Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ 2(3u – v) = 5uv, \\ 2(u + 3v) = 5uv \]

Solution

Step 1: Simplify Both Equations

First equation:

\[ 2(3u – v) = 5uv \]

\[ 6u – 2v = 5uv \quad \text{(1)} \]

Second equation:

\[ 2(u + 3v) = 5uv \]

\[ 2u + 6v = 5uv \quad \text{(2)} \]

Step 2: Express One Expression in Terms of the Other

From equations (1) and (2):

\[ 6u – 2v = 2u + 6v \]

\[ 4u = 8v \]

\[ u = 2v \quad \text{(3)} \]

Step 3: Substitute in Equation (1)

Substitute equation (3) into equation (1):

\[ 6(2v) – 2v = 5(2v)v \]

\[ 12v – 2v = 10v^2 \]

\[ 10v = 10v^2 \]

\[ v = 1 \]

Step 4: Find the Value of u

Substitute \(v = 1\) into equation (3):

\[ u = 2(1) = 2 \]

Conclusion

The solution of the given system of equations is:

\[ u = 2,\quad v = 1 \]

\[ \therefore \quad \text{The solution is } (2,\;1). \]