Question

12t = 4v - 3
-6t = 4v + 6
if (t, v) is the solution to the system of equations, what is the value of t - v?

Explanation:

Step1: Rearrange the equations

We have the system of equations:
\[

$$\begin{cases} 12t-4v=- 3\\ -6t - 4v=6 \end{cases}$$

\]

Step2: Subtract the second - equation from the first one

\[

$$\begin{align*} (12t-4v)-(-6t - 4v)&=-3 - 6\\ 12t-4v + 6t+4v&=-9\\ 18t&=-9\\ t&=-\frac{1}{2} \end{align*}$$

\]

Step3: Substitute \(t =-\frac{1}{2}\) into the first equation

\[

$$\begin{align*} 12\times(-\frac{1}{2})&=4v-3\\ -6&=4v - 3\\ 4v&=-3\\ v&=-\frac{3}{4} \end{align*}$$

\]

Step4: Calculate \(t - v\)

\[

$$\begin{align*} t - v&=-\frac{1}{2}-(-\frac{3}{4})\\ &=-\frac{1}{2}+\frac{3}{4}\\ &=\frac{-2 + 3}{4}\\ &=\frac{1}{4} \end{align*}$$

\]

Answer:

\(\frac{1}{4}\)