$x = 5 - 3 y$

Step 3.3.4

Factor $- 10 y$ out of $- 10 y (- y) - 10 y (3)$ .

$- 10 y (- y + 3) = 0$

$x = 5 - 3 y$

$- 10 y (- y + 3) = 0$

$x = 5 - 3 y$

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 $0$ .

$y = 0$

$- y + 3 = 0$

$x = 5 - 3 y$

Step 3.5

Set $y$ equal to $0$ .

$y = 0$

$x = 5 - 3 y$

Step 3.6

Set $- y + 3$ equal to $0$ and solve for $y$ .

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Step 3.6.1

Set $- y + 3$ equal to $0$ .

$- y + 3 = 0$

$x = 5 - 3 y$

Step 3.6.2

Solve $- y + 3 = 0$ for $y$ .

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Step 3.6.2.1

Subtract $3$ from both sides of the equation.

$- y = - 3$

$x = 5 - 3 y$

Step 3.6.2.2

Divide each term in $- y = - 3$ by $- 1$ and simplify.

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Step 3.6.2.2.1

Divide each term in $- y = - 3$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 3}{- 1}$

$x = 5 - 3 y$

Step 3.6.2.2.2

Simplify the left side.

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Step 3.6.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 3}{- 1}$

$x = 5 - 3 y$

Step 3.6.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 3}{- 1}$

$x = 5 - 3 y$

$y = \frac{- 3}{- 1}$

$x = 5 - 3 y$

Step 3.6.2.2.3

Simplify the right side.

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Step 3.6.2.2.3.1

Divide $- 3$ by $- 1$ .

$y = 3$

$x = 5 - 3 y$

$y = 3$

$x = 5 - 3 y$

$y = 3$

$x = 5 - 3 y$

$y = 3$

$x = 5 - 3 y$

$y = 3$

$x = 5 - 3 y$

Step 3.7

The final solution is all the values that make $- 10 y (- y + 3) = 0$ true.

$y = 0, 3$

$x = 5 - 3 y$

$y = 0, 3$

$x = 5 - 3 y$

Step 4

Replace all occurrences of $y$ with $0$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 5 - 3 y$ with $0$ .

$x = 5 - 3 \cdot 0$

$y = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $5 - 3 \cdot 0$ .

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Step 4.2.1.1

Multiply $- 3$ by $0$ .

$x = 5 + 0$

$y = 0$

Step 4.2.1.2

Add $5$ and $0$ .

$x = 5$

$y = 0$

$x = 5$

$y = 0$

$x = 5$

$y = 0$

$x = 5$

$y = 0$

Step 5

Replace all occurrences of $y$ with $3$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 5 - 3 y$ with $3$ .

$x = 5 - 3 \cdot 3$

$y = 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $5 - 3 \cdot 3$ .

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Step 5.2.1.1

Multiply $- 3$ by $3$ .

$x = 5 - 9$

$y = 3$

Step 5.2.1.2

Subtract $9$ from $5$ .

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

Step 6

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

$(5, 0)$

$(- 4, 3)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(5, 0), (- 4, 3)$

Equation Form:

$x = 5, y = 0$

$x = - 4, y = 3$

Step 8