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Equivalent Expressions

Algebra is the sub division of mathematics that utilizes letters in place of some unknown numbers. Algebraic expressions are nothing but strings consisting numbers, variables in different powers and mathematical operators. We use letters in the place of numbers.

For Example:  consider the area of a square, with side 's' :

$A$  = $S^2$

As soon as we know the measurement length wise of the sides, we can find the area. Algebraic expressions are expressions which involve different mathematical signs and we solve the algebraic expressions step by step. let us see concepts and answers for algebraic expressions.

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Definition

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An algebraic expression consists of numbers, variables, mathematical operators and exponents. An algebraic expression is different from an equation as this does not equate to zero or anything. It is just a string consisting of integers, fractions, variables and operators. They also have exponent terms.

For Example: 2x, 2x + 3, 2xy -3e are all algebraic expressions. When they are made equal to zero they become equations. Equivalent expressions are the algebraic expressions who give same result after substituting the variables by any value.

For Example: 2x + 2, 2(x + 1) are two algebraic expressions.
 
For any value of x, they will give same result. Putting, x = 2

2x + 2 = 6; 2(x + 2) = 6.

Now let us take one more example of two algebraic expressions. 2x + 1 and x + 2. For x = 1, 2x + 1 = 3 and x + 2 = 3. But for x = 0, 2x + 1 = 1 and x + 2 = 2. Hence, these expressions are not equivalent for all values of x. Hence, we cannot take these as equivalent algebraic expressions.

How to solve equivalent expressions ?

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To find if any given algebraic expressions are equivalent or not these two methods can be used:

1) Trial and error: Keep random values for the variables and compare the result of the expressions to check whether they are equal or not.

2)
 Simplifications: Simplify the expressions to check if they are same or not.
Let us take an example and understand. Suppose that we have been given two algebraic expressions $x(x-2)$ and $x^{2}-2x$. We will see how to find if these expressions are equivalent or not.

Trial and error: Put x = 0. $x(x-2)$ = 0, $x^{2}-2x$ = 0.

                          Put x = 1. $x(x-2)$ = -1, $x^{2}-2x$ = -1.

                          Put x = 2. $x(x-2)$ = 0, $x^{2}-2x$ = 0.

                          Put x = 3. $x(x-2)$ = 3, $x^{2}-2x$ = 3.

We can see that for each value of x, both expressions are giving same results only.

Simplification: There is a better way to find if two expressions are equivalent or not. Just expand and simplify the expressions to see if they are same or not.

$x(x-2) = x^{2} - 2x$ which is same as the second expression.

Examples

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Below are the problems on equivalent algebraic expressions:

Example 1:  Find if the given algebraic expressions are equivalent or not.

2x(x + 1) and 2x + 1.

Solution: Simplifying we get $2x(x + 1) = 2x^{2} + 2x$. It is not same as 2x + 1. For x = 1, 2x(x + 1) = 4, 2x + 1 = 3.
 
Hence, they are not equivalent.

Example 2: The expressions 2(3x + 3) and 3(2x + 2) are equivalent expressions or not.

Solution: Simplifying we get: 2(3x + 3) = 6x + 6 and 3(2x + 2) = 6x + 6.

Hence, both the expressions are same only. They are equivalent algebraic expressions.

Example 3: Using simplification, find if 2y(x + 2) and y(2x + 2) are equivalent or not.

Solution: We can see that, 2y(x + 2) = 2xy + 4 and y(2x + 2) = 2xy + 2.

Hence, both these expressions are not same. It can be concluded that they are not equivalent algebraic expressions.

Practice Problems

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Problem 1: Are the expressions $x^2(x + y)$ and $x^3 + xy$ equivalent?

Problem 2: Are $y^2(xy + yx)$ and $2y^3x$ equivalent expressions?

Word Problems

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Problem 1: Does squaring a number and dividing it by 5 is same as dividing a number by 5 and then squaring it?

Solution: Let the number be $x$.

Squaring a number and dividing it by 5 can be expressed as $\frac{x^2}{5}$.

Dividing a number by 5 and then squaring it will give the expression $(\frac{x}{5})^2$ = $\frac{x^2}{25}$

As the denominator of both expressions are different, these two statements are not same.

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