Solutions of Linear Equations in Two Variables

A linear equation in two variables (typically represented as x and y) represents a straight line when graphed on a coordinate plane. Unlike a single linear equation in one variable, which usually has a single solution, a linear equation in two variables has infinitely many solutions. Each solution is an ordered pair (x, y) that satisfies the equation. This means that if you substitute the x and y values of a solution into the equation, the equation will be true.

Formulae

The general form of a linear equation in two variables is:

$ax + by = c$

where ‘a’, ‘b’, and ‘c’ are constants, and ‘x’ and ‘y’ are the variables. Any ordered pair (x, y) that satisfies this equation is a solution.

Examples

Example-1: Consider the equation $x + y = 5$. Here are a few solutions:

• (1, 4) because 1 + 4 = 5
• (2, 3) because 2 + 3 = 5
• (0, 5) because 0 + 5 = 5
• (-1, 6) because -1 + 6 = 5
• (5, 0) because 5 + 0 = 5

And there are infinitely more! You can plug in any value for ‘x’, solve for ‘y’, and get a solution.

Example-2: Consider the equation $2x – y = 1$. Here are a few solutions:

• (0, -1) because 2(0) – (-1) = 1
• (1, 1) because 2(1) – 1 = 1
• (2, 3) because 2(2) – 3 = 1
• (-1, -3) because 2(-1) – (-3) = 1

Theorem with Proof

Theorem: A linear equation in two variables has infinitely many solutions.

Proof:

Let’s consider a general linear equation in two variables: $ax + by = c$, where at least one of ‘a’ or ‘b’ is not zero. (If both ‘a’ and ‘b’ are zero, then the equation is either trivial or inconsistent.)

1. Case 1: If $b \ne 0$, we can solve for ‘y’ as follows: $by = c – ax$ $y = \frac{c – ax}{b}$ For any value of ‘x’ that we choose, we can substitute it into the above equation and compute a corresponding value of ‘y’. For example, if $x = 0$, then $y = \frac{c}{b}$, creating solution $(0, \frac{c}{b})$. If $x = 1$, then $y = \frac{c – a}{b}$, creating solution $(1, \frac{c-a}{b})$. Since we can choose infinitely many values for ‘x’, there are infinitely many solutions.
2. Case 2: If $b = 0$, then $a \ne 0$, and the equation simplifies to $ax = c$. Then $x = \frac{c}{a}$. This still yields infinitely many solutions because the equation will look like $x=k$ where k is a constant. The solutions are of the form ($k, y$), where y can be any value and there are infinite possibilities.

In both cases, we can generate infinitely many ordered pairs (x, y) that satisfy the equation. Therefore, a linear equation in two variables has infinitely many solutions.

Common mistakes by students

• Thinking there’s only one solution: Students often mistakenly believe that a linear equation in two variables has only one solution, similar to a single-variable linear equation.
• Not understanding the concept of an ordered pair: Students may confuse the order of the (x, y) coordinates in a solution, which impacts their ability to correctly substitute and verify solutions.
• Only finding a few solutions: Students might find a couple of solutions and assume they’ve found all of them, not understanding the infinite nature of the solutions.

Real Life Application

Linear equations in two variables are used in many real-life applications. One example includes:

Budgeting: If you have a budget for two items, like groceries and entertainment, you can create a linear equation. For example, $g + e = 100$ where $g$ is the amount spent on groceries and $e$ is the amount spent on entertainment, and 100 is the maximum spend amount. There are infinitely many ways to spend that $100. Every combination of ‘g’ and ‘e’ satisfying the equation is a solution, and each represents a possible spending plan.

Fun Fact

The graph of a linear equation in two variables is a straight line. Each point on the line represents a solution to the equation. Since a line extends infinitely in both directions, it has infinitely many points, and therefore, infinitely many solutions.

Recommended YouTube Videos for Deeper Understanding

Q.1 What is the graphical representation of a linear equation in two variables that has infinitely many solutions?

Check Solution

Ans: B

A linear equation with infinitely many solutions represents the same line.

Q.2 Which of the following systems of equations has infinitely many solutions?

Check Solution

Ans: B

The equations in option B are multiples of each other, representing the same line.

Q.3 If a linear equation in two variables is represented by $3x – 2y = 6$, which of the following equations also represents the same line and therefore has infinitely many solutions with it?

Check Solution

Ans: B

Multiplying the given equation by a constant results in the same line. Multiplying by 2 in option B.

Q.4 The equation $y = 2x + 1$ has infinitely many solutions. Which of the following ordered pairs $(x, y)$ is *not* a solution to this equation?

Check Solution

Ans: D

Substitute the x values into the equation to determine the corresponding y values. In the option D $2(3)+1 = 7 \ne 6$

Q.5 Consider the system of equations: $ax + by = c$ and $dx + ey = f$. What condition guarantees infinitely many solutions for this system, assuming $a, b, c, d, e,$ and $f$ are real numbers?

Check Solution

Ans: B

For infinitely many solutions, the equations must represent the same line, implying proportional coefficients.

Next Topic: Graphing Linear Equations in Two Variables