Given: (5/2)x=-5
Step 1:
Isolate the variable “x.” We are solving for “x,” so we need to isolate it, meaning we need to “undo” any operations on the left side of the equation while maintaining the equality of the equation.
Notice that 5/2 is multiplying x, hence why the term is displayed as (5/2)x. We need to cancel this 5/2 out to isolate “x.” We can do so by taking the reciprocal of 5/2.
The reciprocal of a fraction is the “flipped” version, meaning the numerator and denominator switch places with each other. The reciprocal property can be represented as: in fraction a/b, the reciprocal is b/a. So, in the fraction 5/2, the reciprocal is 2/5.
We take the reciprocal because it cancels out a fraction multiplying a variable. Here’s how:
(5/2)x•2/5 = [(5•2)/(2•5)]x
[(5•2)/(2•5)]x =(10/10)x
(10/10)x=x
So, we first multiply straight across with the two fractions, so the denominators multiply and numerators multiply. For example:
a/b•b/a=(a•b)/(b•a)
This creates a fraction that has a quotient of 1, as a number divided by the same number is always 1. For example:
(ab)/(ab)=1
This is true because ab is the same number for the numerator and denominator.
So, this is why we take the reciprocal of a fraction multiplying a variable. Because it is a flipped fraction, the numerator and denominator of the product will create a fraction that cancels out. Let’s solve the equation:
Take the reciprocal of 5/2, and multiply it on both sides to keep the equation equal.
(2/5)(5/2)x=-5(2/5)
Simplify by multiplying fractions. Multiply the numerators and denominator of each fraction straight across:
(2•5)/(5•2)x=(-5•2)/5
*Note: -5 is = to -5/1 so that it can multiply 2/5.
Simplify:
(10/10)x=-10/5
Simplify:
x=-2
Now, let’s check x=-5 in the original equation:
(5/2)(-2)=-5
(5•-2)/2=-5
(-10)/2=-5
-5=-5
So, x=-2 is the answer.