CONSTRUCTION OF LINEAR EQUATIONS

Now, let us see, how to construct linear equations and solve them.

Consider an expression 7x + 3, where x is the variable.

When x = 2, the value of the expression is

(7 × 2) + 3 = 14 + 3

= 17

Also, this can be written as 7x + 3 = 17, when x = 2.

7x + 3 = 17 is called an equation.

Moreover, no value of x other than 2 satisfies the condition 7x + 3 = 17. Thus x = 2 is called the solution to the equation 7x + 3 = 17.

An equation, is always equated to either a numerical value or another algebraic expression. The equality sign shows that the value of the expression to the left of the ‘=’ sign is equal to the value of the expression to the right of the ‘=’ sign.

In the above example, the expression 7x + 3 on the left side is equal to the constant 17 on the right side.

In general, the right side of an equation is just a number. But, this need not be always be so. The right side may be an expression containing the variable.

For example, the equation 7x + 3 = 3x - 1 has the expression 7x + 3 on the left and 3x - 1 on the right separated by an equality sign.

Let us look at some situations where we have the value of an expression without actually knowing the values of each of the variables.

The price of 10 chairs and 4 tables is $400. We are not given the price of a chair or a table. Let us take the price of one chair as $x and one table as $y. Then the cost of 10 chairs and 4 tables is 10x + 4y.

Therefore, 10x + 4y = 400. This is an equation in two variables x and y taking the value 400.

Let us consider some more examples and find the algebraic equations.

Example 1 :

"The cost of an apple and 2 mangoes is $12"

If the price of an apple is a and one mango is m, then the required equation is

a + 2m = 12

Example 2 :

"A rectangle of some dimensions has perimeter 50 cm"

If l and w are the length and width of the rectangle then we write

2(l + w) = 50

Example 3 :

“Tacey is 4 years elder to her sister Stella and their ages add upto 24”

The ages of Tacey and her sister Stella are treated as variables. Although, their ages are different, we consider this as one variable, because the age of Tacey is associated to the age of Stella.

Suppose, if Stella's age is 10, then the age of Tacey is

10 + 4 = 14

In the same way, if the age of Stella is x, then the age of Tacey is (x + 4).

Thus we can form an equation as

x + (x + 4) = 24

x + x + 4 = 24

2x + 4 = 24

Hence an equation can be looked as algebraic expression equated to either a constant or any other algebraic expression.

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