Question

Eliminate the parameter in the given parametric equations. \(x=2 t+5, \quad y=-3 t+1\)

Short answer

The Cartesian equation is \(y = -\frac{3x}{2} + \frac{17}{2}\).

Step-by-step solution

Solve for t in terms of x

The first step is to express the parameter \(t\) from the equation for \(x\). The equation is given by \(x = 2t + 5\). Solve for \(t\) by subtracting 5 from both sides: \(x - 5 = 2t\). Now, divide both sides by 2 to isolate \(t\): \(t = \frac{x - 5}{2}\).

Substitute the expression for t in terms of x into y

Next, substitute the expression for \(t\) obtained in Step 1 into the equation for \(y\). The equation for \(y\) is \(y = -3t + 1\). Replace \(t\) with \(\frac{x - 5}{2}\): \(y = -3\left(\frac{x - 5}{2}\right) + 1\).

Simplify the equation for y

Now, simplify the expression we have to get \(y\) in terms of \(x\). Start by distributing \(-3\): \(y = -\frac{3(x - 5)}{2} + 1\). This can be simplified to: \(y = -\frac{3x - 15}{2} + 1\). Simplifying further, we get: \(y = -\frac{3x}{2} + \frac{15}{2} + 1\). Finally, combine the constants: \(y = -\frac{3x}{2} + \frac{17}{2}\).

Final Cartesian Equation

Write the equation from Step 3 as the Cartesian equation without parameters. From \(y = -\frac{3x}{2} + \frac{17}{2}\), this is the equation in terms of \(x\) and \(y\) (a Cartesian equation).

Key concepts

Cartesian Equation

A Cartesian equation is one of the most common ways to describe a curve or line using its coordinate values in a plane, usually in terms of variables such as \(x\) and \(y\). Instead of relying on a parameter to describe the position or form of the curve, a Cartesian equation relates \(x\) and \(y\) directly.For example, when you want to describe a straight line, you might use the equation \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. In our exercise, we started with parametric equations and ended up with the Cartesian equation:\[y = -\frac{3x}{2} + \frac{17}{2}\]This final Cartesian equation doesn't have the parameter \(t\) anymore, just \(x\) and \(y\), making it easier to understand and plot in the Cartesian coordinate system.

Eliminate the Parameter

Eliminating the parameter in parametric equations is a common task that reduces the equations to a Cartesian form. The objective is to get an equation that directly relates \(x\) and \(y\), without involving the parameter \(t\).In this particular exercise, you start with the given parametric equations:- \(x = 2t + 5\)- \(y = -3t + 1\)The goal is to eliminate \(t\) and obtain a single equation connecting \(x\) and \(y\). This approach simplifies the expression and clarifies the relationship between \(x\) and \(y\) without the complication of a third variable.

Solve for parameter t

To eliminate the parameter, the first step is often to solve one of the parametric equations for \(t\). By expressing \(t\) in terms of one of the other variables, you prepare to substitute this expression into the other equation.From the equation \(x = 2t + 5\), you solve for \(t\). Start by isolating \(t\) on one side:1. Subtract 5 from both sides: \(x - 5 = 2t\)2. Divide both sides by 2: \(t = \frac{x - 5}{2}\)This expression, \(t = \frac{x - 5}{2}\), can now be used to replace \(t\) in the equation for \(y\). It allows the conversion from the parametric form to the Cartesian form.

Substitute parameter into equation

After solving for \(t\), the next step involves substituting this expression into the other parametric equation to eliminate the parameter and form a Cartesian equation.In our exercise, we substitute \(t = \frac{x - 5}{2}\) into the equation for \(y = -3t + 1\). Replace \(t\):- Start with: \(y = -3\left(\frac{x - 5}{2}\right) + 1\)- Distribute the -3: \(y = -\frac{3(x - 5)}{2} + 1\)- Simplify further to get the equation: \(y = -\frac{3x}{2} + \frac{17}{2}\)This substitution step is crucial as it removes \(t\) and leaves us with the desired Cartesian relationship between \(x\) and \(y\). The result is a clear and simplified equation that represents the curve or line without referring back to the parameter.