Which is equivalent to 80 1/4x

Consider the expressions 3 2 + 1 and 5 × 2 . Both are equal to 10 . That is, they are equivalent expressions.

Now let us consider some expressions that include variables, say 5 x + 2 .

The expression can be rewritten as 5 x + 2 = x + x + x + x + x + 1 + 1 .

We can re-group the right side of the equation to 2 x + 3 x + 1 + 1 or x + 4 x + 2 or some other combination. All these expressions have the same value, whenever the same value is substituted for x . That is, they are equivalent expressions.

Two expressions are said to be equivalent if they have the same value irrespective of the value of the variable(s) in them.

Example 1:

Are the two expressions 2 y + 5 y − 5 + 8 and 7 y + 3 equivalent? Explain your answer.

Combine the like terms of the first expression.

Here, the terms 2 y and 5 y are like terms. So, add their coefficients. 2 y + 5 y = 7 y .

Also, − 5 and 8 can be combined to get 3 .

Thus, 2 y + 5 y − 5 + 8 = 7 y + 3 .

Therefore, the two expressions are equivalent.

Example 2:

Are the two expressions 6 ( 2 a + b ) and 12 a + 6 b equivalent? Explain your answer.

Use the Distributive Law to expand the first expression.

6 ( 2 a + b ) = 6 × 2 a + 6 × b = 12 a + 6 b

Therefore, the two expressions are equivalent.

Example 3:

Check whether the two expressions 2 x + 3 y and 2 y + 3 x equivalent.

The first expression is the sum of 2 x 's and 3 y 's whereas the second one is the sum of 3 x 's and 2 y 's.

Let us evaluate the expressions for some values of x and y . Take x = 0 and y = 1 .

2 ( 0 ) + 3 ( 1 ) = 0 + 3 = 3 2 ( 1 ) + 3 ( 0 ) = 2 + 0 = 2

So, there is at least one pair of values of the variables for which the two expressions are not the same.

Therefore, the two expressions are not equivalent.

Example 4:

Check whether the two expressions 3 × m × m m and m + m + m equivalent.

Consider the first expression for any non-zero values of the variable.

Cancel the common terms.

3 × m × m m = 3 m

Combine the like terms of the second expression.

m + m + m = 3 m

So, 3 × m × m m = m + m + m when m ≠ 0 .

When m = 0 , the expression 3 × m × m m is not defined.

That is, the expressions are equivalent except when m = 0 . They are not equivalent in general.

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     x+1/4*x-(80)=0 

Step by step solution :

Step  1  :

            1
 Simplify   —
            4

Equation at the end of step  1  :

         1          
  (x +  (— • x)) -  80  = 0 
         4          

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  4  as the denominator :

          x     x • 4
     x =  —  =  —————
          1       4  

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 x • 4 + x     5x
 —————————  =  ——
     4         4 

Equation at the end of step  2  :

  5x    
  —— -  80  = 0 
  4     

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  4  as the denominator :

          80     80 • 4
    80 =  ——  =  ——————
          1        4   

Adding fractions that have a common denominator :

 3.2       Adding up the two equivalent fractions

 5x - (80 • 4)     5x - 320
 —————————————  =  ————————
       4              4    

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   5x - 320  =   5 • (x - 64)

Equation at the end of step  4  :

  5 • (x - 64)
  ————————————  = 0 
       4      

Step  5  :

When a fraction equals zero :

 5.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  5•(x-64)
  ———————— • 4 = 0 • 4
     4    

Now, on the left hand side, the  4  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   5  •  (x-64)  = 0

Equations which are never true :

 5.2      Solve :    5   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 5.3      Solve  :    x-64 = 0 Add  64  to both sides of the equation : 
                      x = 64

One solution was found :

 x = 64

Is equivalent to expression?

Which is equivalent to 64 to the power of 1 4?

Which is equivalent to log_2 N 4?

Which expression is equivalent to the root of 80?