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Simplifying Algebraic Expressions with Numerical

Key Concepts

  • Combine like terms to simplify algebraic expressions.
  • Simplify algebraic expressions with fractions.
  • Simplify algebraic expressions with parenthesis and decimals

Properties of Operations 

Identity Property  

of addition                 a + 0 = a = 0 + a 

of multiplication      a x 1 = a = 1 x a 

Distributive Property 

across addition a(b + c) =  a(b) + a(c) 

across subtraction a(b – c) = a(b) – a(c) 

Example 1: Terms that have the same variable part, such as y and 2y, are like terms. To simplify algebraic expressions, use properties of operations to write equivalent expressions that have no like terms and no parentheses.  

Write simplified equivalent expressions for  

x + x + x and 2y – y. 

Solution: 

Case1: 

Combine the like terms in x + x + x. 

So, 3x is equivalent to x + x + x. 

Case2: 

Combine the like terms in 2y – y. 

= 1y or y  

So, y is equivalent to 2y – y. 

TRY IT! 

Simplify the expression 4z – z + z – 2z. 

Solution: 

The simplified expression is 2z. 

Example 2: 

A new hiking trail includes three sections. There is a flat section, a hilly section, and a winding section. The park ranger marked the sections of the trail in relation to the length of the flat stretch of the trail, which is n kilometers. What is the simplified expression that describes the length of the new trail? 

Solution: Length of flat section = n 

Length of hilly section = 2n 

Length of winding section =2/3n + 4 

 Total length of new hiking trail = ? 

 Total length of new hiking trail = length of flat section + length of hilly section + length of winding section 

Total length of new hiking trail  

                             = length of flat section + length of hilly section + length of winding section 

TRY IT! 

Park rangers add another section to the trail, represented by the expression

1/2n + n +1/2 . Write an expression for the new total length of the trail. Then write a simplified equivalent expression. 

Solution: 

Given expression is

1/2n + n +1/2 n  

So, the simplified expression is = 1

1 1/2n +1/2

Example 3: 

This summer, Vanna wants to charge 2.5 times as much for mowing and ranking, but her expenses ($ 10 per weekend) will also increase 2.5 times. The expression below can be used to find how much Vanna will make this summer mowing and ranking x lawns in a weekend.  

2.5(20. 50 x + 5.50 x – 10) 

How can you use properties of operations to write a simplified equivalent expression without parentheses? 

Solution: 

TRY IT! 

Suppose Vanna increases her rate by 3.5 times, and her expenses also increase by 3.5 times this summer. Write two equivalent expressions to represent how much she can earn mowing and raking grass. 

Solution: 

In Example 3, the expression 2.5(20.50x + 5.50x – 10) represented her total earnings, where 2.5 was how many times the amount she charged and her expenses increased by. 

If the amount she charges and her expenses increase by 3.5 times, then we need to replace the 2.5 in the expression with 3.5.  

The first expression is then 3.5(20.50x + 5.50x – 10) 

Using the properties of operations, an equivalent expression is: 

Practice and Problem solving: 

1.Henry wrote  4z²– z² as 4. Are  4z²– z² and 4 equivalent expressions? Explain. 

Solution: 

 4z²– z² =  z² (4 – 1) 

=z²(3) 

=3z²

The equivalent expression for 4z²– z² is 3z².

So, the statement “ 4z²– z²” and 4 are equivalent expressions”, which is given by Henry is wrong. 

2. Write an algebraic expression for the perimeter of the swimming pool. 

Solution: 

Given that, 

Length of the pool = 2y + 1  

Width of the pool = y 

Perimeter of the rectangle = 2(l + b) 

=2 [(2y + 1) + y] 

=2[2y + 1 + y] 

=2[3y + 1] 

=6y + 2 

∴The algebraic expression for the perimeter of the swimming pool is 6y + 2 

Let’s check our knowledge: 

  1. 3 + 3y – 1 + y – Simplify the expression. 
  1. 3.2x + 6.5 – 2.4x – 4.4 – Simplify the expression. 
  1. 3/4x+2+3x – 1/2 – Simplify the expression. 
  1. Rodney rewrote the expression 1/2(2x + 7) as x +3½. Which property of operations did Rodney use? 

Answers: 

1.3+ 3y – 1 + y 

= 3y + y + 3 – 1  

= 3y + 1y + 2 

= 4y+ 2 

2. 2x + 6.5 – 2.4x – 4.4 = 3.2x – 2.4x + 6.5 – 4.4 

= 0.8 x + 2.1  

3. 3/4x+2+3x-1/2= 3/4x+3x+2 – 1/2   

=(3/4+3)x+3/2

=15/4x+3/2

=3¾x+1½

4. Rodney rewrote the expression 1/2(2x + 7) as x +3½ . Which property of operations did Rodney use? 

Solution: 

Given that, 

1/2(2x+ 7) = x +3½

LHS =1/2(2x+ 7)  

=1/2 x 2x + 1/2 x 7 

=x + 7/2

=x + 3½

LHS = RHS  

∴ Here Rodney used distributive property to solve this expression. 

Key concept 

You can combine like terms to write equivalent expressions. Like terms have the same variable part. 

2x + 6 + 5x + 4 ………………………………… Identify like terms. 

= 2x + 5x + 6 + 4 ……… Commutative property of addition. 

= 7x + 10 ……. Add like terms  

2x + 6 + 5x + 4 = 7x + 10 

Exercise:

  1. What is the value of 5x + 10x + 20y + 100y?
  2. The price of a burger is $10, and the price of a pizza is $20. Robert bought x burgers and y pizzas. Write a numerical expression and solve it.
  3. Simplify 5.5x + 5.5y + 1x + 2y
  4. What is the value of 100x + 100y + (-100y) + (-100x)
  5. Prove that 2(5x + 10x) = 60 when the value of ‘x’ is 2?

Concept map: 

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