Linear Equations in Two - Class 10 Haryana Board Mathematics

Linear Equations in Two Variables

A linear equation in two variables is an equation that can be written in the form $ax + by + c = 0$, where $x$ and $y$ are variables and $a$, $b$, $c$ are real numbers with $a$ and $b$ not both zero. The degree of each variable is one. When two such equations are considered together, they form a pair of linear equations in two variables, often referred to as a system or pair.

Meaning and Structure of Linear Equations

A linear equation in one variable has the form $ax + b = 0$,…

Linear Equations in Two Variables

A linear equation in two variables is an equation that can be written in the form $ax + by + c = 0$, where $x$ and $y$ are variables and $a$, $b$, $c$ are real numbers with $a$ and $b$ not both zero. The degree of each variable is one. When two such equations are considered together, they form a pair of linear equations in two variables, often referred to as a system or pair.

Meaning and Structure of Linear Equations

A linear equation in one variable has the form $ax + b = 0$, where the variable has degree one. In two variables, the equation takes the form $ax + by + c = 0$. The solution to such an equation is a pair of values $(x, y)$ that satisfy the equation.

• Example (One Variable): $5x + 10 = 0$ (degree of $x$ is one)
• Example (Two Variables): $5x + 2y - 10 = 0$ (degrees of $x$ and $y$ are both one)

General Form of a Pair of Linear Equations

A pair of linear equations in two variables can be written as:

Here, $a_1, b_1, c_1, a_2, b_2, c_2$ are real numbers, and $x, y$ are variables.

Methods to Solve a Pair of Linear Equations

There are three main methods to solve a pair of linear equations in two variables:

• Graphical Method (ग्राफीय विधि): Plot both equations on the coordinate plane and find their point of intersection.
• Substitution Method (प्रतिस्थापन विधि): Express one variable in terms of the other using one equation, then substitute into the second equation.
• Elimination Method (विलोपन विधि): Add or subtract the equations to eliminate one variable, then solve for the other.

Graphical Interpretation and Types of Solutions

The graphical method involves plotting both equations as straight lines. The nature of their intersection determines the number of solutions:

• If the lines intersect at a single point, there is one solution (consistent pair).
• If the lines coincide (are the same), there are infinitely many solutions (consistent pair).
• If the lines are parallel and distinct, there is no solution (inconsistent pair).

Criteria for Types of Solutions

• Intersecting Lines: $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ — one solution.
• Coincident Lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ — infinitely many solutions.
• Parallel Lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ — no solution.

Consistent and Inconsistent Pairs

A pair of equations is called consistent if it has at least one solution (either one or infinitely many). If there is no solution, the pair is called inconsistent.

• Consistent Pair: Lines intersect at one point or coincide.
• Inconsistent Pair: Lines are parallel and do not intersect.

Stepwise Solution: Substitution Method

The substitution method involves the following steps:

1. Express one variable in terms of the other using one equation.
2. Substitute this expression into the second equation to get an equation in one variable.
3. Solve for the variable.
4. Substitute back to find the other variable.

Example: Given $x + y = 14$ and $x - y = 4$:

• From the second equation: $x = 4 + y$
• Substitute into the first: $(4 + y) + y = 14 \implies 2y = 10 \implies y = 5$
• Substitute $y = 5$ into $x = 4 + y$: $x = 9$

Stepwise Solution: Elimination Method

The elimination method involves the following steps:

1. Arrange both equations in standard form.
2. Multiply one or both equations by suitable numbers to make the coefficients of one variable equal.
3. Add or subtract the equations to eliminate one variable.
4. Solve for the remaining variable.
5. Substitute back to find the other variable.

Example: Given $x + y = 14$ and $x - y = 4$:

• Add both equations: $(x + y) + (x - y) = 14 + 4 \implies 2x = 18 \implies x = 9$
• Substitute $x = 9$ into $x + y = 14$: $9 + y = 14 \implies y = 5$

Application to Word Problems

Word problems involving pairs of linear equations require translating the problem statement into equations.

• Assign variables to unknowns.
• Form equations based on the relationships described.
• Solve using any method.

Example: "The difference between two numbers is 26. One number is three times the other."

• Let greater number = $x$, smaller number = $y$
• $x - y = 26$
• $x = 3y$
• Substitute: $3y - y = 26 \implies 2y = 26 \implies y = 13$, $x = 39$

Special Cases and Homework Examples

Some problems require forming a second equation to ensure a specific type of solution (intersecting, parallel, or coincident lines).

• To ensure intersecting lines: Choose coefficients so that $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$.
• To ensure parallel lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.
• To ensure coincident lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

Example: Given $2x + 3y - 8 = 0$, to make the pair intersecting, choose another equation such as $4x + 9y - 16 = 0$ (ratios of $a$ and $b$ are not equal).

Key Ratios and Decision Table

The following table summarizes the relationship between the ratios of coefficients and the type of solution:

Practice and Exam-Focused Takeaways

• Always check ratios of coefficients to determine the nature of solutions.
• For graphical solutions, plot points by assigning values to one variable and solving for the other.
• In substitution, always isolate one variable completely before substituting.
• In elimination, align coefficients before adding or subtracting equations.
• Translate word problems stepwise: assign variables, form equations, solve systematically.
• Consistency and inconsistency are determined by the existence of solutions, not by the method used.
• All methods (graphical, substitution, elimination) yield the same solution if applied correctly.