# Consider the parametric equations x = 3t - 5 and y = 2t + 3 how do you eliminate the parameter to find a Cartesian equation of the curve?

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To eliminate the parameter ( t ) and find the Cartesian equation of the curve, we can isolate ( t ) in one of the parametric equations and substitute it into the other equation.

Given the parametric equations: [ x = 3t - 5 ] [ y = 2t + 3 ]

From the first equation: [ t = \frac{x + 5}{3} ]

Substitute ( t ) into the second equation: [ y = 2\left(\frac{x + 5}{3}\right) + 3 ]

Simplify: [ y = \frac{2x + 10}{3} + 3 ]

Multiply through by 3 to clear the fraction: [ 3y = 2x + 10 + 9 ]

Combine like terms: [ 3y = 2x + 19 ]

Finally, rearrange to obtain the Cartesian equation: [ 2x - 3y + 19 = 0 ]

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