Solving Equations – Method One (Balancing Method)

Also called the “balancing” or “do-the-same-to-both-sides” method.
View an equation as a set of scales: whatever you put on / remove from one pan must be matched on the other pan to keep it level.
Formal rule: Any legal arithmetic operation carried out on one side of the equality symbol must be carried out on the other side.
Operations allowed:
- Adding or subtracting the same number or algebraic expression.
- Multiplying or dividing by the same non-zero number.
- More advanced: applying the same function (e.g., squaring, taking a root) provided it is bijective on the domain under consideration.
Goal in linear (one-variable) equations: isolate the unknown so that it appears as $x = \text{(some value)}$ .
Letters for the unknown can be any alphabet symbol except those easily confused with numbers or with each other (e.g., avoid $o, l, I, e$ ).

Addition/Subtraction
- Acts on the entire side, but because a side is already written as a single expression, you may simply tag the number on: $2x-1+10 = 3+10$ .
- No need to distribute over individual terms because you are adding to the whole side.
Multiplication/Division
- Must be distributed to every term on that side:
  $2x-1\;\xrightarrow{\times 2}\;2\bigl(2x-1\bigr) = 2(2x) - 2(1)=4x-2$
- Safer notation: place the original side in brackets first, then apply the factor: $2(\;2x-1\;)$ .
- Division follows the same rule: each term is divided, or place the whole side over the divisor: $\tfrac{2x-1}{2}$ .
Fractions As Division
- Reminder: the division bar is simply a large fraction; $\frac{6}{2}$ is identical to “6 ÷ 2.”
Order of Undoing (for two-step linear equations):
1. Clear the added/subtracted constant term.
2. Clear the coefficient by multiplying/dividing.
- Mnemonic: “Unwrap the gift from the outside in.”

Starting equation: $2x-1=3$
- Adding $10$ to one side and to the other (purely demonstration):
  $2x-1+10 = 3+10 \;\Rightarrow\; 2x+9 = 13$ (not helpful but mathematically legit).
- Doubling both sides:
  $2(2x-1)=2\cdot3\;\Rightarrow\;4x-2 = 6$ .
- Halving the line above:
  $\tfrac{4x-2}{2}=\tfrac{6}{2}\;\Rightarrow\;2x-1 = 3$ (returns to the original).
Key takeaway from these “detours”: legality ≠ efficiency; you want efficient steps that move toward isolation of $x$ .

Starting base equation in the video: $2x = 6$ . Requested modifications:

(a) Add 10: $2x+10 = 16$
(b) Subtract 5: $2x-5 = 1$ (or $2x+1-5=7$ simplified to $2x-4=7$ as the presenter did)
(c) Multiply by 3: $3(2x-3)=12 \;\Rightarrow\;6x-9=12$
(d) Divide by 2: $\tfrac{2x-8}{2}=\tfrac{20}{2} \;\Rightarrow\;x-4=10$

• Example 1: $2x-1=5$

• Example 2: $3x+1=10$

• Example 3: $4x-3 = 11$

• Example 4 (Fractional left side): $\dfrac{a}{3}-1 = 6$

• Example 5 (Left constant first): $2 + 4b = 12$

• Example 6 (Right side constant): $10 = 3x - 1$

Solve for $x$ :

Forgetting to apply multiplication/division to every term on a side.
Sign errors when adding/subtracting; write the operation explicitly to minimize mistakes.
Re-ordering anxiety: $3=2x-1$ is syntactically equivalent to $2x-1=3$ —just pick a side and work.
Stopping too soon: remember to simplify coefficients all the way to $x = \dots$ .

The balancing concept mirrors physical scales—critical when modelling forces, chemical equations (stoichiometry), or financial ledgers.
Method One lays the groundwork for:
- Solving equations containing brackets.
- Equations with unknowns on both sides.
- Fractional, decimal, or integer coefficients.
- Later algebraic techniques such as simultaneous equations, inequalities, and algebraic manipulation in calculus.
Upcoming videos (as announced) will treat:
1. Equations with brackets (distribution first).
2. Unknowns on both sides.
3. Dedicated sessions on handling fractions more extensively.

Always check your solution by substitution to enforce metacognitive verification—valuable in exams and real-life problem-solving.
Encourage seeking help (“come see me at lunchtime”)—recognises diverse learning paces and promotes academic honesty.