7.1 Basic Algebra
Key Takeaways
- OPM names basic algebra as Occupational Math content.
- Solving a linear equation means isolating the variable by applying inverse operations to both sides in reverse order.
- Translating a word problem into an equation is the harder half; define the variable in words before writing anything.
- Substituting the solution back into the original equation is a decisive check that costs a few seconds.
- Formula rearrangement uses the same inverse-operation logic as equation solving.
7.1 Basic Algebra
Quick Summary: OPM lists basic algebra among the Occupational Math topics. In practice that means three things: solving a linear equation for one unknown, evaluating or rearranging a formula you are given, and translating a word problem into an equation. The third is where items are won or lost, and the competency definition supports that emphasis — solves practical problems by choosing appropriately from a variety of mathematical techniques such as formulas and percentages.
Solving linear equations
To isolate a variable, undo the operations applied to it, in reverse order, doing the same thing to both sides. Each operation has one inverse:
| Operation applied to the variable | Undo it by |
|---|---|
| Added a number | Subtract that number from both sides |
| Subtracted a number | Add that number to both sides |
| Multiplied by a number | Divide both sides by that number |
| Divided by a number | Multiply both sides by that number |
| Wrapped in parentheses | Expand, or divide both sides by the outside factor |
Solve 4x + 7 = 31.
Subtract 7 from both sides: 4x = 24. Divide both sides by 4: x = 6. Check: 4(6) + 7 = 24 + 7 = 31. ✓
The order matters. The variable was multiplied by 4 and then had 7 added, so undoing runs backwards: subtract first, divide second.
With the variable on both sides
Collect variable terms on one side and constants on the other.
Solve 5x − 12 = 2x + 9.
Subtract 2x: 3x − 12 = 9. Add 12: 3x = 21. Divide by 3: x = 7. Check: 5(7) − 12 = 23, and 2(7) + 9 = 23. ✓
With parentheses
Expand first.
Solve 3(x + 4) = 27.
Expand: 3x + 12 = 27. Subtract 12: 3x = 15. Divide by 3: x = 5. Check: 3(5 + 4) = 3 × 9 = 27. ✓
Alternatively divide both sides by 3 first: x + 4 = 9, so x = 5. Either route works; pick whichever produces cleaner numbers.
Translating word problems
This is the skill that transfers. Follow a fixed order.
Step 1 — Define the variable in words. Write it out: let x = the number of forms processed on Monday. A variable without a written definition is the leading cause of solving correctly and answering the wrong quantity.
Step 2 — Express other quantities in terms of the variable.
Step 3 — Find the sentence that states an equality — a total, a difference, a "same as". That sentence becomes the equation.
Step 4 — Solve, then return to Step 1 and confirm you have reported the quantity actually asked for.
An office processed 312 forms over two days. Tuesday's count was 24 more than Monday's. How many were processed on Tuesday?
Step 1. Let x = forms processed on Monday. Step 2. Tuesday = x + 24. Step 3. Total: x + (x + 24) = 312. Step 4. 2x + 24 = 312 → 2x = 288 → x = 144. That is Monday. Tuesday = 144 + 24 = 168. Check: 144 + 168 = 312, and 168 − 144 = 24. ✓
The number 144 will be among the options. It is correct algebra answering the wrong question, which is exactly why Step 1 is written down.
Formulas
Some items give you a formula and ask you to use it. No memorisation is required — the formula is supplied, consistent with OPM's rule that all needed information is in the question text.
Evaluating: substitute and compute.
Given C = 45n + 120, find C when n = 16.
C = 45(16) + 120 = 720 + 120 = 840.
Rearranging: the same inverse-operation logic.
Given C = 45n + 120, find n when C = 1,065.
1,065 = 45n + 120 → 945 = 45n → n = 21. Check: 45(21) + 120 = 945 + 120 = 1,065. ✓
Rearranging symbolically: to make n the subject, subtract 120 from both sides and divide by 45, giving n = (C − 120) ÷ 45.
Two useful patterns
Fixed plus variable cost
A great many practical problems have the shape total = fixed amount + rate × quantity.
A service charges a $75 setup fee plus $12 per unit. How many units can be bought for $459?
75 + 12n = 459 → 12n = 384 → n = 32. Check: 75 + 12(32) = 75 + 384 = 459. ✓
Comparing two options
Set the two expressions equal to find the break-even point.
Plan A costs $200 plus $8 per unit. Plan B costs $320 plus $5 per unit. At how many units do they cost the same?
200 + 8n = 320 + 5n → 3n = 120 → n = 40. Check: A = 200 + 320 = 520; B = 320 + 200 = 520. ✓
Below 40 units Plan A is cheaper; above 40, Plan B is.
Always substitute back
Substitution is the most decisive check available in algebra, and it takes seconds. If your solution does not satisfy the original equation, it is wrong — no judgement required. Given that answers cannot be revised after RECORD ANSWER (Section 2.2), spending fifteen seconds on a check you can trust completely is time well spent.
An office processed 312 forms over two days, and Tuesday's count was 24 more than Monday's. How many were processed on Tuesday?
Plan A costs $200 plus $8 per unit and Plan B costs $320 plus $5 per unit. At how many units do the two plans cost the same?
Given the formula C = 45n + 120, what is n when C = 1,065?