A linear equation is an equation where the highest power of the variable(s) is 1. In other words, the equation represents a straight line when plotted on a graph. The solutions of a linear equation are the values of the variables that satisfy the equation, meaning when substituted into the equation, both sides of the equation are equal.

Example of a Linear Equation:

2x + 3 = 7

To find the solution of this equation, we solve for $x$:

1. Subtract 3 from both sides:

   2x=7−3   2x=4

2. Divide both sides by 2:

   x=4/2     x = $$2$$

So, the solution to 2x + 3 = 7 is x = 2. This means that when x = 2, the equation holds true.

Key Concepts About Solutions of Linear Equations:

1. Single Solution (Unique Solution):
   For equations with one variable, the solution is typically a single value of the variable. For example, 3x – 5 = 10 has the solution x = 5.

2. Infinite Solutions:
   In some cases, a linear equation can represent an identity (e.g., 2x + 4 = 2x + 4). These equations have infinitely many solutions because both sides are always equal, regardless of the value of $x$.

3. No Solution:
   If the equation simplifies to a contradiction (e.g., 3x + 2 = 3x + 5), it has no solution because there is no value of $x$ that can make both sides equal.

4. Graphical Interpretation:

   • In the case of a single-variable linear equation (e.g., 2x + 3 = 7), the solution is the point where the graph of the equation intersects the x-axis.

   • For two-variable linear equations (e.g., 2x + 3y = 6), the solution is the point (x, y) where the two lines representing the equations intersect on a graph.

Linear Equation in Two Variables

A linear equation in two variables has the form:

ax + by = c

Where:

• $a$, $b$, and $c$ are constants,

• $x$ and $y$ are variables.

The solution to this equation is any pair of values for x and $y$ that makes the equation true. For example, the equation 2x+3y=6 has many solutions such as (x = 0, y = 2), (x = 3, y = 0), etc.