Unit: Algebra – 1
Chapter: Introduction to Two variables
Reference: – Introduction to Linear Equations in Two Variables, General Form, Solutions of an Equation, Graph of a Linear Equation, Infinite Solutions, Intercepts, Standard Form, Slope-Intercept Form, Applications, Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- Introduction to Linear Equation in Two Variable
- General Form & Solution Concepts
- Infinite Solutions Property
- Graphical Representation
Introduction to Linear Equations in Two Variables
Definition
A linear equation in two variables is an equation that can be written in the form ax + by + c = 0 (or ax + by = c), where a, b, and c are real numbers, and a and b are not both zero. The variables (usually x and y) have the highest power of 1.
When we study linear equations in two variables, we essentially ask: "What are all the pairs of (x, y) that satisfy this equation?"
Unlike equations in one variable (which have a single solution), equations in two variables have infinitely many solutions.
Importance of Linear Equations in Two Variables
- Models’ real-world relationships between two quantities (cost and quantity, distance and time, etc.)
- Foundation for graphing lines on a coordinate plane
- Essential for solving systems of equations
- Used extensively in economics, physics, and engineering
Example
Equation: x + y = 5
Some solutions: (1,4), (2,3), (3,2), (4,1), (0,5), (5,0), (2.5, 2.5)
Common Property: In every solution, the sum of x and y is 5. So, if we are given (6, -1), it also satisfies because 6 + (-1) = 5.
Subtopics
1. General Form
The general form of a linear equation in two variables is ax + by + c = 0, where a, b, c are real numbers, and at least one of a or b is non-zero.
Examples of General Form:
- 2x + 3y – 6 = 0 (here a=2, b=3, c=-6)
- x – y = 4 can be written as x – y – 4 = 0 (a=1, b=-1, c=-4)
- y = 2x + 1 can be written as -2x + y – 1 = 0 (a=-2, b=1, c=-1)
- x = 5 can be written as x – 5 = 0 (a=1, b=0, c=-5) → vertical line
- y = -3 can be written as y + 3 = 0 (a=0, b=1, c=3) → horizontal line
Special Cases:
- If b = 0, the equation becomes ax + c = 0 → x = constant (vertical line)
- If a = 0, the equation becomes by + c = 0 → y = constant (horizontal line)
2. Solution of a Linear Equation in Two Variables
A solution is an ordered pair (x, y) that satisfies the equation (makes LHS equal to RHS).
Key Property: A linear equation in two variables has infinitely many solutions.
Example of Finding Solutions: For the equation 2x + y = 7
If we choose x = 0, then y = 7 → solution (0, 7)
If we choose x = 1, then y = 5 → solution (1, 5)
If we choose x = 2, then y = 3 → solution (2, 3)
If we choose x = 3, then y = 1 → solution (3, 1)
If we choose x = 4, then y = -1 → solution (4, -1)
If we choose x = -1, then y = 9 → solution (-1, 9)
If we choose x = 2.5, then y = 2 → solution (2.5, 2)
Note: You can choose any value for one variable and calculate the corresponding value of the other variable.
3. Graphical Representation
When we plot all solutions of a linear equation in two variables on a coordinate plane, they form a straight line. This is why it's called a "linear" equation.
Key Points about the Graph:
- Every point on the line is a solution.
- Any point not on the line is not a solution.
- A line is determined by just two points.
Steps to Graph a Linear Equation:
- Step 1: Find two solutions (ordered pairs)
- Step 2: Plot them on the coordinate plane
- Step 3: Draw a straight line through them
Example – Graphing x – y = 2:
One solution: if x = 0, then y = -2 → point (0, -2)
Another solution: if x = 2, then y = 0 → point (2, 0)
Plot (0,-2) and (2,0) and draw the line through them.
4. Standard Forms of Linear Equations
Linear equations in two variables can be written in different forms, each useful for different purposes.
Standard Form: ax + by = c
Example: 3x + 4y = 12
Slope-Intercept Form: y = mx + b
Here, m = slope (steepness of the line) and b = y-intercept (where the line crosses the y-axis)
Example: y = 2x + 3 → slope = 2, y-intercept = 3
Intercept Form: x/a + y/b = 1
Here, a = x-intercept and b = y-intercept
Example: x/4 + y/3 = 1 → x-intercept = 4, y-intercept = 3
5. Intercepts
Intercepts are the points where the line crosses the axes.
x-intercept: Set y = 0, then solve for x. The coordinates are (a, 0).
y-intercept: Set x = 0, then solve for y. The coordinates are (0, b).
Example – Finding Intercepts for 2x + 3y = 12:
To find x-intercept: set y = 0 → 2x = 12 → x = 6 → point (6, 0)
To find y-intercept: set x = 0 → 3y = 12 → y = 4 → point (0, 4)
Solved Examples
Example 1: Check if (2, 3) is a solution of 2x – y = 1.
Solution: LHS = 2(2) – 3 = 4 – 3 = 1 = RHS. Yes, it is a solution.
Example 2: Find four solutions of the equation x + 2y = 8.
Solution: Choose any four values for x and find y.
If x = 0, then y = (8 – 0)/2 = 4 → solution (0, 4)
If x = 2, then y = (8 – 2)/2 = 3 → solution (2, 3)
If x = 4, then y = (8 – 4)/2 = 2 → solution (4, 2)
If x = 6, then y = (8 – 6)/2 = 1 → solution (6, 1)
Example 3: Write the equation 3x + 2y = 6 in slope-intercept form.
Solution: Solve for y.
3x + 2y = 6
2y = -3x + 6
y = (-3/2)x + 3
Example 4: Find the x-intercept and y-intercept of 5x – 3y = 15.
Solution:
x-intercept: set y = 0 → 5x = 15 → x = 3 → point (3, 0)
y-intercept: set x = 0 → -3y = 15 → y = -5 → point (0, -5)
Example 5 – Odd One Out (Ordered Pairs):
Examine the five ordered pairs below. Exactly one is NOT a solution of the equation 2x + y = 10. Identify it.
Pairs: (2, 6), (3, 4), (4, 2), (5, 0), (1, 7)
Solution: Check each pair in the equation 2x + y = 10.
For (2, 6): 2(2) + 6 = 4 + 6 = 10 ✓
For (3, 4): 2(3) + 4 = 6 + 4 = 10 ✓
For (4, 2): 2(4) + 2 = 8 + 2 = 10 ✓
For (5, 0): 2(5) + 0 = 10 + 0 = 10 ✓
For (1, 7): 2(1) + 7 = 2 + 7 = 9 ≠ 10 ✗
Three reasons why (1, 7) is the odd one out:
(A) It does not satisfy the equation 2x + y = 10 (gives 9 instead of 10).
(B) In all other pairs, the sum 2x + y equals 10; in (1,7), it equals 9.
(C) On a graph, (1,7) lies on a different line (2x + y = 9), not on the line 2x + y = 10.
Conclusion: (1, 7) is the odd one out.
Example 6 – Odd One Out (Equations):
Examine the five equations below. Exactly one is NOT a linear equation in two variables. Identify it.
Equations: 2x + 3y = 12, y = 4x – 5, x² + y = 9, 3x – y = 7, x = 2y + 1
Solution:
2x + 3y = 12 → linear (powers of x and y are 1)
y = 4x – 5 → linear (powers of x and y are 1)
x² + y = 9 → NOT linear (x has power 2, which is quadratic)
3x – y = 7 → linear (powers of x and y are 1)
x = 2y + 1 → linear (powers of x and y are 1)
Three reasons why x² + y = 9 is the odd one out:
(A) Its degree is 2 (because of x²), while all others have degree 1.
(B) Its graph is a parabola (curved), while all others graph as straight lines.
(C) The variable x has exponent 2, violating the definition of a linear equation.
Conclusion: x² + y = 9 is the odd one out.
Common Mistakes to Avoid
Mistake 1 – Thinking only one solution exists
Why it's wrong: A linear equation in two variables has infinitely many solutions. For any value of x, we can find a corresponding y.
Correct understanding: Solutions are infinite; we can only list a few.
Mistake 2 – Confusing (x,y) with (y,x)
Why it's wrong: The order matters. (2,3) and (3,2) are different points unless x equals y.
Correct understanding: In an ordered pair, the first number is always x and the second is always y.
Mistake 3 – Forgetting negative signs when finding intercepts
Why it's wrong: Setting x=0 gives y-intercept, but solving incorrectly may give the wrong sign.
Correct understanding: Carefully solve the equation after substituting zero.
Mistake 4 – Assuming line is vertical or horizontal without checking
Why it's wrong: The equation ax + by = c is vertical only if b = 0, and horizontal only if a = 0.
Correct understanding: Check the coefficients before deciding.
Quick Reference Summary
General Form: ax + by + c = 0 (a and b not both zero)
Solutions: Infinitely many ordered pairs (x, y) that satisfy the equation
Graph: Always a straight line
x-intercept: Set y = 0, solve for x → point (a, 0)
y-intercept: Set x = 0, solve for y → point (0, b)
Slope-Intercept Form: y = mx + b (m = slope, b = y-intercept)
Standard Form: ax + by = c