- 2. • 5x • 2x – 3 • 3x + y • 2xy + 5 • xyz + x + y + z • x2 + 1 • y + y2 Some examples of expressions
- 4. • 5x = 25 • 2x – 3 = 9 • 2y + 𝟓 𝟐 = 8 • 6z + 10 = -7 • 9x – 11 = 8 Some examples of equations
- 5. Equations use the equality (=) sign If yes = equation If no = equation
- 6. These are linear expressions: • 2x • 2x + 1 • 3y – 7 • 12 – 5z These are not linear expressions • 𝑥2 + 1 • y+𝑦2 • 1+z+𝑧2 +𝑧3 (since highest power of variable > 1) A linear expression is an expression whose highest power of the variable is one only.
- 7. Linear Equations The equation of a straight line is the linear equation. It could be in one variable or two variables. Linear Equation in One Variable If there is only one variable in the equation then it is called a linear equation in one variable. The general form is ax + b = c, where a, b and c are real numbers and a ≠ 0.
- 8. Example x + 5 = 10 y – 3 = 19 These are called linear equations in one variable because the highest degree of the variable is one and there is only one variable.
- 9. • We assume that the two sides of the equation are balanced. • We perform the same mathematical operations on both sides of the equation, so that the balance is not disturbed. How to find the solution of an equation?
- 10. X + 3 = 7 X = 7 The value which when substituted for the variable in the equation, makes its two sides equal, is called a solution ( or root ) of the equation.
- 11. REMEMBER • The same number can be added to both the sides of the equation. • The same number can be subtracted from both the sides of the equation. • We can multiply or divide both the sides of the equation by the same non-zero number.
- 13. • Solve the linear equation: 4x + 3 = 15 – 2x Solution: 4x + 3 = 15 – 2x On subtracting 3 from both the sides, we get 4x + 3 – 3 = 15 – 2x – 3 Or, 4x = 12 – 2x 4x + 2x = 12 – 2x + 2x 6x = 12 𝟔𝒙 𝟔 = 𝟏𝟐 𝟔 X = 2 Hence, x = 2 is the solution of the given solution.
- 14. MATHEMATICS – I CHAPTER – 1 LINEAR EUATIONS IN TWO VARIABLES CLASS X
- 15. • ax + b = c, where a, b and c are real numbers and a ≠ 0. Example , 2x + 3 = 6 • ax + by + c = 0, where a, b and are real numbers, such that a ≠ 0, b ≠ 0. Example, 2x + 5y + 8 = 0 Linear equation in one variable Linear equation in two variable
- 16. ax + by + c = 0, where a, b and c are real numbers, such that a ≠ 0, b ≠ 0. Example: a) 4x + 3y = 4 b) -3x + 7 = 5y c) X = 4y d) Y = 2 – 3x
- 17. • Compare the equation ax + by + c = 0 with equation 2x + 3y = 4.37 and find the values of a, b and c? a)a = 2, b = 3 and c = 4.37 b)a = 2, b = 3 and c = - 4.37 c)a = 2, b = -3 and c = - 4.37 d)a = -2, b = 3 and c = 4.37 Solution: b) a = 2, b = 3 and c = - 4.37 ax + by + c = 0 2x + 3y + (-4.37) = 0
- 18. Solution of a linear equations in two variable The solution of a linear equation in two variables is an ordered pair of numbers, which satisfies the equation. The values x = m and y = n are said to be the solution of the linear equation. ‘ax + by + c = 0’ if am + bn + c = 0
- 19. • Show that x = 4 and y = -1 satisfy the equation x + 3y – 1 = 0 Solution: On substituting x = 4 and y = -1 in equation x + 3y – 1 = 0, we get L. H. S = 4 + 3 X (-1) – 1 = 4 – 3 – 1 = 0 = R. H. S Hence, x = 4 and y = -1 satisfy the equation x + 3y – 1 = 0.
- 20. Oranges + Apples = 10
- 21. Oranges Apples Fruits 1 9 10 2 8 10 3 7 10 4 6 10 5 5 10 6 4 10 7 3 10 8 2 10 9 1 10 Hit and Trial Method
- 22. Hint – Apples are 2 more than oranges 6 Apples + 4 Oranges = 10 6 Apples – 4 Oranges = 2 x + y = 10 x – y = 2 (2 – Equations, 2 – Variables or 2 – Unknowns) This pair of equation is called linear of equation in two variables or System of equation or simultaneous equations. Algebraic or Graphical Method.
- 24. • If x = 1, y = 2 is a solution of the equation 3x + 2y = k, then the value of k is a) 7 b) 6 c) 5 d) 4 Solution: 3x + 2y = k 3 (1) + 2 (2) = k 3 + 4 = k K = 7
- 25. • 𝑻𝒉𝒆 𝒄𝒐𝒎𝒎𝒐𝒏 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏 𝒐𝒇 𝟑𝒙 + 𝟐𝒚 = 𝟔 𝒂𝒏𝒅 𝟓𝒙 − 𝟐𝒚 = 𝟏𝟎 𝒊𝒔 a)(0,3) b)(0,-5) c)(2,0) d)(1,0) Solution: 3x + 2y = 6 5x – 2y = 10 + ------------------------------- 8 x + 0 = 16 x = 𝟏𝟔 𝟖 = 2 3x + 2y = 6 3 (2) + 2y = 6 6 + 2y = 6 2y = 6 – 6 2y = 0 Y = 0 (x, y) = (2,0)
- 26. • Solution of the linear equation (x-1) = ( 𝟑 𝟒 )(x+1) – ( 𝟏 𝟐 ) will be a) x = 5 b)x = 4 c) x = 3 d)x = 1 Solution: (x-1) = ( 𝟑 𝟒 )(x+1) – ( 𝟏 𝟐 ) x – 1 = 𝟑 𝟒 x + 𝟑 𝟒 - 𝟏 𝟐 x - 𝟑 𝟒 x = 𝟑 𝟒 - 𝟏 𝟐 + 1 𝒙(𝟒−𝟑) 𝟒 = (𝟔−𝟒) 𝟖 + 1 𝒙 𝟒 = 𝟐 𝟖 + 1 2x = 2 + 8 2x = 10 X = 5
- 27. • What is the value of x in the equation 𝟑 x – 2 = 2 𝟑 + 4 a) 2 ( 1 - 𝟑 ) b) 2 ( 1 + 𝟑 ) c) 1 + 𝟑 d) 1 - 𝟑 Solution – 𝟑 x – 2 = 2 𝟑 + 4 • 𝟑 x = 2 𝟑 + 4 + 2 • 𝟑 x = 2 𝟑 + 6 • 𝟑 𝟑 x = 𝟐 𝟑+ 𝟔 𝟑 • x = 2 + 𝟔 𝟑 • x = 2 + ( 𝟔 𝟑 x 𝟑 𝟑 ) • x = 2 + ( 𝟔 𝟑 x 𝟑 ) • x = 2 + 2 𝟑 • x = 2 ( 1 + 𝟑 )
- 28. • The value of x, which satisfies the equation 2.8 x – 0.8 x = 9 is a) 𝟓 𝟐 b) 𝟗 𝟐 c) 𝟐 𝟓 d) 𝟐 𝟗 Solution: 8 x – 0.8 x = 9 2 x = 9 a) 2 ( 𝟓 𝟐 ) = 𝟗 5 9 b) 2 ( 𝟗 𝟐 ) = 9 9 = 9
- 29. • One solution of the equation 2x + 3y = 10 a) x = 3 and y = 2 b) x = 2 and y = 3 c) x = 2 and y = 2 d) x = 10 and y = -10 Solution: a) x = 3 and y = 2 2 (3) + 3(2) = 10 6 + 6 = 10 12 10 b) x = 2 and y = 3 2 (2) + 3 (3) = 10 4 + 9 = 10 13 ≠ 𝟏𝟎 c) x = 2 and y = 2 2 (2) + 3 (2) = 10 4 + 6 = 10 10 = 10
- 30. • The solution of the equation x + 3y = 12 will be a) (2,3) b) (1,2) c) (3,3) d) (4,2) Solution: a) (x, y) = (2,3) x + 3y = 12 → 2 + 3(3) = 12 → 2 + 9 = 12 → 𝟏𝟏 ≠ 𝟏𝟐 b) (x, y) = (1,2) x + 3y = 12 → 𝟏 + 3(2) = 12 → 1 + 6 = 12 → 𝟕 ≠ 𝟏𝟐 c) (x, y) = (3,3) x + 3y = 12 → 𝟑 + 3(3) = 12 → 3 + 9 = 12 → 𝟏𝟐 = 𝟏𝟐
- 31. Summary • An equation of the form ax + by + c = 0, where a, b, c are real numbers, such that a ≠ 0, b ≠ 0. Is called linear equation in two variable. • Substitution method - The substitution method is the algebraic method to solve simultaneous linear equations. As the word says, in this method, the value of one variable from one equation is substituted in the other equation.
- 32. Glossary Equation Equation is a mathematical statement which shows that the value of two expression are equal. Linear equation in one variable An equation which has the variable with the highest power one is called a linear equation in one variable Linear equation in two variable An equation which can be put in the form ax + by + c = 0. where a, b and c are real numbers and a and b both are not equal to zero, is called a linear equation in two variables. Solution of a linear equation in one variable The value of a variable for which the given equation becomes true is known as “Solution” or “Root’ of the equation. Solution of a linear equation in two variable A solution of the linear equation in two variables is an ordered pair of numbers, which satisfies the equations.
- 35. Example Find the four solutions of the equation 2x + 3y – 12 = 0 Solution – The given equation can be written as : y = 𝟏𝟐 −𝟐𝒙 𝟑 y = 𝟏𝟐 −𝟐𝒙 𝟑 . . . . . . . . . . ( 1 ) On putting x = 0 in equation (1), we get y = 𝟏𝟐 −𝟐 ( 𝟎 ) 𝟑 y = 𝟏𝟐 𝟑 = 4 On putting x = 3 in equation (1), we get y = 𝟏𝟐 −𝟐 (𝟑) 𝟑 y = 𝟔 𝟑 y = 2
- 36. On putting x = 6 in equation (1), we get y = 𝟏𝟐 −𝟐 ( 𝟔 ) 𝟑 y = 𝟏𝟐 −𝟏𝟐 𝟑 y = 0 On putting x = 9 in equation (1), we get y = 𝟏𝟐 −𝟐 (𝟗) 𝟑 y = 𝟏𝟐 −𝟏𝟖 𝟑 y = −𝟔 𝟑 y = -2
- 37. Hence, the four solutions of the given equation are: (i) x = 0, y = 4 (ii) x = 3, y = 2 (iii) x = 6, y = 0 (iv) x = 9, y = -3 A linear equation in two variables has infinitely many solutions.
- 38. Test Your Knowledge Equation p = 2q + 3 has _____________ a)only one solution. b)only two solutions. c)infinitely many solutions. d)no solution.