To write the set of parametric equations [tex]\( x = \frac{1}{2} t + 3 \)[/tex] and [tex]\( y = 2t - 1 \)[/tex] in rectangular form by eliminating [tex]\( t \)[/tex], we need to solve each equation for [tex]\( t \)[/tex] and then compare these expressions to find a consistent value for [tex]\( t \)[/tex].
First, let’s solve the equation [tex]\( x = \frac{1}{2} t + 3 \)[/tex] for [tex]\( t \)[/tex]:
1. Subtract 3 from both sides of the equation:
[tex]\[ x - 3 = \frac{1}{2} t \][/tex]
2. Multiply both sides by 2 to solve for [tex]\( t \)[/tex]:
[tex]\[ t = 2(x - 3) \][/tex]
So, one possible equation to help eliminate [tex]\( t \)[/tex] is:
[tex]\[ t = 2(x - 3) \][/tex]
Now, let’s solve the equation [tex]\( y = 2t - 1 \)[/tex] for [tex]\( t \)[/tex]:
1. Add 1 to both sides of the equation:
[tex]\[ y + 1 = 2t \][/tex]
2. Divide both sides by 2 to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{y + 1}{2} \][/tex]
So, another possible equation to help eliminate [tex]\( t \)[/tex] is:
[tex]\[ t = \frac{y + 1}{2} \][/tex]
Given the choices provided:
1. [tex]\( t = 2(x - 3) \)[/tex]
2. [tex]\( t = 2(x + 3) \)[/tex]
3. [tex]\( t = \frac{y - 1}{2} \)[/tex]
4. [tex]\( t = 2(y + 1) \)[/tex]
We can see that the correct equations that match our solved values for [tex]\( t \)[/tex] are:
1. [tex]\( t = 2(x - 3) \)[/tex]
3. [tex]\( t = \frac{y + 1}{2} \)[/tex]
Thus, the correct choices are:
- [tex]\( t = 2(x - 3) \)[/tex]
- [tex]\( t = \frac{y + 1}{2} \)[/tex]
Hence, Jimmy can use these equations to eliminate [tex]\( t \)[/tex] for writing the parametric equations in rectangular form. The correct choices are:
(1) [tex]\( t = 2(x - 3) \)[/tex] and (3) [tex]\( t = \frac{y + 1}{2} \)[/tex].
