Depth · one concept from first principles · step 2 of 4

Write the solution in a, then filter by wholeness

In the equation $x-4=\frac{1}{5}(x-2a)$ in $x$ , both the solution $x$ and the constant $a$ are natural numbers (positive whole numbers). Find all possible values of $a$ .

Expressions

Your answer

How to solve

Strategy Make a Systematic List — First get x as a formula in a. Two filters then cut the natural numbers down: x must be whole (so a must be even) and x must be positive (so a must be small). A short list of the survivors is the answer.

1STEP 1

Clear the fraction

Multiply both sides by 5.

5x-20=x-2a

2STEP 2

Solve for x in terms of a

Collect the x terms on the left and the rest on the right, then divide by 4.

4x=20-2a → x=5-a/2

3STEP 3

x must be whole

x=5-a/2 is a whole number only when a/2 is whole, so odd values of a are out: a must be even.

4STEP 4

x must be positive

x ＞ 0 means 5-a/2 ＞ 0, so a/2 ＜ 5, that is, a ＜ 10.

a/2 ＜ 5 → a ＜ 10

5STEP 5

List the survivors

Even natural numbers less than 10, with the matching x for each.

a=2,4,6,8 → x=4,3,2,1

Answer

2, 4, 6, 8

Take a=8, x=1: left side 1-4=-3; right side (1/5)(1-16)=-3. Equal. And a=10 would give x=0, which is not natural, so the list stops at 8.

Takeaway

Solve for x with the letter still in it, then let "whole" and "positive" do the sorting.

Clear the fraction
Solve for x in terms of a
x must be whole
x must be positive
List the survivors

Where next?

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▶ Practice — 12 problems