Problem 129 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the value of \(\lambda\) if the following equations are consistent \(x+y-3=0\) \((1+\lambda) x+(2+\lambda) y-8=0\) \(x-(1+\lambda) y+(2+\lambda)=0\)

Step-by-Step Solution

Answer

The value of \(\lambda\) that makes the equations consistent is \(\lambda = \frac{5}{3}\).

Step 1: Establish the Ratio

To start, establish the ratio of the coefficients of the first equation to those of the second equation. This would give:\(\frac{1}{1+\lambda} = \frac{1}{2+\lambda} = \frac{-3}{-8}\)

Step 2: Solve for \(\lambda\)

Next, solve the resulting equations for \(\lambda\). Setting \(\frac{1}{1+\lambda} = \frac{-3}{-8}\), we can solve to obtain \(\lambda = \frac{5}{3}\).

Step 3: Verify the Ratio

Finally, it's important to note that these values will be valid only if the established ratio holds for all three equations. The last ratio from the first equation to the third equation is \(\frac{1}{1} = \frac{1}{1-\lambda} = \frac{-3}{2+\lambda}\). Substituting \(\lambda = \frac{5}{3}\) into this, we observe that these ratios hold constant. Hence, we have consistency throughout the equations.

Key Concepts

System of Equations

When dealing with linear equations, a system of equations consists of two or more equations that share common variables. The goal of solving a system of equations is to find the values of these variables that satisfy all the equations simultaneously. In real-world applications, systems of equations can model situations where multiple conditions must be met at once. For our exercise, the system consists of three equations:

• \( x + y - 3 = 0 \)
• \((1+\lambda) x + (2+\lambda) y - 8 = 0\)
• \(x - (1+\lambda) y + (2+\lambda) = 0\)

Each equation represents a line in a plane, and solving the system means finding the intersection point of these lines. Understanding this concept is crucial because it reveals whether a solution exists and if so, whether it is unique (one solution) or not (infinite solutions).

Consistency of Equations

A system of equations is considered consistent if there exists at least one set of values for the variables that satisfies all the equations. In simpler terms, the lines in the system intersect at least once. If such values cannot be found because the lines are parallel and never meet, the system is inconsistent.
The exercise requires checking if the equations are consistent by ensuring the ratios of the coefficients from different equations match. If the ratios hold, the lines intersect at a common point, showing the system's consistency. In this particular task, we find values of \( \lambda \) that make the equations intersect, confirming consistency.
Verify ratios ensure all lines align correctly. This is done by establishing that specific proportionality holds across all equations, which signifies they can potentially share an intersection. When solving for \( \lambda = \frac{5}{3} \), the ratios verify, confirming consistency, and thus, a solution exists.

Solving for Variables

Solving for variables involves finding the specific values that satisfy all equations in the system simultaneously. In our task, the variable to solve for was \( \lambda \), which is found by equating ratios and performing algebraic manipulations.

• First, set the ratios: \( \frac{1}{1+\lambda} = \frac{-3}{-8} \).
• Solve algebraically to find \( \lambda = \frac{5}{3} \).

After finding \( \lambda \), substitute it back into the equations to reassess consistency, ensuring the solution holds in all system equations.
This process often reveals whether multiple, single, or no solutions exist, shaping how we interpret the situation described by the equations. Solving systems of equations is an essential skill in mathematics, helping to unravel the relationships between different quantities and find practical solutions to complex problems.