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Higher Algebra

How to use linear equations in everyday life Linear equations in one variable can be used to model real life situations. For example: we burn approximately 118 calories per hour of walking. How long should we walk to burn 354 calories ? To determine this, we use equation: 118t = 354, where 't' is the time. The answer is 3 hours.

Linear Equation in One Variable

Equation: An equation is a statement in which two algebraic expressions are equal. In other words, an equation is a sentence that expresses the equality of two algebraic expressions. Ex: 4x – 3y = 2; x2 + y2 + z2 = 10; p2 + 17 = 2p2 + 4q2. In the first example 4x – 3y = 2,  4x – 3y is called left hand side (L.H.S.) of the equation and 2 is called right hand side (R.H.S.) of the equation.

Linear equation in one variable: An equation which involves only one variable with exponent 1, is called a linear equation in that variable. The general form of a linear equation in one variable is:
ax + b = 0
where 'a', 'b' are real numbers, ‘x’ is a variable and a ≠ 0.
Each one of the equations: 3x + 2 = 0; 4p – 5 = p; 2y + 3 = 3y + 4 is a linear equation in one variable.

Solution of a Linear Equation

A value of the variable which when substituted for the variable in the equation, makes its two sides (LHS and RHS) equal, is called a solution (or root) of the equation.
To solve a linear equation in one variable, we have to find the value of the variable satisfying the given equation. Consider the equation 3x + 2 = 8. Because 3(2) + 2 = 8 is true, we say that 2 satisfies the equation. Therefore, 2 is the solution or root for given linear equation.

Rules for solving a linear equation:
The equality of a linear equation is not changed, when:
(i) the same number is added to both sides of the equation
(ii) the same number is subtracted from both sides of the equation
(iii) both sides of the equation are multiplied by the same non-zero number
(iv) both sides of the equation are divided by the same non-zero number.

  • Ex: Determine whether ‘5’ is a solution of the linear equation 4x – 3 = 3x + 2.

    Sol: Substitute 5 for 'x' and evaluate each side of the equation. Given linear equation is: 4x – 3 = 3x + 2. By substituting x = 5, we get: L.H.S. = 17 and R.H.S. = 17. The two sides of the equation have same values. So, 5 is the solution for the given equation.
Transposition and Cross-multiplication

Transposition: Any term of an equation may be taken to the other side with its sign changed, without affecting the equality. This process is called transposition.
For example: 4x – 2 = 2x – 6 4x – 2x = – 6 + 2.
Here, the term involving ‘x’ has been transposed to L.H.S. and the constant term (– 2) from L.H.S. has been transposed to R.H.S.

Cross-multiplication: If , then (a × d) = (b × c), that is, ad = bc. The numerator on L.H.S. is multiplied with denominator on R.H.S. and numerator on R.H.S. is multiplied with denominator on L.H.S., with "=" sign in tact. This process is called cross-multiplication.
Ex: Consider the equation . Then, by cross-multiplication, we get: 2(2x + 3) = 5(x – 4).

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