Math

Question Find the standard form of the rectangular equation given the equations x=42tx=4-2t and y=3ty=3-t.

Studdy Solution

STEP 1

Assumptions
1. We have two parametric equations in terms of the parameter tt.
2. The equations are x=42tx=4-2t and y=3ty=3-t.
3. We need to eliminate the parameter tt to find the standard form of the rectangular equation.

STEP 2

To eliminate the parameter tt, we need to solve one of the equations for tt and then substitute it into the other equation.

STEP 3

Solve the equation x=42tx=4-2t for tt.
t=4x2t = \frac{4-x}{2}

STEP 4

Now, substitute the expression for tt from the xx equation into the yy equation.
y=3ty = 3 - t

STEP 5

Replace tt with the expression found in STEP_3.
y=34x2y = 3 - \frac{4-x}{2}

STEP 6

Simplify the equation by distributing the negative sign and combining like terms.
y=342+x2y = 3 - \frac{4}{2} + \frac{x}{2}

STEP 7

Continue simplifying by performing the arithmetic operations.
y=32+x2y = 3 - 2 + \frac{x}{2}

STEP 8

Combine the constant terms.
y=1+x2y = 1 + \frac{x}{2}

STEP 9

Multiply through by 2 to clear the fraction and to obtain the standard form.
2y=2+x2y = 2 + x

STEP 10

Rearrange the terms to get the standard form of the rectangular equation, which is usually written as Ax+By=CAx + By = C.
x2y=2x - 2y = -2
The standard form of the rectangular equation is x2y=2x - 2y = -2.

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