Whole Numerical Equations

Whole numerical equations are equations where the variable has a numerical coefficient and never appears in the denominator.

In plain language, an equation is both numerical and whole if it meets two simple conditions:

• An equation is "whole" when the variable doesn’t appear in the denominator.

  Note. The word “whole” refers to the variable, not to the coefficients - they can still be fractions.

• An equation is "numerical" when the coefficient of the variable is a number, not a letter or symbol.

Understanding with an example

Let’s start with a simple equation:

$$ \frac{1}{2} x + 3 = \frac{15}{2} $$

This equation is whole because x doesn’t appear in the denominator.

It’s also numerical because the coefficient of x is a number (½).

So, it’s a whole numerical equation.

Example 2

Now look at this one:

$$ \frac{1}{2x} + 3 = \frac{15}{2} $$

Here, the variable x is in the denominator - so it’s not whole.

Therefore, it’s not a whole numerical equation.

Example 3

Consider this equation:

$$ \frac{a}{2} x + 3 = \frac{15}{2} $$

It’s whole because x isn’t in the denominator, but it’s not numerical because the coefficient of x includes a literal (a).

So again, this isn’t a whole numerical equation.

First-degree whole numerical equations

A first-degree whole numerical equation with one variable has x raised to the power of 1. For example: $$ 2x = 8 $$

These equations are always reducible to the standard form ax = b:

$$ ax = b $$

Here, “a” is the coefficient of x and “b” is the constant term. Equations like these are the simplest to solve.

How to solve them

If the coefficient “a” is not zero (a≠0), the equation is called determinate because it has exactly one solution:

$$ x = \frac{b}{a} $$

Note. To find x, we use the second principle of equivalence: divide both sides by “a.” $$ ax = b $$ $$ \frac{ax}{a} = \frac{b}{a} $$ Simplify: $$ \require{cancel} \frac{\cancel{a}x}{\cancel{a}} = \frac{b}{a} $$ $$ x = \frac{b}{a} $$

Special cases

If “a” is zero (a=0), two different situations may occur:

• Indeterminate equation: infinite solutions, when b is also zero (b=0).

Example. $$ 0x = 0 $$ Whatever value you assign to x (2, 3, or 100), both sides are still 0=0. So, the equation is true for every x.

• Impossible equation: no solution, when b is not zero (b≠0).

Example. $$ 0x = 2 $$ No matter what x is, the left side is always 0, and the equation becomes 0=2 - which is false. So, it has no solution.

Worked example

Let’s solve this first-degree whole numerical equation:

$$ 5x - 1 = 2x + 5 $$

We apply the first principle of equivalence: subtract 2x from both sides.

$$ 5x - 1 - 2x = 2x + 5 - 2x $$

$$ 3x - 1 = 5 $$

Then add 1 to both sides:

$$ 3x - 1 + 1 = 5 + 1 $$

$$ 3x = 6 $$

Now the equation is in the form ax = b.

Since both the coefficient and the constant are nonzero, the equation is determinate - it has one solution.

Apply the second principle of equivalence again and divide both sides by 3:

$$ \frac{3x}{3} = \frac{6}{3} $$

$$ \frac{\cancel{3}x}{\cancel{3}} = 2 $$

$$ x = 2 $$

So, the solution is x = 2.

Check your result. Substitute x=2 into the original equation: $$ 5x - 1 = 2x + 5 $$ $$ 5 \cdot 2 - 1 = 2 \cdot 2 + 5 $$ Simplify: $$ 10 - 1 = 4 + 5 $$ $$ 9 = 9 $$ The equation holds true - x=2 is correct.

And that’s how you solve whole numerical equations step by step.