Linear differentiation equation is a form of the equation that is linear polynomial in nature. It is a part of the new derivatives and functions of an equation within a structure. A linear equation usually consists of derivatives of multiple variables. When a result and variable is partial, then it is also stated as a linear partial differential equation.
Let’s take an example where the linear equation is in one variable 4x = 2x + 2. A student can quickly figure out that the value of x here is one. When using matrix inverse to solve a system of 3 linear equations, simultaneous concepts are employed to find variables. If a linear equation consists of several variables, then ways to solve using the inverse matrix becomes complex.
This section tries to clear the concepts of applying matrix inverse methods for solving a system of equations. These concepts have been explained in detail with examples for an easy understanding.
By understanding the concepts, students will know how to use the given inverse to solve the system of equations.
How to Use an Inverse Matrix to Solve the System in Linear Equations
Here is a detailed explanation on the way to use an inverse matrix to solve the linear system.
Equations needed to solve using inverse matrix require a system like-
e1x+f1y+g1z=d1
e2x+f2y+g2z=d2
e3x+f3y+g3z=d3
Students will see that here d1, d2, d3 don’t have the value as zero
If we take A as | [|e1e2e3 f1f2f3 g1g2g3|] |
Now we can consider B’s value to be |[|XYZ|] |
And C here is ⎡⎣⎢d1d2d3|] |
Ideally, when taking the inverse matrix to solve a linear system, one requires a coefficient matrix. Here B can be considered as a coefficient of a linear system.
This makes the following equation as |[|e1e2e3f1f2f3g1g2g3|]| xyz |[|d1d2d3|]|
By considering the above system, one can conclude the equation of a system to be Ax = B
This further gives two visuals or cases that can be explained when one uses inverse to solve a system of equations.
Visual 1: Here one can see that A is a matrix in the non-singular form
We know that the value of a matrix’s determinant is not zero. This fact proves that the inverse of a matrix subsists in an equation.
Following the above-said statement, we find Ax = B
To find the ultimate value, we multiply both of its sides with A-1
This gives the value A-1(Ax) which is equal to A -1B
Finally using associative property, it gives (A-1A) x which is again equal to A -1B
Ultimately, the value of x will be A-1B
Here an individual can find an exclusive solution due to the inverse nature of the matrix in a linear equation.
Visual 2: In the second visual we see A as a singular matrix
It is known that a singular matrix has a value zero in case of determinants. Therefore, the value of adjacent (A)B will be found.
Here adjacent (A)B is not equal to 0, which makes the solution to be void making an equation of a system to be inconsistent.
Suppose there is a case where adj (A)B is equal to zero, which gives the value of a linear equation to be consistent or inconsistent. This can be based on multiple solutions or lack of one.
Apart from solving systems using inverse matrices, students need to practice equations based on linear equations. They should follow quality test papers and exercise materials, offering multiple questions on the inverse method to solve the system of equations.
One can check Vedantu, which is a reliable education portal offering multiple benefits. They provide solutions and tips to use the inverse of a matrix to solve linear systems. Moreover, students can check their live classes and training sessions available for a budget-friendly price. To learn how to solve using the inverse matrix, download the official app today!
FAQs on How to Solve the System of Linear Equations in Two Variables or Three Variables?
1. What is a system of linear equations?
A system of linear equations is a set of two or more linear equations that involve the same variables. The solution to the system is the set of values for the variables that satisfies all equations simultaneously. For two variables, this solution is the point where their lines intersect on a graph.
2. What are the main methods to solve linear equations in two variables?
For systems with two variables, as covered in the CBSE Class 10 syllabus, the primary methods are:
- Substitution Method: Solving one equation for one variable and substituting that expression into the other equation.
- Elimination Method: Adding or subtracting the equations to eliminate one variable.
- Graphical Method: Graphing both equations and finding their point of intersection.
- Cross-Multiplication Method: A formula-based approach to directly find the values of the variables.
3. How can we solve a system of linear equations with three variables?
Solving a system with three variables (x, y, and z) requires three equations. You can extend the elimination or substitution methods by first reducing the three equations to a system of two equations with two variables. For more complex systems, methods like Cramer's Rule (using determinants) and the Matrix Method (using the inverse of a matrix), taught in Class 12, are more efficient.
4. What is the difference between a consistent and an inconsistent system of equations?
A consistent system of equations is one that has at least one solution. This solution can be either unique (one set of values) or have infinitely many solutions. In contrast, an inconsistent system has no solution at all. Graphically, the lines of an inconsistent system are parallel and never intersect.
5. How can you tell if a system of equations will have a unique, infinite, or no solution without solving it?
For a pair of linear equations in the form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, you can compare the ratios of their coefficients:
- If a₁/a₂ ≠ b₁/b₂, the system has a unique solution (consistent).
- If a₁/a₂ = b₁/b₂ = c₁/c₂, the system has infinitely many solutions (consistent).
- If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the system has no solution (inconsistent).
6. When is it better to use the substitution method instead of the elimination method?
The substitution method is often easier and quicker when one of the variables in an equation has a coefficient of 1 or -1, making it simple to isolate. The elimination method is generally more efficient when the coefficients of one variable are already the same or are opposites, allowing for direct addition or subtraction to eliminate it.
7. What are some real-world examples where we use systems of linear equations?
Systems of linear equations are used to model and solve many real-world problems. For example, they can help a business find its break-even point (where cost equals revenue), determine the right quantities for a mixture in chemistry, or calculate speeds, distances, and times in problems involving travel.
8. How does the matrix method work for solving three linear equations?
The matrix method, often taught in Class 12 Maths, involves representing the system of equations in the form AX = B. Here, 'A' is the matrix of coefficients, 'X' is the matrix of variables (x, y, z), and 'B' is the matrix of constants. The solution is found by calculating X = A⁻¹B, where A⁻¹ is the inverse of the coefficient matrix A.
