Linear equations in two variables are equations of the form ax + by + c = 0, where a, b, and c are constants, and x and y are variables. The highest power of both x and y in such equations is 1.
Key Concepts
- Solution of a Linear Equation: A pair of values (x, y) that satisfies the equation is its solution.
- Graphical Representation: The graph of a linear equation in two variables is a straight line.
- Number of Solutions: A linear equation in two variables has infinitely many solutions.
- Consistency:
- Consistent system: Two or more linear equations having at least one common solution.
- Inconsistent system: Two or more linear equations having no common solution.
- Dependent system: Two or more linear equations having infinitely many solutions.
- Algebraic Methods of Solution: Elimination and substitution methods are used to solve a pair of linear equations.
Applications
Linear equations in two variables have wide applications in various fields, including science, economics, and engineering. They can be used to model real-world situations and solve problems involving two unknown quantities.
Exercise 4.1
1. The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.
(Take the cost of a notebook to be Rs. x and that of a pen to be Rs.y).
Ans :
- x as the cost of a notebook
- y as the cost of a pen
Given:
The cost of a notebook is twice the cost of a pen.
Equation:
x = 2y
This is the required linear equation in two variables representing the given statement.
2. Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case:
(i) 2x + 3y = 9.35
(ii) x−y/5−10=0
(iii) – 2x + 3y = 6
(iv) x = 3y
(v) 2x = -5y
(vi) 3x + 2 = 0
(vii) y – 2 = 0
(viii) 5 = 2x
Ans :
i) 2x + 3y = 9.35
a = 2,
b = 3,
c = -9.35
ii) 5x – y – 50 = 0
a = 5, b = -1, c = -50
iii) -2x + 3y = 6
a = -2,
b = 3,
c = -6
iv) x = 3y
Rewrite the equation as x – 3y = 0
a = 1, b = -3, c = 0
v) 2x = -5y
Rewrite the equation as 2x + 5y = 0
a = 2, b = 5, c = 0
vi) 3x + 2 = 0
Rewrite the equation as 3x + 0y + 2 = 0
a = 3,
b = 0,
c = 2
vii) y – 2 = 0
Rewrite the equation as 0x + y – 2 = 0
a = 0, b = 1, c = -2
viii) 5 = 2x
Rewrite the equation as -2x + 0y + 5 = 0
a = -2, b = 0, c = 5
Exercise 4.2
1. Which one of the following options is true, and why?
y = 3x + 5 has
(i) a unique solution,
(ii) only two solutions,
(iii) infinitely many solutions
Ans :
The equation y = 3x + 5 has (iii) infinitely many solutions.
Why?
- A linear equation in two variables, like y = 3x + 5, represents a straight line on a graph.
- Every point on this line corresponds to a solution to the equation.
- Since a line consists of infinitely many points, there are infinitely many pairs of values for x and y that satisfy the equation.
2. Write four solutions for each of the following equations:
(i) 2x + y = 7
(ii) πx + y = 9
(iii) x = 4y
Ans :
i) 2x + y = 7
- Let x = 0, then y = 7: (0, 7)
- Let x = 1, then y = 5: (1, 5)
- Let x = 2, then y = 3: (2, 3)
- Let x = 3, then y = 1: (3, 1)
ii) πx + y = 9
- Let x = 0, then y = 9: (0, 9)
- Let x = 1, then y = 9 – π: (1, 9 – π)
- Let x = 2, then y = 9 – 2π: (2, 9 – 2π)
- Let x = 3, then y = 9 – 3π: (3, 9 – 3π)
iii) x = 4y
- Let y = 0, then x = 0: (0, 0)
- Let y = 1, then x = 4: (4, 1)
- Let y = 2, then x = 8: (8, 2)
- Let y = 3, then x = 12: (12, 3)
3. Check which of the following are solutions of the equation x – 2y = 4 and which are not:
(i) (0,2)
(ii) (2,0)
(iii) (4, 0)
(iv) (√2, 4√2)
(v) (1, 1)
Ans :
(i) (0, 2):
- Substituting x = 0 and y = 2, we get: 0 – 2(2) = -4 ≠ 4
- Therefore, (0, 2) is not a solution.
(ii) (2, 0):
- Substituting x = 2 and y = 0, we get: 2 – 2(0) = 2 ≠ 4
- Therefore, (2, 0) is not a solution.
(iii) (4, 0):
- Substituting x = 4 and y = 0, we get: 4 – 2(0) = 4
- Therefore, (4, 0) is a solution.
(iv) (√2, 4√2):
- Substituting x = √2 and y = 4√2, we get: √2 – 2(4√2) = √2 – 8√2 = -7√2 ≠ 4
- Therefore, (√2, 4√2) is not a solution.
(v) (1, 1):
- Substituting x = 1 and y = 1, we get: 1 – 2(1) = -1 ≠ 4
- Therefore, (1, 1) is not a solution.
4. Find the value of k, if x = 2, y = 1 ¡s a solution of the equation 2x + 3y = k.
Ans :
Given:
- Equation: 2x + 3y = k
- Solution: (x, y) = (2, 1)
To find: Value of k
Solution: Since (2, 1) is a solution to the equation, it must satisfy the equation. Substitute x = 2 and y = 1 into the equation:
2(2) + 3(1) = k 4 + 3 = k k = 7
Therefore, the value of k is 7.
