An algebraic expression consists of variables (or literals) and /or constants connected by mathematical operations. Let us observe the following expressions.
2x + 3y + 19, 7a – 8b + 11 and 12xy + 7yz – 8zr are examples of algebraic expressions.
2x – 7 = 11, 4x + 3y = 24, 3a – 5b =15 and 11m = 55 are examples of algebraic equations.
Two algebraic expressions connected by an equality (=) sign make an equation. An equation contain one or more variables. An equation is a statement of equality that involves one or more literals or variables and /or constants. In the equation 4x + 3y = 24, is called the coefficient of x, 3 is called the coefficient of y and 24 is called constant term.
LINEAR EQUATION IN ONE VARIABLE:
One way of describing an equation is in terms of its degree. The degree of an equation is equal to the highest power of the variable in it. If an equation has only one variable and the highest power of the variable is one, then it is called a linear equation in one variable.
The following algebraic expressions are not linear expressions since the highest power of there is 2 i.e., more than 1.
2x2 + 5y2, 3x2 + 5, a2 + 3a – 1
The following equations are linear equations in one variable, their variables being x, a and n respectively.
3x + 5 = 11, 12a – 4 = 20, 5n =35.
FEATURES OF AN EQUATION
While working with equations, keep the following important points in mind.
1.
RHS and LHS: In an equation, the expression on the left of the equality sing is called left hand side (LHS) and the expression on the right of the equality sig is called the right hand side (RHS).
RHS and LHS: In an equation, the expression on the left of the equality sing is called left hand side (LHS) and the expression on the right of the equality sig is called the right hand side (RHS).
For example: 3x + 5 = 11
3x + 5 = LHS and 11 = RHS
2. SOLUTION OF AN EQUATION: In an equation, we can substitute different values for the variable, but the values of LHS and RHS expressions will be equal only for certain values of the variable.
These values of the variable for which the LHS is equal to the RHS are called solutions of the equation. All linear equations will have only one solution. The solution can be rational number too.
Let us check whether x = 5 is the solution for 3x + 5 = 11.
Substituting x =5 in 3x + 5, we get
LHS = (3 ´ 5) + 5 = 15 + 5 = 20
This is not equal to the RHS which is 11.
3. Mathematical Operations on an equation: An equation is like a balance. It means if you perform the same mathematical operation on both sides of an equation, then the equality remains unchanged.
Solving Linear Equations
1. When Variable is only on One Side of the Equation: Let us consider on some example, solutions of linear equations having a variable only one side of equality. In this example we will observe that during transposing of a number or expression from one side of the equation to the other, their sign is changed, i.e., negative sign becomes positive and positive sign becomes negative.
1. Find the solution of 5x + 4 = 34
Solution: Step 1. Subtract 4 from both sides.
5x + 4 – 4 = 34 – 4
Implies 5x = 30
Step 2. Divide both sides by 5.
5x/5 = 30/5
Implies x = 6, which is required solution.
2. Find he solution for 4x – 3 = 20
Solution: 4x – 3 = 20
Step 1. Transpose – 3 from LHS to RHS.
4x = 20 + 3
4x = 23
Step 2. Divide both sides by 4.
4x/4 = 23/4
CHECK: We can verify whether our answer is correct by substituting the newly found values of the variable in the equation.
Solution: x =23/4 in the LHSof 4x – 3 = 20
LHS = (4 ´ 23/4) – 3
= 23 – 3 = 20
RHS = 20
LHS = RHS
So, the solution is correct.
When Variable is one Both sides of the Equality: Observe the following example that deal with the solution of equations having a variable on both sides of the equality.
Example: 5(x – 7) = 4(x + 3)
Solution: Step 1. Simplifying both side of the equality.
5x – 35 = 4x + 12
Step2. Bringing all the terms with variables to one side or to the LHS and taking the contants to the other side.
5x – 4x = 12 + 35
x = 47
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