Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph
traced by the particle and the direction of motion.
-$x=5 t+5, y=15 t+9,-\leq t \leq \infty$

A) $y = 3 x - 6$; Entire line, from right to left
B) $y = 3 x + 6$; Entire line, from right to left
C) $y = 3 x - 6$; Entire line, from left to right
D) $y = 3 x + 6$; Entire line, from left to right

Question

Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph
traced by the particle and the direction of motion.
-$x=5 t+5, y=15 t+9,-\leq t \leq \infty$

A) $y = 3 x - 6$; Entire line, from right to left 
B) $y = 3 x + 6$; Entire line, from right to left 
C) $y = 3 x - 6$; Entire line, from left to right 
D) $y = 3 x + 6$; Entire line, from left to right

Quizplus · Accepted Answer

To find a Cartesian equation, we need to eliminate the parameter t. Solving for t in terms of y from the second equation gives us $t=\frac{y-9}{15}$. Substitution into the first equation gives us $x=5 \left(\frac{y-9}{15}\right)+5$ or $x=\frac{y}{3}$. This is the equation of a line with slope 3 passing through the origin. Therefore, the particle's path is a line passing through the origin with a slope of 3. Since the parameter interval is $-\leq t \leq \infty$, the particle covers the entire line from left to right.

The correct choice is C.

Parametric Equations and And a Parameter Interval for the Motion x=5t+5,y=15t+9,−≤t≤∞x=5 t+5, y=15 t+9,-\leq t \leq \inftyx=5t+5,y=15t+9,−≤t≤∞

Parametric Equations and And a Parameter Interval for the Motion $x=5 t+5, y=15 t+9,-\leq t \leq \infty$