Calculus Examples

$y^{2} - x y - 10 = 0$ , $x = 3$

Step 1

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

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Step 1.1

Replace all occurrences of $x$ in $y^{2} - x y - 10 = 0$ with $3$ .

$y^{2} - 3 \cdot y - 10 = 0$

$x = 3$

Step 1.2

Simplify the left side.

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Step 1.2.1

Multiply $- 3$ by $y$ .

$y^{2} - 3 y - 10 = 0$

$x = 3$

$y^{2} - 3 y - 10 = 0$

$x = 3$

$y^{2} - 3 y - 10 = 0$

$x = 3$

Step 2

Solve for $y$ in $y^{2} - 3 y - 10 = 0$ .

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Step 2.1

Factor $y^{2} - 3 y - 10$ using the AC method.

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Step 2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 10$ and whose sum is $- 3$ .

$- 5, 2$

$x = 3$

Step 2.1.2

Write the factored form using these integers.

$(y - 5) (y + 2) = 0$

$x = 3$

$(y - 5) (y + 2) = 0$

$x = 3$

Step 2.2

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

$y - 5 = 0$

$y + 2 = 0$

$x = 3$

Step 2.3

Set $y - 5$ equal to $0$ and solve for $y$ .

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Step 2.3.1

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

$x = 3$

Step 2.3.2

Add $5$ to both sides of the equation.

$y = 5$

$x = 3$

$y = 5$

$x = 3$

Step 2.4

Set $y + 2$ equal to $0$ and solve for $y$ .

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Step 2.4.1

Set $y + 2$ equal to $0$ .

$y + 2 = 0$

$x = 3$

Step 2.4.2

Subtract $2$ from both sides of the equation.

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

Step 2.5

The final solution is all the values that make $(y - 5) (y + 2) = 0$ true.

$y = 5, - 2$

$x = 3$

$y = 5, - 2$

$x = 3$

Step 3

Solve the system of equations.

$y = 5, x = 3$

Step 4

Solve the system of equations.

$y = - 2, x = 3$

Step 5

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

$(3, 5)$

$(3, - 2)$