How to Solve a System of Linear Equations Using Substitution Elimination and Graphical Method
The Solution of System of Linear Equations is a core topic in algebra, essential for students preparing for school exams (Class 10, 12), competitive tests (like JEE), and those interested in how mathematics solves real-life problems. Understanding how to solve systems helps in topics ranging from physics to economics and technology.
What is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations involving the same set of variables. The solution to a system is the set of variable values that makes every equation in the system true at the same time.
- The most common systems are in two variables (2x2) or three variables (3x3).
- Systems are used to solve real-world problems where several conditions must be satisfied together, such as budgeting, mixture problems, or finding points of intersection in geometry.
Types of Solutions in Systems of Equations
A system of linear equations can have:
- Unique Solution: The equations intersect at exactly one point. This is called a consistent and independent system.
- Infinite Solutions: The equations represent the same line (coincident), so every point on the line is a solution (consistent and dependent system).
- No Solution: The equations represent parallel lines that never meet (inconsistent system).
Geometrically, two lines on a graph will either intersect at a point (unique), lie on top of each other (infinite), or never meet (no solution).
Methods to Solve a System of Linear Equations
Several methods are used to solve systems of linear equations:
- Substitution Method: Solve one equation for one variable and substitute it in the other.
- Elimination Method: Add or subtract equations to eliminate one variable.
- Graphical Method: Graph each equation and find the intersection point(s).
- Matrix Methods: Use matrix algebra, including:
- Inverse Matrix Method
- Cramer’s Rule (uses determinants)
- Gauss Elimination (row operations)
The choice of method depends on the number of variables, the context of the problem, and your confidence with algebra or matrices.
Important Formulas for System of Linear Equations
Here are some crucial formulas and matrix representations:
- Matrix Form: \(AX = B\) where
- \(A\) is the coefficient matrix
- \(X\) is the column of variables
- \(B\) is the constant column
- Inverse Matrix Method: If \(A\) is invertible, then \(X = A^{-1}B\)
- Cramer’s Rule (2x2 Example):
| For system: \( a_1x + b_1y = c_1 \) \( a_2x + b_2y = c_2 \) |
\( x = \dfrac{\begin{vmatrix}c_1 & b_1\\c_2 & b_2 \end{vmatrix}}{\begin{vmatrix}a_1 & b_1\\a_2 & b_2 \end{vmatrix}} \) \( y = \dfrac{\begin{vmatrix}a_1 & c_1\\a_2 & c_2 \end{vmatrix}}{\begin{vmatrix}a_1 & b_1\\a_2 & b_2 \end{vmatrix}} \) |
Where \( \begin{vmatrix} \cdot \end{vmatrix} \) denotes the determinant.
Worked Example: Solving a 2x2 System by Elimination
Solve the system:
\( 2x + 3y = 13 \)
\( x - 2y = -4 \)
- Multiply the second equation by 2:
\( 2x - 4y = -8 \) - Subtract this from the first equation:
\( (2x + 3y) - (2x - 4y) = 13 - (-8) \)
\( 7y = 21 \)
\( y = 3 \) - Substitute \( y = 3 \) in \( x - 2y = -4 \):
\( x - 2(3) = -4 \implies x - 6 = -4 \implies x = 2 \) - Solution: \( x = 2, y = 3 \)
Practice Problems
- Solve by substitution: \( x + y = 7; \; 2x - y = 4 \)
- Use elimination: \( 3x + 2y = 11; \; 2x - y = 1 \)
- Express in matrix form and solve: \( x + 2y = 9; \; 4x - y = 5 \)
- Use Cramer’s Rule: \( 2x - 3y = 1; \; 5x + 2y = 12 \)
- If a system has no solution, what does it mean graphically?
Common Mistakes to Avoid
- Incorrectly aligning variables when using elimination or substitution.
- Arithmetic errors in adding/subtracting or in matrix calculations.
- Forgetting to check if the determinant is zero before applying Cramer’s Rule or inverse matrix method.
- Confusing unique, infinite, and no solution cases.
- Missing the geometric meaning: not graphing or visualizing for quick checks.
Real-World Applications
Systems of linear equations are used in budgeting (comparing two buying options), physics (motion with multiple forces), chemistry (mixing solutions), and business (solving for price and profit). For instance, in a mixture problem, you might have two unknowns—quantity and concentration—which you find using two equations.
Page Summary
In this topic, you learned about the solution of system of linear equations using various methods like substitution, elimination, and matrix techniques. Recognizing the solution type and avoiding common errors are crucial for quick and accurate problem-solving in both exams and real-life scenarios. At Vedantu, we break down each concept into simple steps, helping you master linear systems and excel in mathematics.
You may also like to explore related topics: Linear Equations in One Variable, Matrices, Cramer's Rule, Gauss Elimination Method.
FAQs on Solution of System of Linear Equations in Two Variables
1. What is a system of linear equations?
A system of linear equations is a set of two or more linear equations involving the same variables that are solved simultaneously. Each equation represents a straight line (in two variables) or a plane (in three variables), and the solution is the common point where they intersect. For example:
- 2x + y = 5
- x − y = 1
2. How do you solve a system of linear equations by substitution?
To solve a system by substitution method, express one variable in terms of the other and substitute it into the second equation. Steps:
- Solve one equation for one variable.
- Substitute that expression into the other equation.
- Solve the resulting single-variable equation.
- Substitute back to find the second variable.
- From x − y = 2, get x = y + 2.
- Substitute: (y + 2) + y = 6 → 2y + 2 = 6 → y = 2.
- Then x = 4.
3. How do you solve a system of linear equations by elimination?
The elimination method solves a system by adding or subtracting equations to eliminate one variable. Steps:
- Make the coefficients of one variable equal (if needed).
- Add or subtract the equations to eliminate that variable.
- Solve for the remaining variable.
- Substitute back to find the other variable.
- Add both equations: 4x = 10 → x = 2.5.
- Substitute back: 2(2.5) + y = 7 → y = 2.
4. What is the graphical method of solving a system of linear equations?
The graphical method solves a system by plotting each linear equation and finding their point of intersection. Steps:
- Rewrite each equation in slope-intercept form (y = mx + c).
- Plot both lines on the same coordinate plane.
- Identify the intersection point.
5. What are the possible types of solutions of a system of linear equations?
A system of linear equations can have one solution, no solution, or infinitely many solutions. These cases are:
- Consistent and independent: One unique solution (lines intersect).
- Inconsistent: No solution (parallel lines).
- Consistent and dependent: Infinitely many solutions (same line).
6. What is the formula for solving a system of linear equations using determinants?
The Cramer's Rule formula for two variables uses determinants: x = Dx/D and y = Dy/D. For the system:
- a₁x + b₁y = c₁
- a₂x + b₂y = c₂
- D = a₁b₂ − a₂b₁
- Dx = c₁b₂ − c₂b₁
- Dy = a₁c₂ − a₂c₁
7. How do you know if a system of linear equations has no solution?
A system has no solution when the lines are parallel and never intersect. In algebraic terms, for two equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂:
- If a₁/a₂ = b₁/b₂ but a₁/a₂ ≠ c₁/c₂,
8. Can you give an example of a system of linear equations with infinitely many solutions?
A system has infinitely many solutions when both equations represent the same line. Example:
- 2x + 4y = 8
- x + 2y = 4
9. What is the difference between consistent and inconsistent systems of linear equations?
A consistent system has at least one solution, while an inconsistent system has no solution. Specifically:
- Consistent and independent → one solution.
- Consistent and dependent → infinitely many solutions.
- Inconsistent → no solution.
10. Why are systems of linear equations important in real life?
Systems of linear equations are important because they model real-world problems involving multiple variables and constraints. Common applications include:
- Business: calculating cost and revenue.
- Physics: solving motion and force problems.
- Economics: supply and demand models.
- Engineering: circuit analysis.
