Calculus Examples

$x = 3 t$ , $y = 3 - t$ , $t = 5$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Replace all occurrences of $t$ with $5$ in each equation.

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Step 1.1.1

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

$x = 3 (5)$

$t = 5$

$y = 3 - t$

Step 1.1.2

Simplify the right side.

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Step 1.1.2.1

Multiply $3$ by $5$ .

$x = 15$

$t = 5$

$y = 3 - t$

$x = 15$

$t = 5$

$y = 3 - t$

Step 1.1.3

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

$y = 3 - (5)$

$x = 15$

$t = 5$

Step 1.1.4

Simplify the right side.

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Step 1.1.4.1

Simplify $3 - (5)$ .

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Step 1.1.4.1.1

Multiply $- 1$ by $5$ .

$y = 3 - 5$

$x = 15$

$t = 5$

Step 1.1.4.1.2

Subtract $5$ from $3$ .

$y = - 2$

$x = 15$

$t = 5$

$y = - 2$

$x = 15$

$t = 5$

$y = - 2$

$x = 15$

$t = 5$

$y = - 2$

$x = 15$

$t = 5$

Step 1.2

List all of the solutions.

$y = - 2, x = 15, t = 5$

Step 2

Reorder $3$ and $- t$ .

$y = - t + 3$