Problem 154 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the value of \(x, y, z\) and \(a\) which satisfy the matrix equation \(\left[\begin{array}{cc}x+3 & 2 y+x \\ z-1 & 4 a-6\end{array}\right]=\left[\begin{array}{cc}0 & -7 \\ 3 & 2 a\end{array}\right]\)

Step-by-Step Solution

Answer

After solving, we find that \(x = -3\), \(y = -2\), \(z = 4\) and \(a = 3\).

Step 1: Form Equations

Form four equations from the matrix equation \(\begin{bmatrix}x+3 & 2y+x \ z-1 & 4a-6\end{bmatrix}=\begin{bmatrix}0 & -7 \ 3 & 2a\end{bmatrix}\) \[x+3=0, 2y+x=-7, z-1=3, 4a-6=2a\].

Step 2: Solve for x

Using the equation \(x + 3 = 0\), we can subtract 3 from both sides to get \(x = -3\).

Step 3: Substitute x into the Second Equation

Substitute \(x=-3\) into the second equation: \(2y + x = -7\) to obtain \(2y + (-3) = -7\). Simplifying this gives \(2y = -4\), thus, \(y = -2.\)

Step 4: Solve for z

Solve the third equation \(z - 1 = 3\) by adding 1 to both sides to obtain \(z = 4\).

Step 5: Solve for a

Solve the fourth equation \(4a - 6 = 2a\) by subtracting \(2a\) from both sides to get \(2a = 6\). Thus, \(a = 3\).

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear mappings, and the systems of linear equations. It is essential for solving problems where multiple variables interact with each other, and it forms the foundation for many areas of mathematics and applied science. Matrices are a key tool in linear algebra, providing a compact way to represent and manipulate linear systems. In the context of the exercise, the matrix equation is used to represent a system of linear equations with multiple variables \( x, y, z,\text{and} a\). By equating two matrices, we set up a system of linear equations that can be solved using algebraic operations to find the values of the unknowns.

Systems of Equations

A system of equations is a set of multiple equations that are solved together to find common solutions for the variables involved. For a system to have a unique solution, the number of distinct equations should be equal to the number of unknowns. Solving systems of equations can be approached in different ways, such as substitution, elimination, or matrix methods like row reduction. In the exercise, we obtained a system of equations from the matrix equation and solved it systematically: by finding the value of one variable and then substituting it into other equations to find the others. This is an example of the substitution method, which is particularly effective for systems where one equation can be easily isolated for a specific variable.

Algebraic Operations

Algebraic operations include addition, subtraction, multiplication, and division applied to algebraic expressions. In solving systems of equations, these operations are used to manipulate the equations to isolate and solve for the unknowns. In the step-by-step solution from the exercise, we applied subtraction to isolate \(x\), followed by substitution and further arithmetic to find \(y\), \(z\), and \(a\). Understanding how to properly apply these operations is vital to reach the correct solution for the system of equations. Additionally, one must perform the same operation on both sides of the equation to maintain its balance, which is a fundamental concept in algebra.