Meaning of 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,
• (a, b,) and (c) are real numbers, and
• (a) and (b) are not both zero.

Examples:

• (2x + 3y = 7)
• (x - y = 4)
• (3x + 2y - 6 = 0)

2. Graphical Representation

Each linear equation in two variables represents a straight line on the coordinate plane.
A pair of such equations represents two lines, which can:

• Intersect at one point → one unique solution,
• Be parallel → no solution, or
• Coincide (overlap) → infinitely many solutions.

Example:
For (2x + y = 7) and (x - y = 1), the two lines intersect at one point — the solution to the system.

3. Methods of Solving Linear Equations
4. Elimination Method

We eliminate one variable by adding or subtracting the equations.

Example 1:
Solve:
2x + y = 7 \quad \text{(i)}
x - y = 1 \quad \text{(ii)}

Step 1: Add the equations to eliminate (y):
(2x + y) + (x - y) = 7 + 1 \Rightarrow 3x = 8
x = \frac{8}{3}

Step 2: Substitute (x = \frac{8}{3}) into (ii):
\frac{8}{3} - y = 1 \Rightarrow y = \frac{5}{3}

✅ Solution: (x = \frac{8}{3},\ y = \frac{5}{3})

1. Substitution Method

We make one variable the subject in one equation and substitute into the other.

Example 2:
Solve:
x + y = 10
2x - y = 4

Step 1: From the first equation, (y = 10 – x).
Step 2: Substitute into the second:
2x - (10 - x) = 4 \Rightarrow 3x = 14 \Rightarrow x = \frac{14}{3}
Step 3: Substitute (x) into (y = 10 – x):
y = 10 - \frac{14}{3} = \frac{16}{3}

✅ Solution: (x = \frac{14}{3},\ y = \frac{16}{3})

1. Graphical Method

Plot both equations on the same axes.
The point of intersection gives the values of (x) and (y).

Example 3:
For (x + y = 6) and (2x - y = 3):

• First line: (y = 6 - x)
• Second line: (y = 2x - 3)
  Their intersection point gives the solution.

4. Word Problems on Simultaneous Equations

Example 4:
A pen and a pencil cost ₦250 together. Two pens and one pencil cost ₦400. Find the cost of each item.

Let:

• Cost of pen = (x)
• Cost of pencil = (y)

Then:
x + y = 250 \quad \text{(i)}
2x + y = 400 \quad \text{(ii)}

Subtract (i) from (ii):
x = 150
Substitute (x = 150) into (i):
150 + y = 250 \Rightarrow y = 100

✅ Pen = ₦150, Pencil = ₦100