Solve the System of Equations by the Method of Cross-Multiplication

Video Explanation

Question

Solve the following system of equations by the method of cross-multiplication:

\[ \frac{57}{x+y} + \frac{6}{x-y} = 5,\quad \frac{38}{x+y} + \frac{21}{x-y} = 9 \]

Solution

Step 1: Substitute Variables

Let

\[ \frac{1}{x+y} = u,\quad \frac{1}{x-y} = v \]

Then the given equations become:

\[ 57u + 6v = 5 \quad \text{(1)} \]

\[ 38u + 21v = 9 \quad \text{(2)} \]

Step 2: Apply Cross-Multiplication Method

Comparing with the standard form:

\[ a_1u + b_1v = c_1,\quad a_2u + b_2v = c_2 \]

We get:

\[ a_1 = 57,\ b_1 = 6,\ c_1 = 5 \]

\[ a_2 = 38,\ b_2 = 21,\ c_2 = 9 \]

Using cross-multiplication:

\[ \frac{u}{(b_1c_2 – b_2c_1)} = \frac{v}{(a_2c_1 – a_1c_2)} = \frac{1}{(a_1b_2 – a_2b_1)} \]

\[ \frac{u}{(6\cdot9 – 21\cdot5)} = \frac{v}{(38\cdot5 – 57\cdot9)} = \frac{1}{(57\cdot21 – 38\cdot6)} \]

\[ \frac{u}{(-51)} = \frac{v}{(-323)} = \frac{1}{969} \]

Step 3: Find u and v

\[ u = -\frac{17}{323},\quad v = -\frac{1}{3} \]

Step 4: Find x and y

\[ \frac{1}{x+y} = -\frac{17}{323} \Rightarrow x + y = -19 \]

\[ \frac{1}{x-y} = -\frac{1}{3} \Rightarrow x – y = -3 \]

Solving the two equations:

\[ x = -11,\quad y = -8 \]

Conclusion

The solution of the given system of equations is:

\[ x = -11,\quad y = -8 \]

\[ \therefore \quad \text{The solution is } (-11,\;-8). \]