Linear Equations in Two Variables Solved Examples (9 Mathematics)
Linear equations in two variables represent straight lines on a graph. These examples demonstrate how to solve, graph, and verify solutions to equations of
TL;DR: Linear equations in two variables represent straight lines on a graph. These examples demonstrate how to solve, graph, and verify solutions to equatio…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
Linear equations in two variables represent straight lines on a graph. These examples demonstrate how to solve, graph, and verify solutions to equations of
Linear Equations in Two Variables — Solved Numerical Examples (Step by Step)
Example 1: Check if x = 2, y = 1 is a solution to 2x + 3y = 7.
Solution: Substitute x = 2 and y = 1 into the equation:
2(2) + 3(1) = 4 + 3 = 7
Yes, the equation is satisfied.
Example 2: Find three solutions of the equation x + 2y = 5.
Solution: Rearrange: x = 5 - 2y
Solution 1: Put y = 0
x = 5 - 2(0) = 5
Solution: (5, 0)
Solution 2: Put y = 1
x = 5 - 2(1) = 3
Solution: (3, 1)
Solution 3: Put y = 2
x = 5 - 2(2) = 1
Solution: (1, 2)
All three satisfy the equation.
Example 3: Express 2x + 3y - 6 = 0 in the form x/a + y/b = 1 (intercept form).
Solution: Start with: 2x + 3y - 6 = 0
2x + 3y = 6
Divide by 6:
2x/6 + 3y/6 = 1
x/3 + y/2 = 1
This is the intercept form where a = 3 (x-intercept) and b = 2 (y-intercept).
Example 4: Draw the graph of x + y = 4. Find where it intersects the axes.
Solution: Rearrange: y = 4 - x
X-intercept (when y = 0):
0 = 4 - x
x = 4
Point: (4, 0)
Y-intercept (when x = 0):
y = 4 - 0 = 4
Point: (0, 4)
Other points for graphing:
When x = 1: y = 3, Point (1, 3)
When x = 2: y = 2, Point (2, 2)
When x = 3: y = 1, Point (3, 1)
The line passes through (4, 0) and (0, 4).
Example 5: Find the equation of a line passing through points (2, 3) and (4, 7).
Solution: Step 1: Find the slope m.
m = (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 4 / 2 = 2
Step 2: Use point-slope form with point (2, 3).
y - y1 = m(x - x1)
y - 3 = 2(x - 2)
y - 3 = 2x - 4
y = 2x - 1
Or in standard form: 2x - y - 1 = 0
Verification with (4, 7):
y = 2(4) - 1 = 8 - 1 = 7 ✓
Example 6: A mother is 20 years older than her son. Write the linear equation in two variables for this relation.
Solution: Let the son's age be x years and mother's age be y years.
The condition states: Mother's age = Son's age + 20
y = x + 20
In standard form: x - y + 20 = 0 or x - y = -20
This represents a linear relationship between the two ages.
Example 7: For the equation 3x - 2y = 6, find the slope and y-intercept.
Solution: Convert to slope-intercept form y = mx + c:
3x - 2y = 6
-2y = -3x + 6
y = (3/2)x - 3
Comparing with y = mx + c:
Slope m = 3/2
Y-intercept c = -3
The line intersects y-axis at (0, -3) and has a slope of 3/2 (rises 3 units for every 2 units to the right).
Tips
- Always arrange the equation in standard form ax + by + c = 0 or slope-intercept form y = mx + c.
- To find solutions, assign any value to one variable and solve for the other.
- A line has infinitely many solutions, each represented by a point (x, y) on the line.
- The slope m = (y2 - y1) / (x2 - x1) tells you the steepness and direction of the line.
Frequently Asked Questions
How many solutions does a linear equation in two variables have?
A linear equation in two variables has infinitely many solutions. Each solution is a point (x, y) that satisfies the equation. All these points lie on a straight line when plotted on a graph.
What is the difference between a linear equation in one variable and two variables?
A linear equation in one variable (like 2x + 3 = 7) has a unique solution. A linear equation in two variables (like 2x + y = 7) has infinitely many solutions, graphically represented by a straight line.
More Mathematics Solved Examples
- Real Numbers (HCF LCM Euclid)
- Pair of Linear Equations in Two Variables
- Circles
- Statistics
- Heron's Formula
- Integrals (Indefinite and Definite)
🤖 Stuck on any of these? Ask Syllab's free AI Tutor to explain step by step →