# Algebraic Equations in One Variable

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## Linear Equations in One Variable

 What is an Algebraic Equation? - A condition in which two expressions have equal values. General form of a linear equation in one variable: ax + b = c where a, b, c are real numbers and x is a variable i.e. Quantity whose value has to be found. Examples of Linear Equations: 3x + 2 = 8 16 = 4x x/3 = 4 3x + 2 = 4 Linear Equations are those polynomials whose degree is 1. Points to note: 1. There must be a variable in one of the two expressions. 2. Even if we interchange the expressions, the value remains unchanged i.e. ax + b = c and c = ax + b mean the same thing. Let us discuss Algebraic Expressions in detail before learning how to solve linear equations. Below are some examples in which we will learn how to form an equation Write the following statements in the form of equation: Question 1 - Four times of 'x' is 12 Solution - Four times of 'x' = 4x The resultant equation we get; 4x = 12 Question 2 – Sum of 3 times 'x' and 2 is 14 Solution - Three times of 'x' = 3x. Sum of 3x and 2 = 3x + 2 The resultant equation we get; 3x + 2 = 14 Question 3 – Subtract 3 from 4 times of “x” and we get 9 Solution - Four times of 'x' = 4x. Subtract 3 from 4x = 4x - 3 4x - 3 = 9 Question 4 – One third of “x” is 15 Solution - This proceeds as : One third of “x” = 1/3x The resultant equation we get; 1/3x = 15 Now let us learn how to solve linear equations: Example 1: 3x - 4 = 11 Step are as follows - STEP 1: Term containing the variable should be on LHS. 3x = 11 + 4 Or 3x = 15 STEP 2: Coefficient of the variable should be 1. Or x = 15 / 3 x = 5 Verification: Put x = 5 in LHS LHS = 3x - 4 LHS = (3 X 5) - 4 LHS = 15 - 4 LHS = 11 RHS = 18 So, LHS = RHS Let's study some more examples and verify whether the given value of variable is a solution to the equation Example - 2: Solve the Equation 2y + 3 = 3y and verify whether LHS = RHS Solution: Steps to solve this equation are as follows - 2y + 3 = 3y or 3 = 3y - 2y or 3 = y i.e y = 3 Verification: LHS = 2y + 3 LHS = (2 X 3) + 3 LHS = 9 RHS = 3y RHS = 3 X 3 RHS = 9 So, LHS = RHS Example - 3: Solve Equation 3y + 4 = 13 and verify whether LHS = RHS Solution: Steps to solve this equation are as follows - 3y + 4 = 13 3y = 13 - 4 3y = 9 y = 9 / 3 y = 3 Verification: LHS = 3y + 4 LHS = (3 X 3) + 4 LHS = 9 + 4 LHS = 13 RHS = 13 So, LHS = RHS Example - 4: Solve 3x + 5 = 5x - 1 Solution: steps are as follows - 3x + 5 = 5x - 1 5 + 1 = 5x - 3x 6 = 2x x = 3 Example - 5: Solve 2(p - 3) = 4(p + 1) - 8 Solution: steps are as follows - 2(p - 3) = 4(p + 1) - 8 2p - 6 = 4p + 4 - 8 2p - 4p = 6 + 4 - 8 -2p = 2 -p = 1 p = -1