Question

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y\) $$x=3-t, y=3+t$$

Short answer

Question: Eliminate the parameter t from the parametric equations x = 3 - t and y = 3 + t. Answer: y = -x + 6

Step-by-step solution

Solve one of the parametric equations for \(t\)

Let's solve the first parametric equation, \(x = 3 - t\), for \(t\). To do this, subtract 3 from both sides and then multiply by -1: $$t = -x + 3$$ Now we have found an expression for \(t\) in terms of \(x\).

Substitute the expression for t into the second parametric equation

Now that we have an expression for \(t\), we can substitute it into the second parametric equation \(y = 3 + t\). Replace \(t\) with the expression we found in step 1: $$y = 3 + (-x + 3)$$

Simplify the equation to get the final result

Now, we just need to simplify the equation we found in step 2: $$y = 3 - x + 3$$ So, the equation relating \(x\) and \(y\) without the parameter \(t\) is: $$y = -x + 6$$

Key concepts

Eliminating Parameters

To convert parametric equations into a more familiar form, we often need to eliminate the parameter. The parameter is a third variable, usually denoted as \( t \), that is used to express the equations in terms of two other variables, commonly \( x \) and \( y \). By eliminating this parameter, we transition from two equations involving \( t \) to a single equation involving only \( x \) and \( y \). In the exercise provided, we began with

• \( x = 3 - t \)
• \( y = 3 + t \)

To eliminate \( t \), solve one equation for the parameter. Here, we solved \( x = 3 - t \) for \( t \) and found: \( t = -x + 3 \). This step is crucial because it allows us to express \( t \) in terms of \( x \) or \( y \), enabling further substitution.

Parametric to Cartesian Conversion

Once the parameter has been expressed in terms of \( x \) or \( y \), the next step is to substitute it into the counterpart equation. This transforms the parametric equations into a Cartesian form, which is a single equation involving only \( x \) and \( y \). Using our previous find, \( t = -x + 3 \), we substitute into the equation \( y = 3 + t \), resulting in: \[ y = 3 + (-x + 3) \] This gives us an equation that directly shows the relationship between \( x \) and \( y \) without involving the parameter \( t \). This type of conversion is highly useful in calculus and analytical geometry, as it simplifies understanding the relationships between variables.Finally, simplify the equation. For example,

• \( y = 3 - x + 3 \)

Simplifying further, you obtain \( y = -x + 6 \), a clear representation of the relationship in Cartesian form.

Linear Equations

The result of eliminating the parameter and converting the equation brings us to a linear equation. A linear equation is one that can be graphed as a straight line. In our solution, after simplification, we arrived at the linear equation: \[ y = -x + 6 \]This equation is in the slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

• Slope \( m \): This indicates the direction and steepness of the line. Here, \( m = -1 \), meaning the line slopes downward from left to right.
• Y-Intercept \( b \): The point where the line crosses the y-axis is \( y = 6 \).

Linear equations such as this are fundamental in algebra and help describe simple relationships between two variables. They provide a straightforward way to anticipate one variable based on another.