Pair of Linear Equations in Two Variables Class 10 Chapter Notes

Key Points to Remember

• A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero.
• A pair of linear equations in two variables is a set of two such equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.
• Each linear equation represents a straight line when graphed on a Cartesian plane.
• The solution to a pair of linear equations is the point (x, y) that satisfies both equations simultaneously.
• Graphical Method: The solution corresponds to the point(s) of intersection of the two lines.
• Algebraic Methods: Substitution, Elimination, and (historically) Cross-multiplication are used to find solutions.
• A system of equations is consistent if it has at least one solution (unique or infinitely many).
• A system of equations is inconsistent if it has no solution.
• Word problems require careful formulation of two linear equations from the given statements.

Key Definitions

Linear Equation in Two Variables

An equation of the form ax + by + c = 0, where a, b, c are real numbers and a² + b² ≠ 0. Its graph is always a straight line.

Pair of Linear Equations

Two linear equations involving the same two variables, typically x and y.

Consistent System

A system of linear equations that has at least one solution (either a unique solution or infinitely many solutions).

Inconsistent System

A system of linear equations that has no solution.

Unique Solution

A single pair of values (x, y) that satisfies both equations in a system. Graphically, the lines intersect at exactly one point.

Infinitely Many Solutions

Every point on the line is a solution because the two equations represent the same line. Graphically, the lines are coincident.

No Solution

There is no pair of values (x, y) that satisfies both equations simultaneously. Graphically, the lines are parallel and never intersect.

Understanding Solutions: Graphical & Algebraic Approaches

Solving a pair of linear equations means finding the values of the variables (usually x and y) that satisfy both equations simultaneously. There are two primary approaches for this: Graphical Method and Algebraic Methods.

Graphical Method

Each linear equation in two variables represents a straight line on a Cartesian coordinate plane. When you graph both equations in a pair, three outcomes are possible, which directly determine the nature of the solution:

1. Intersecting Lines: If the two lines intersect at exactly one point, then that point of intersection (x, y) is the unique solution to the system of equations. In this case, the system is consistent.
2. Parallel Lines: If the two lines are parallel and never intersect, there is no common point satisfying both equations. Therefore, the system has no solution, and it is considered inconsistent.
3. Coincident Lines: If the two lines completely overlap, meaning they are essentially the same line, then every point on the line is a common solution. In this scenario, there are infinitely many solutions, and the system is consistent (and dependent).

Algebraic Methods

Algebraic methods provide exact solutions without needing to plot graphs. The two most common methods are:

1. Substitution Method: In this method, we express one variable in terms of the other from one equation and substitute this expression into the second equation. This reduces the problem to solving a linear equation in a single variable. Once that variable is found, its value is substituted back into the expression to find the other variable.
2. Elimination Method: This method aims to eliminate one of the variables by making its coefficients equal (or additive inverses) in both equations. This is achieved by multiplying one or both equations by suitable non-zero constants. Then, by adding or subtracting the modified equations, one variable is eliminated, leaving a single variable equation to solve. This is often the quickest method for many problems.

Understanding when to use which method and interpreting the results in terms of consistent/inconsistent systems is key to mastering this chapter.

Conditions for Solvability (Ratio Comparison)

Steps for Algebraic Methods

1. Substitution Method — To solve the system (1) a₁x + b₁y + c₁ = 0 and (2) a₂x + b₂y + c₂ = 0:
2. Step 1: Express one variable — From one equation (e.g., equation (1)), express one variable (say, y) in terms of the other variable (x). So, y = ( -c₁ - a₁x ) / b₁.
3. Step 2: Substitute — Substitute this expression for y into the other equation (equation (2)). This will result in an equation with only one variable (x).
4. Step 3: Solve for the variable — Solve the resulting linear equation for x.
5. Step 4: Find the other variable — Substitute the value of x found in Step 3 back into the expression for y (from Step 1) to find the value of y.
6. Elimination Method — To solve the system (1) a₁x + b₁y = -c₁ and (2) a₂x + b₂y = -c₂:
7. Step 1: Make coefficients equal — Multiply one or both equations by suitable non-zero constants so that the coefficients of one variable (e.g., x) become numerically equal (or additive inverses).
8. Step 2: Add or Subtract — Add or subtract the modified equations to eliminate the variable whose coefficients were made equal. This leaves a linear equation in the other single variable.
9. Step 3: Solve for the variable — Solve the resulting equation for the remaining variable.
10. Step 4: Find the other variable — Substitute the value of the found variable into either of the original equations to find the value of the second variable.

Worked Examples

• {"title":"Example 1: Substitution Method","bodyMarkdown":"Problem: Solve: x + y = 14 (1) and x - y = 4 (2)\n\nSolution:\nFrom (1), y = 14 - x. Substitute this into (2):\nx - (14 - x) = 4\nx - 14 + x = 4\n2x = 18\nx = 9\nSubstitute x = 9 back into y = 14 - x:\ny = 14 - 9 = 5\nSolution: (x, y) = (9, 5)"}
• {"title":"Example 2: Elimination Method","bodyMarkdown":"Problem: Solve: 2x + 3y = 11 (1) and 2x - 2y = -4 (2)\n\nSolution:\nNotice the coefficient of x is already the same. Subtract equation (2) from equation (1):\n(2x + 3y) - (2x - 2y) = 11 - (-4)\n2x + 3y - 2x + 2y = 11 + 4\n5y = 15\ny = 3\nSubstitute y = 3 into (1):\n2x + 3(3) = 11\n2x + 9 = 11\n2x = 2\nx = 1\nSolution: (x, y) = (1, 3)"}

Exam Tips & Common Mistakes

1. Word Problems: Read carefully! Identify the two unknown quantities and assign variables. Formulate two correct linear equations from the given conditions. A common mistake is forming only one equation or incorrect equations. Practice is key.
2. Graphical Solutions: When asked for a graphical solution, draw neat graphs with properly labeled axes and scales. Use a sharp pencil and ruler. The point of intersection must be clearly marked and its coordinates accurately read. Label lines with their equations.
3. Algebraic Accuracy: Be careful with signs, especially during substitution or when multiplying equations in the elimination method. A small sign error can lead to a completely wrong answer.
4. Verification: After finding x and y, always substitute these values back into both original equations to verify your solution. This takes a minute but guarantees correctness.
5. Conditions for Solvability: Memorize the a₁/a₂ ratios and their implications for unique, no, or infinite solutions. These are frequently asked in MCQs and short-answer questions.

Practice Questions with Solutions

• Q: What is the condition for a pair of linear equations to have a unique solution? A: The condition is a₁/a₂ ≠ b₁/b₂.
• Q: If the lines represented by a pair of linear equations are parallel, what does it imply about the solutions? A: It implies that the system has no solution and is inconsistent.
• Q: Which algebraic method typically involves multiplying equations by constants to make coefficients equal? A: The Elimination Method.
• Q: For the equations x + 2y = 3 and 2x + 4y = 6, how many solutions exist? A: a₁/a₂ = 1/2, b₁/b₂ = 2/4 = 1/2, c₁/c₂ = 3/6 = 1/2. Since a₁/a₂ = b₁/b₂ = c₁/c₂, there are infinitely many solutions.

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