What is Pair of Linear Equations in Two Variables Class 10: Definition & Basics

Definition of Pair of Linear Equations in Two Variables

A pair of linear equations in two variables refers to two equations where each is linear and involves exactly two variables, usually $x$ and $y$. The general form is:

$$ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} $$

Here, $a_1$, $b_1$, $c_1$, $a_2$, $b_2$, and $c_2$ are constants, and $x$, $y$ are variables. Each equation represents a straight line when plotted on the Cartesian plane.

The solution to this pair is the set of values $(x, y)$ that satisfy both equations simultaneously.

Graphical Representation and Interpretation

Graphically, each linear equation in two variables corresponds to a straight line on the coordinate plane.

• Plot each equation as a line using intercepts or slope-intercept form.
• The solution to the pair is the point(s) where the two lines intersect.

There are three possibilities:

Understanding these helps in visualising the nature of solutions.

Methods to Solve Pair of Linear Equations

Class 10 NCERT Mathematics teaches three main methods to solve a pair of linear equations:

1. Graphical Method

• Plot both lines on graph paper.
• Identify the intersection point.

2. Substitution Method

• Solve one equation for one variable.
• Substitute into the other equation.
• Solve the resulting single-variable equation.

3. Elimination Method

• Multiply equations to align coefficients.
• Add or subtract equations to eliminate one variable.
• Solve for the remaining variable.

Worked Example (Substitution Method):

Solve:

$$ \begin{cases} 2x + 3y = 12 \\ x - y = 3 \end{cases} $$

• From second equation: $x = y + 3$
• Substitute into first:

$$2(y + 3) + 3y = 12$$ $$2y + 6 + 3y = 12$$ $$5y + 6 = 12$$ $$5y = 6$$ $$y = \frac{6}{5}$$

• Substitute $y$ back:

$$x = \frac{6}{5} + 3 = \frac{6}{5} + \frac{15}{5} = \frac{21}{5}$$

Solution: $\left(\frac{21}{5}, \frac{6}{5}\right)$

Applications of Pair of Linear Equations in Real Life

Pair of linear equations is not just a theoretical topic but has practical uses:

• Business: Calculating cost and profit where two variables affect outcomes.
• Physics: Solving problems involving speed, distance, and time.
• Chemistry: Balancing chemical equations with two unknowns.
• Everyday problems: Sharing money, mixing solutions, or comparing quantities.

Example: If two shops sell pens and pencils at different prices, pair of linear equations can determine the number of pens and pencils bought given total cost and quantity.

Comparison of Methods to Solve Pair of Linear Equations

Choosing the right method depends on the problem type and ease of calculation.

Students should practice all methods to handle diverse problems confidently.