Comprehensive theory, key formulas, diagrams, and memory aids for Linear Equations in Two Variables.
Equations are the backbone of algebra, allowing us to find unknown values when given specific conditions. While you have previously studied linear equations in one variable, many real-life situations involve two unknown quantities. In this chapter, we will explore linear equations in two variables, their solutions, and their representation on the Cartesian plane.
An equation is a mathematical statement asserting that two expressions are equal. A linear equation in one variable is an equation of the form $ax + b = 0$, where $a$ and $b$ are real numbers and $a \neq 0$. Examples: $2x + 5 = 0$, $\sqrt{3}y - 7 = 0$.
Linear equations in one variable have a unique (one and only one) solution. For instance, the solution to $2x + 5 = 0$ is $x = -5/2$. This solution can be represented as a unique point on the number line.
However, consider a scenario: "In a cricket match, two batsmen together scored 176 runs." If we let the runs scored by the first batsman be $x$ and the runs scored by the second batsman be $y$, we can represent this situation algebraically as: $x + y = 176$ This is a linear equation in two variables.
Any equation which can be put in the form: $ax + by + c = 0$ where $a, b,$ and $c$ are real numbers, and $a$ and $b$ are not both zero ($a^2 + b^2 \neq 0$), is called a linear equation in two variables.
Example: Express $2x = 3y + 5$ in the general form. Subtracting $3y$ and $5$ from both sides, we get: $2x - 3y - 5 = 0$. Here, $a = 2$, $b = -3$, and $c = -5$.
Equations like $x = 5$ or $y = -2$ can also be expressed as linear equations in two variables: $x = 5$ can be written as $1 \cdot x + 0 \cdot y - 5 = 0$. $y = -2$ can be written as $0 \cdot x + 1 \cdot y + 2 = 0$.
A solution to a linear equation in two variables is a pair of values, one for $x$ and one for $y$, which makes the two sides of the equation equal. This is often written as an ordered pair $(x, y)$.
Let's find a solution for the equation $2x + 3y = 12$. * Let $x = 0$. Then $2(0) + 3y = 12 \Rightarrow 3y = 12 \Rightarrow y = 4$. So, $(0, 4)$ is a solution. * Let $y = 0$. Then $2x + 3(0) = 12 \Rightarrow 2x = 12 \Rightarrow x = 6$. So, $(6, 0)$ is a solution. * Let $x = 3$. Then $2(3) + 3y = 12 \Rightarrow 6 + 3y = 12 \Rightarrow 3y = 6 \Rightarrow y = 2$. So, $(3, 2)$ is a solution.
Since we can choose any real number for $x$ and find a corresponding real number for $y$ (or vice versa), there is no end to the number of solutions we can find.
Important Conclusion: A linear equation in two variables has infinitely many solutions.
Every solution $(x, y)$ of a linear equation can be plotted as a point on the Cartesian plane. When we plot these points and join them, they always lie on a straight line.
The graph of every linear equation in two variables is a straight line. Conversely, every point on the line is a solution of the linear equation.
Linear equations in two variables are incredibly useful for modeling real-life word problems.
Example: The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement. Solution: Let the cost of a notebook be $x$ rupees. Let the cost of a pen be $y$ rupees. According to the problem, the cost of a notebook is twice the cost of a pen. Therefore, $x = 2y$. This can be written in the general form as $x - 2y + 0 = 0$.
Example: In a class, the number of girls is 5 more than the number of boys. Solution: Let the number of girls be $y$ and the number of boys be $x$. The equation is $y = x + 5$, or $-x + y - 5 = 0$.
While a single linear equation in two variables has infinitely many solutions, a system of two such equations (called simultaneous linear equations) generally has a unique solution. You will study methods to solve pairs of linear equations in detail in Class 10. The fundamental concept remains the same: the solution to the system is the point of intersection of their two corresponding straight lines on the graph.
Understanding linear equations in two variables and their graphical representations provides a visual connection to algebraic concepts. It teaches us that algebra and geometry are deeply interconnected, with straight lines perfectly mapping the infinite solutions of these equations.