Generating equivalent expressions :
An algebraic expression is a mathematical sentence involving constants (any real number), variables and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).
To generate equivalent expression to another expression, we have to be aware of the parts of an algebraic expression.
We can use properties to combine like terms in an expression.
For example, let us consider the algebraic expression
3x + 2x + 4
Use the Distributive Property to add the coefficients of the like terms (3x and 2x).
3x + 2x + 4 = (3 + 2)x + 4 = 5x + 4
When we rewrite an expression so that it has no parentheses and so that all like terms are combined, we are simplifying the expression.
Example 1 :
Simplify the expression 6x² - 4x²
In the given expression, 6x² and 4x² are like terms.
Use the distributive property for like terms
6x² - 4x² = x²(6 - 4)
Subtract inside the parentheses.
6x² - 4x² = x²(2)
Use commutative property of multiplication
6x² - 4x² = 2x²
Hence, the expression equivalent to 6x² - 4x² is 2x²
Example 2 :
Simplify the expression 3a + 2(b + 5a)
Use distributive property in 2(b + 5a)
3a + 2(b + 5a) = 3a + 2b + 2(5a)
Use the associative property of multiplication in 2(5a)
3a + 2(b + 5a) = 3a + 2b + (5·2)a
Multiply 2 and 5 in (5·2)a
3a + 2(b + 5a) = 3a + 2b + 10a
3a + 2(b + 5a) = 3a + 10a + 2b
Use the distributive property in 3a + 10a
3a + 2(b + 5a) = (3 + 10)a + 2b
Add inside the parentheses
3a + 2(b + 5a) = 13a + 2b
Hence, the expression equivalent to 3a+2(b+5a) is 13a+2b
Example 3 :
Simplify the expression y + 11x + 7y - 7x
In the given expression, 11x & 7x are like terms and y & 7y are like terms.
Use commutative property of addition in 11x + 7y
y + 11x + 7y - 7x = y + 7y + 11x - 7x
Use distributive property in y + 7y and 11x - 7x
y + 11x + 7y - 7x = (1 + 7)y + (11 - 7)x
Perform operations inside parentheses.
y + 11x + 7y - 7x = 8y + 4x
Use commutative property of addition in 8y + 4x
y + 11x + 7y - 7x = 4x + 8y
Hence, the expression equivalent to y + 11x + 7y - 7x is 4x+8y
After having gone through the stuff given above, we hope that the students would have understood "How to generate equivalent expressions".
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