Solve the simultaneous equation using elimination method

To solve the simultaneous equations using the elimination method, we have the following equations:

1. \( 2u + 7v = 10 \) (Equation 1)
2. \( 3u + v = 6 \) (Equation 2)

We can eliminate one of the variables by making the coefficients of \(v\) in both equations equal. Let's eliminate \(v\).

First, we can multiply Equation 2 by 7 to match the coefficient of \(v\) in Equation 1:

\[
7(3u + v) = 7(6)
\]

This gives us:

\[
21u + 7v = 42 \quad \text{(Equation 3)}
\]

Now we have:

1. \( 2u + 7v = 10 \) (Equation 1)
2. \( 21u + 7v = 42 \) (Equation 3)

Next, we can subtract Equation 1 from Equation 3:

\[
(21u + 7v) - (2u + 7v) = 42 - 10
\]

This simplifies to:

\[
21u - 2u + 7v - 7v = 32
\]
\[
19u = 32
\]

Now, solve for \(u\):

\[
u = \frac{32}{19}
\]

Next, we substitute \(u\) back into one of the original equations to find \(v\). We can use Equation 2:

\[
3u + v = 6
\]

Substituting \(u\):

\[
3\left(\frac{32}{19}\right) + v = 6
\]
\[
\frac{96}{19} + v = 6
\]

To isolate \(v\), we subtract \(\frac{96}{19}\) from both sides:

\[
v = 6 - \frac{96}{19}
\]

Converting 6 into a fraction with a denominator of 19:

\[
v = \frac{114}{19} - \frac{96}{19}
\]
\[
v = \frac{114 - 96}{19}
\]
\[
v = \frac{18}{19}
\]

Thus, the solution to the simultaneous equations is:

\[
u = \frac{32}{19}, \quad v = \frac{18}{19}
\]

Final Answer:

\((u, v) = \left( \frac{32}{19}, \frac{18}{19} \right)\)