The practical math behind tech and STEM work: numbers, algebra, functions, geometry, logic and counting, with short code exercises.

Math for Tech & STEM

## Why this matters

A spreadsheet formula, a line of code and a physics equation all share one skill: writing a calculation with **variables** so it works for any input. Once you can read and evaluate expressions, formulas stop looking like secret codes and start looking like recipes.

## The core idea

A **variable** is a letter (or a name) that stands for a number that can change. An **expression** combines numbers, variables and operations, such as `3x + 5` or `2πr`. An expression does not have an equals sign; it describes a value. (An *equation* says two expressions are equal; that is the next lesson.)

Some vocabulary:

- In `4x² − 7x + 2`, the **terms** are `4x²`, `−7x` and `2`.
- `4` and `−7` are **coefficients**; `2` is a **constant**.
- `3x` means 3 × x. Writing two things side by side means multiply.

In code, multiplication is never implied: you must write `3 * x`, and powers use `**` in Python.

## Order of operations

Everyone must evaluate an expression the same way, so there is an agreed order:

1. **Parentheses** (brackets) first
2. **Exponents** (powers and roots)
3. **Multiplication and division**, left to right
4. **Addition and subtraction**, left to right

You may know this as PEMDAS or BODMAS. Note that multiplication does not beat division; they share a level and you work left to right. So 12 ÷ 3 × 2 = 4 × 2 = 8, not 12 ÷ 6 = 2.

## Worked example 1: evaluating

Evaluate `3x² − 2xy + 5` when x = 2 and y = 1.

1. Substitute, using parentheses to be safe: 3(2)² − 2(2)(1) + 5
2. Exponent: 3(4) − 2(2)(1) + 5
3. Multiply: 12 − 4 + 5
4. Add and subtract left to right: **13**

Substituting with parentheses matters most with negatives. If x = −3, then x² = (−3)² = 9. Writing −3² without parentheses means −(3²) = −9.

## Worked example 2: combining like terms

**Like terms** have exactly the same variable part, so they can be added: `5x + 3x = 8x`. Terms like `5x` and `3x²` are not alike and cannot be combined.

Simplify `4a + 3b − a + 2b − 6`:

- a-terms: 4a − a = 3a
- b-terms: 3b + 2b = 5b
- constant: −6
- Result: **3a + 5b − 6**

## Worked example 3: the distributive property

Multiply everything inside the parentheses by what is outside:

`3(2x − 4) = 6x − 12`

`−2(x + 5) = −2x − 10` (the negative sign is distributed too)

Simplify `2(x + 3) + 4(x − 1)`: 2x + 6 + 4x − 4 = **6x + 2**.

You can check a simplification by plugging in a number. With x = 1: the original gives 2(4) + 4(0) = 8 and the simplified version gives 6 + 2 = 8. Matching values at a couple of points is strong evidence the algebra is right.

## Expressions in code

```python
x = 2
y = 1
value = 3 * x**2 - 2 * x * y + 5
print(value)   # 13
```

Python follows the same order of operations, with `**` done before `*`. When in doubt, add parentheses; they cost nothing and make intent clear.

## A real example

A cloud service costs $0.023 per GB stored plus $0.09 per GB downloaded. Monthly cost as an expression: `0.023s + 0.09d`. For s = 500 and d = 120: 11.5 + 10.8 = **$22.30**. The expression is reusable every month; only the inputs change.

## Common mistakes

- **Squaring a negative without parentheses.** (−3)² = 9 but −3² = −9.
- **Combining unlike terms**, such as turning 2x + 3x² into 5x³.
- **Distributing to only the first term**: 3(x + 4) is 3x + 12, not 3x + 4.
- **Doing multiplication before division** regardless of position. They are equal priority; go left to right.

## Quick recap

- Variables stand for numbers; expressions combine them without an equals sign.
- Follow the order of operations; use parentheses when substituting.
- Combine only like terms; distribute to every term in the parentheses.
- Check a simplification by substituting a test value.

## Try it

In the **Do** tab, translate `3x² − 2xy + 5` into Python so the function returns the right value for any x and y.


Read, simplify and evaluate algebraic expressions using the order of operations.

Variables and expressions

Algebra Basics: What Is Algebra? - Math Antics

Intro to Evaluating Algebraic Expressions | How to Evaluate Algebraic Expressions | Math with Mr. J

## Why this matters

Many practical questions are really "find the missing number" problems. How many months until savings reach a target? What current gives a certain voltage drop? How many servers handle a given load? Each one becomes a linear equation, and solving it is a short, reliable routine.

## The core idea: keep the balance

An equation says two expressions are equal, like a balanced scale: `3x + 5 = 20`. You may do anything you like to one side **as long as you do the same thing to the other side**. The goal is to get the variable alone (isolate it).

To isolate the variable, undo what has been done to it using **inverse operations**:

- addition ↔ subtraction
- multiplication ↔ division

Undo in the reverse order of operations: deal with added or subtracted numbers first, then multiplied or divided ones.

## Worked example 1: one step

`x − 7 = 12`

Add 7 to both sides: x = 19.

Check: 19 − 7 = 12. (checks out)

## Worked example 2: two steps

`3x + 5 = 20`

1. Subtract 5 from both sides: 3x = 15
2. Divide both sides by 3: x = 5

Check: 3(5) + 5 = 20. (checks out)

## Worked example 3: negative coefficient

`−4x + 6 = −10`

1. Subtract 6: −4x = −16
2. Divide by −4: x = 4

Check: −4(4) + 6 = −16 + 6 = −10. (checks out)

Dividing by a negative is fine in equations. (It only flips the sign in *inequalities*, such as `<` and `>`.)

## Worked example 4: variables on both sides

`5x − 3 = 2x + 9`

1. Subtract 2x from both sides: 3x − 3 = 9
2. Add 3: 3x = 12
3. Divide by 3: x = 4

Check: left 5(4) − 3 = 17, right 2(4) + 9 = 17. (checks out)

## Worked example 5: a word problem

A freelancer charges a $40 call-out fee plus $55 per hour. An invoice is $315. How many hours?

1. Name the unknown: h = hours.
2. Write the equation: 40 + 55h = 315
3. Subtract 40: 55h = 275
4. Divide by 55: h = 5

Answer: 5 hours. Check: 40 + 55 × 5 = 315. (checks out)

## A general formula

Every equation of the form `ax + b = c` (with a not zero) is solved the same way:

x = (c − b) / a

That is just the two-step routine written once. Code can use it directly:

```python
def solve_linear(a, b, c):
    return (c - b) / a

x = solve_linear(3, 5, 20)
print(x)                 # 5.0
print(3 * x + 5 == 20)   # True: always check
```

If a = 0 there is no x to solve for: the equation is either always true (b = c) or never true. Real code should handle that case instead of dividing by zero.

## Why checking matters

Substituting your answer back into the original equation takes seconds and catches sign errors, arithmetic slips and setup mistakes. In professional work, an unchecked answer is a guess. Make the check a habit, not an extra.

## Common mistakes

- **Doing an operation to one side only.** Every step must apply to both sides.
- **Undoing in the wrong order**, such as dividing by 3 before subtracting 5 in 3x + 5 = 20 (possible, but you must divide *every* term: x + 5/3 = 20/3).
- **Losing a negative sign** when moving terms. Write out each step rather than "moving" terms in your head.
- **Checking against your rewritten equation** instead of the original one.

## Quick recap

- Do the same operation to both sides to keep the balance.
- Undo addition/subtraction first, then multiplication/division.
- For ax + b = c: x = (c − b) / a, provided a ≠ 0.
- Always substitute back into the original equation.

## Try it

In the **Do** tab, write the one-line formula inside `solve_linear` and let the checks confirm it works for positive, negative and fractional answers.


Solve one- and two-step linear equations with inverse operations, and check every answer.

Solving linear equations

Algebra Basics: Solving 2-Step Equations - Math Antics

Solving two-step equations | Linear equations | Algebra I | Khan Academy

## Why this matters

Formulas are usually printed solved for one variable, but the question you face often asks for a different one. Ohm's law is written V = IR, yet an electronics tech often needs the resistance. A finance formula gives interest, but you want to know how long a loan runs. Rearranging (also called solving a *literal equation*) lets you get any variable you need from one formula, instead of memorising several versions.

## The core idea

Rearranging a formula is exactly the same as solving an equation. The only difference is that the other "numbers" are letters. Pick the variable you want, then use inverse operations on both sides until it is alone.

Treat every other letter as if it were a known number. If it helps, mentally replace them with numbers first, work out the steps, then repeat the steps with letters.

## Worked example 1: Ohm's law

V = I × R. Solve for R.

R is multiplied by I, so divide both sides by I:

**R = V / I**

Check with numbers: V = 12 volts, I = 0.5 amps gives R = 24 ohms, and 0.5 × 24 = 12. (checks out)

## Worked example 2: speed, distance and time

d = v × t. Solve for t: divide both sides by v, so **t = d / v**.

A 450 km trip at 90 km/h takes 450 / 90 = 5 hours.

## Worked example 3: temperature

The formula from Celsius to Fahrenheit is F = (9/5)C + 32. Solve for C.

1. Subtract 32 from both sides: F − 32 = (9/5)C
2. Multiply both sides by 5/9 (the reciprocal of 9/5): **C = (5/9)(F − 32)**

Check: F = 212 gives C = (5/9)(180) = 100. Water boils at 100 °C. (checks out)

Note the parentheses. (5/9)(F − 32) is not the same as (5/9)F − 32. In code, write `(f - 32) * 5 / 9`.

## Worked example 4: simple interest

I = P × r × t, where P is the principal, r the yearly rate as a decimal, and t the time in years. Solve for t.

P and r both multiply t, so divide both sides by P × r:

**t = I / (P × r)**

How long until $2,000 at 4% earns $240 in simple interest? t = 240 / (2000 × 0.04) = 240 / 80 = 3 years.

## Worked example 5: a variable inside a power

The area of a circle is A = πr². Solve for r.

1. Divide by π: r² = A / π
2. Take the square root: **r = √(A / π)**

A 50 cm² circle has radius √(50 / π) ≈ 3.99 cm. (We take the positive root because a length cannot be negative.)

## A general strategy

1. Circle the variable you want.
2. List what has been done to it, in order (multiplied by, then added to, then squared, and so on).
3. Undo those steps in reverse order, doing each step to both whole sides.
4. Check by substituting simple numbers into the original formula and your new one.

## Rearranged formulas in code

```python
def resistance(voltage, current):
    return voltage / current

def celsius(fahrenheit):
    return (fahrenheit - 32) * 5 / 9

print(resistance(12, 0.5))   # 24.0
print(celsius(98.6))         # about 37.0
```

## Common mistakes

- **Dividing only part of a side.** From F − 32 = (9/5)C, dividing gives (F − 32) ÷ (9/5), not F − 32 ÷ (9/5).
- **Dropping parentheses** when turning a formula into code.
- **Forgetting units** when you check. A rearranged formula should still give sensible units: V / I gives volts per amp, which is ohms.
- **Dividing by a quantity that could be zero**, such as t = d / v when v = 0. Note those limits.

## Quick recap

- Rearranging a formula is solving an equation with letters.
- Undo operations in reverse order, applying each to both sides.
- Keep parentheses around whole sides when you divide or multiply.
- Check with easy numbers in both versions of the formula.

## Try it

In the **Do** tab, rearrange the Fahrenheit formula for Celsius and write it as Python. The checks test freezing, boiling, body temperature and the one temperature where both scales agree.


Isolate any variable in a physics or finance formula using the same steps as solving equations.

Rearranging formulas

Solve Literal Equations (Step by Step)

Variables, expressions, linear equations and rearranging formulas so you can solve for any unknown.