A linear equation in two variables is an equation that can be written in the form: ax + by + c = 0 where a, b, and c are real numbers, and a and b are not both zero. The variables x and y are the two unknowns.
The solution of such an equation is a pair of values (x, y) which satisfies the equation. The graph of the solution is a straight line in the Cartesian plane. Every point on this line represents a solution.
A linear equation in two variables always has infinitely many solutions unless a second equation is provided.
Example 1: Show that (2, 3) is a solution of the equation 2x + y = 7.
Solution: Substitute x = 2 and y = 3 into the equation:
2(2) + 3 = 4 + 3 = 7
Hence, (2, 3) is a solution.
Example 2: Find 3 solutions of the equation x + 2y = 6.
| x | y |
|---|---|
| 0 | 3 |
| 2 | 2 |
| 4 | 1 |
Example 3: Write a linear equation whose solution set includes the point (-1, 5).
Solution: Using x + y = c form:
-1 + 5 = 4 ⇒ x + y = 4.
Example 4: Find the value of k if (1, k) satisfies 3x – 2y = 4.
Solution: 3(1) – 2k = 4 ⇒ -2k = 1 ⇒ k = -1/2.
Example 5: Does the point (3, -2) lie on the line 4x + 5y = 2?
Solution: 4(3) + 5(-2) = 12 – 10 = 2 ⇒ Yes, it lies on the line.
- Verify whether (4, -1) is a solution of 3x + 2y = 10.
- Find four different solutions of x – y = 2.
- Find y when x = -3 in the equation 2x + y = 5.
- Write a linear equation whose graph passes through the point (2, -2).
- Find value of m if (m, 1) satisfies 5x – 3y = 7.
- Find the linear equation whose graph passes through (0, 3) and (3, 0).
- Write two more solutions for the equation 2x + 3y = 12.
- State true or false: The equation x = 5 has infinitely many solutions.
3. Real Life Applications: Age and Number Problems
Linear equations in two variables are widely used to model real-life problems involving age, numbers, time, etc. For example:
- If a father's age is 3 years more than twice his son's age, and their total age is 39 years, we can form equations using variables and solve them.
- If the sum of two numbers is known and their relation is described, a linear equation helps find them.
Such problems can be solved using:
- Defining variables
- Formulating the equation from the statements
- Solving or interpreting graphically
Example 1: The sum of two numbers is 15. Let one be x, the other is 15 - x. Find three such pairs.
| x | y = 15 - x |
|---|---|
| 5 | 10 |
| 8 | 7 |
| 12 | 3 |
Example 2: A father is 5 years more than twice the age of his son. Express this as an equation.
Let son's age = x, Then father's age = 2x + 5 → y = 2x + 5
- Two numbers add up to 30. Write a linear equation.
- A father's age is 4 times his son. Together they are 50. Write the equation.
- The equation 2x + 3y = 6 has:
- (A) No solution
- (B) Exactly one solution
- (C) Infinitely many solutions (Correct)
- (D) Two solutions
- Which of the following is a linear equation in two variables?
- (A) x² + y = 4
- (B) x + y = 7 (Correct)
- (C) x + y² = 10
- (D) x³ + y³ = 1
- The graph of a linear equation in two variables is a:
- (A) Circle
- (B) Parabola
- (C) Straight line (Correct)
- (D) Ellipse
- Which of these equations represents a vertical line?
- (A) y = x
- (B) x = 2 (Correct)
- (C) y = -x + 3
- (D) y = 0
- What is the slope of the line y = 3x + 1?
- (A) 1
- (B) 3 (Correct)
- (C) -3
- (D) 0
Linear equations in two variables are widely used to model real-life problems involving age, numbers, time, etc. For example:
- If a father's age is 3 years more than twice his son's age, and their total age is 39 years, we can form equations using variables and solve them.
- If the sum of two numbers is known and their relation is described, a linear equation helps find them.
Such problems can be solved using:
- Defining variables
- Formulating the equation from the statements
- Solving or interpreting graphically
Example 1: The sum of two numbers is 15. Let one be x, the other is 15 - x. Find three such pairs.
| x | y = 15 - x |
|---|---|
| 5 | 10 |
| 8 | 7 |
| 12 | 3 |
Example 2: A father is 5 years more than twice the age of his son. Express this as an equation:
Let son's age = x, then father's age = 2x + 5, so the equation is y = 2x + 5.
Example 3: The difference between two numbers is 7. Let x and y be the numbers:
x - y = 7
Example 4: The sum of ages of Amit and his father is 50. Amit is 20 years younger than his father:
Let father = x, Amit = x - 20
Total: x + (x - 20) = 50 ⇒ 2x = 70 ⇒ x = 35.
Example 5: The cost of 1 pen is Rs. 7 and 1 notebook is Rs. 10. Total cost is Rs. 51:
Let pens = x, notebooks = y.
Then: 7x + 10y = 51
- Two numbers add up to 30. Write a linear equation.
- A father's age is 4 times his son. Together they are 50. Write the equation.
- A woman has Rs. 100 in Rs. 10 and Rs. 20 notes. Write the equation.
- If x is twice y and their sum is 21, find the equation.
- A bag contains x red and y blue balls, total 50.
- Cost of 3 pens and 2 pencils is Rs. 38.
- A train covers a distance in x hours at speed y. Total distance = 240 km.
- A boat travels 2 hrs downstream and 3 hrs upstream. Speed downstream = x, upstream = y. Total distance = 100 km.
Choose the correct option:
- The equation \( 2x + 3y = 6 \) has:
- (A) No solution
- (B) Exactly one solution
- (C) Infinitely many solutions
- (D) Two solutions
- Which of the following is a linear equation in two variables?
- (A) \( x^2 + y = 4 \)
- (B) \( x + y = 7 \)
- (C) \( x + y^2 = 10 \)
- (D) \( x^3 + y^3 = 1 \)
- The graph of a linear equation in two variables is a:
- (A) Circle
- (B) Parabola
- (C) Straight line
- (D) Ellipse
- The point \( (3, 2) \) lies on the line \( x + y = \):
- (A) 4
- (B) 5
- (C) 6
- (D) 7
- How many solutions does the equation \( x = 4 \) have?
- (A) None
- (B) One
- (C) Two
- (D) Infinitely many
- Which of the following pairs is a solution to \( 2x + y = 10 \)?
- (A) \( (2, 5) \)
- (B) \( (3, 4) \)
- (C) \( (4, 2) \)
- (D) \( (5, 1) \)
- The x-intercept of the line \( 3x + 4y = 12 \) is:
- (A) 0
- (B) 3
- (C) 4
- (D) 4
- Which point lies on the line \( x - 2y = 0 \)?
- (A) \( (1, 1) \)
- (B) \( (2, 1) \)
- (C) \( (0, 2) \)
- (D) \( (2, 2) \)
- If \( x + y = 0 \), what is the value of \( y \) when \( x = -4 \)?
- (A) 4
- (B) -4
- (C) 0
- (D) 2
- The line \( y = 5 \) is:
- (A) Horizontal
- (B) Vertical
- (C) Passing through origin
- (D) A parabola
- Which of the following represents a linear equation?
- (A) \( y = 2x + 1 \)
- (B) \( y = x^2 + 1 \)
- (C) \( y = \frac{1}{x} \)
- (D) \( y = \sqrt{x} \)
- What is the y-intercept of \( x + y = 6 \)?
- (A) 0
- (B) 3
- (C) 6
- (D) -6
- The line passing through \( (0, 0) \) and \( (2, 2) \) has the equation:
- (A) \( y = x \)
- (B) \( y = 2x \)
- (C) \( x = y + 1 \)
- (D) \( y = x + 1 \)
- A linear equation in two variables has:
- (A) One solution
- (B) Two solutions
- (C) Infinitely many solutions
- (D) No solution
- If \( 3x + y = 12 \), and \( x = 2 \), then \( y = \)?
- (A) 4
- (B) 6
- (C) 8
- (D) 2
- What is the slope of the line \( y = 3x + 1 \)?
- (A) 1
- (B) 3
- (C) -3
- (D) 0
- A line parallel to the x-axis is:
- (A) \( x = a \)
- (B) \( y = b \)
- (C) \( x = y \)
- (D) None of these
- A line with equation \( x = 3 \) is:
- (A) Vertical
- (B) Horizontal
- (C) Diagonal
- (D) None
- Which of these is not a solution of \( x + y = 8 \)?
- (A) \( (3, 5) \)
- (B) \( (4, 4) \)
- (C) \( (2, 6) \)
- (D) \( (5, 2) \)
- The line \( y = -x \) passes through:
- (A) \( (1, -1) \)
- (B) \( (-1, -1) \)
- (C) \( (1, 1) \)
- (D) \( (-1, 1) \)
- The general form of linear equation in two variables is:
- (A) \( ax + by + c = 0 \)
- (B) \( ax^2 + by + c = 0 \)
- (C) \( ax + b = 0 \)
- (D) \( ax^2 + bxy = 0 \)
- Which of these equations represents a vertical line?
- (A) \( y = x \)
- (B) \( x = 2 \)
- (C) \( y = -x + 3 \)
- (D) \( y = 0 \)
- The line \( y = x + 2 \) intersects the y-axis at:
- (A) (0, 2)
- (B) (2, 0)
- (C) (0, 0)
- (D) (2, 2)
- Which equation has (4, 1) as a solution?
- (A) \( 2x + y = 9 \)
- (B) \( x + y = 5 \)
- (C) \( 3x + y = 10 \)
- (D) \( x - y = 3 \)
- A line passes through \( (0, 4) \) and \( (4, 0) \). Its equation is:
- (A) \( x + y = 4 \)
- (B) \( x + y = 8 \)
- (C) \( x - y = 4 \)
- (D) \( x = y \)
Linear Equations in Two Variables
Part 3: Self-Assessment Paper
Class 9 Mathematics
Choose the correct option:
- The equation of a vertical line passing through \( x = 5 \) is:
- (A) \( y = 5 \)
- (B) \( x = 5 \) (Correct)
- (C) \( y = x \)
- (D) \( x = y \)
- The number of solutions of the equation \( x + y = 10 \) is:
- (A) 0
- (B) 1
- (C) 2
- (D) Infinite (Correct)
- The graph of the equation \( y = 3 \) is a:
- (A) Vertical Line
- (B) Horizontal Line (Correct)
- (C) Diagonal Line
- (D) None of the above
- The equation \( x + 2y = 6 \) has y-intercept:
- (A) 2 (Correct)
- (B) 3
- (C) 6
- (D) 0
- If \( y = mx + c \), then the graph is:
- (A) Curved line
- (B) Straight line (Correct)
- (C) Circle
- (D) Parabola
- The coordinates of the point where the line \( y = x \) intersects the x-axis is:
- (A) (0, 0) (Correct)
- (B) (1, 1)
- (C) (1, 0)
- (D) (0, 1)
- Which of the following equations is not linear?
- (A) \( x + y = 7 \)
- (B) \( 3x + 2y = 5 \)
- (C) \( xy = 6 \) (Correct)
- (D) \( 2x - y = 3 \)
- The slope of the line \( y = 2x + 1 \) is:
- (A) 1
- (B) 2 (Correct)
- (C) 0
- (D) -1
- Write the linear equation whose graph passes through points \( (1, 2) \) and \( (3, 6) \).
- Find the solution of \( 2x + y = 7 \) if \( x = 2 \).
- Draw the graph of the equation \( x + y = 4 \).
- Find the y-intercept of the line \( 5x + 2y = 10 \).
- Write any two solutions of the equation \( x - y = 3 \).
- Determine whether the point \( (2, 3) \) lies on the line \( 3x + y = 9 \).
- Solve: Two numbers differ by 3. The sum of twice the smaller and thrice the larger is 19. Find the numbers.
- A number consists of two digits. If the digits are reversed, the number becomes 9 more than the original number. Find the number.
- Draw the graphs of \( 2x + y = 6 \) and \( x - y = 2 \) on the same pair of axes and find the point of intersection.
- The ages of two persons differ by 4 years. Four years ago, the elder one was 3 times as old as the younger. Find their present ages.
Case Study: Meera and Raj are friends who started a small food delivery service in their locality. They offer lunchboxes. Meera prepares vegetarian lunchboxes while Raj prepares non-vegetarian ones. Each vegetarian lunchbox costs Rs.50 and each non-vegetarian one costs Rs.70. In one day, they sold a total of 30 boxes and collected Rs.1800 in revenue.
Let the number of vegetarian boxes sold be \( x \), and non-vegetarian boxes be \( y \).
- What is the equation representing total boxes sold?
- (A) \( x + y = 30 \) (Correct)
- What is the equation representing the total money collected?
- (A) \( 50x + 70y = 1800 \) (Correct)
- What type of equations are \( x + y = 30 \) and \( 50x + 70y = 1800 \)?
- (C) Linear equations in two variables (Correct)
- If 10 vegetarian boxes were sold, how many non-veg boxes were sold?
- (B) 20 (Correct)
- If \( x = 15 \), find value of \( y \) such that \( 50x + 70y = 1800 \)
- (C) 12 (Correct)
Section A: 1.(B), 2.(D), 3.(B), 4.(A), 5.(B), 6.(A), 7.(C), 8.(B)
Section B: Students should solve these short questions.
Section C: Students should solve long questions based on understanding.
Section D: (i)-(A), (ii)-(A), (iii)-(C), (iv)-(B), (v)-(C)