Solve the Pair of Linear Equations

Video Explanation

Question

Solve the following pair of equations:

\[ \frac{x}{10} + \frac{y}{5} – 1 = 0, \quad \frac{x}{8} + \frac{y}{6} = 15 \]

Hence find the value of \(\lambda\) if \[ y = \lambda x + 5. \]

Solution

Step 1: Convert Equations into Standard Form

First equation:

\[ \frac{x}{10} + \frac{y}{5} = 1 \]

Multiply by 10:

\[ x + 2y = 10 \quad (1) \]

Second equation:

\[ \frac{x}{8} + \frac{y}{6} = 15 \]

Multiply by 24:

\[ 3x + 4y = 360 \quad (2) \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 2y = 10 – x \]

\[ y = \frac{10 – x}{2} \quad (3) \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 3x + 4\left(\frac{10 – x}{2}\right) = 360 \]

\[ 3x + 2(10 – x) = 360 \]

\[ 3x + 20 – 2x = 360 \]

\[ x = 340 \]

Step 4: Find the Value of y

Substitute \(x = 340\) into equation (3):

\[ y = \frac{10 – 340}{2} \]

\[ y = -165 \]

Step 5: Find the Value of λ

Given:

\[ y = \lambda x + 5 \]

Substitute \(x = 340\) and \(y = -165\):

\[ -165 = 340\lambda + 5 \]

\[ 340\lambda = -170 \]

\[ \lambda = -\frac{1}{2} \]

Conclusion

The solution of the given pair of equations is:

\[ x = 340,\quad y = -165 \]

The value of \(\lambda\) is:

\[ \lambda = -\frac{1}{2} \]

\[ \therefore \quad \text{The required value of } \lambda \text{ is } -\frac{1}{2}. \]