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Algebra Basics: Understanding Variables and Mathematical Language

Capítulo 1

Estimated reading time: 5 minutes

Algebra as a Practical Language

Algebra is a way to write math when a quantity is unknown, not fixed yet, or can change. Instead of always working with specific numbers, algebra lets you describe a situation using symbols so you can solve it later or see how it changes.

Everyday idea: If the price of one notebook is unknown, you can still describe the total cost of buying 3 notebooks. You write 3n where n is the price of one notebook.

Quick identification

In 3n, circle the symbol that stands for the unknown.
Check
Variable: n
In d + 5, circle what could change.
Check
Variable: d

Variables: Placeholders and Changing Quantities

A variable is a symbol (often a letter) that represents a number. It can be:

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A placeholder for an unknown value (you don’t know it yet).
A quantity that can vary (it can take different values depending on the situation).

Common variable letters include x, y, n, t. The letter itself doesn’t matter; what matters is what it represents.

Step-by-step: reading an expression with a variable

Find the variable(s).
Read any number attached to it as multiplication.
Translate into words.

Example: 4x + 7

Variable: x
4x means 4 × x
In words: “four times x, plus seven”

Quick identification (circle the variable)

9y
Check
Variable: y
a - 12
Check
Variable: a
5m + 2n
Check
Variables: m and n

Subscripts: When You Need More Than One of the Same Kind

Sometimes you need several related variables, like several test scores or several items. Subscripts help you name them clearly: x₁, x₂, x₃. You can read x₁ as “x one.”

Important: x₁ is a single variable name, not x × 1.

Example situation: Three quiz scores: q₁, q₂, q₃. The total is q₁ + q₂ + q₃.

Quick identification

In x₁ + x₂, how many variables are there?
Check
Two variables: x₁ and x₂
In 3x₁, circle the variable name.
Check
Variable: x₁

Constants, Coefficients, and Terms

Constants

A constant is a fixed number in an expression. It does not change.

Example: In 2x + 9, the constant is 9.

Coefficients

A coefficient is the number multiplying a variable.

Example: In 2x + 9, the coefficient of x is 2.

If a variable has no number written in front, the coefficient is 1. For example, x means 1x.

If there is a minus sign, the coefficient is negative. For example, -x means -1x.

Terms

A term is a piece of an expression separated by + or - signs.

Example: In 5x - 3y + 8, the terms are 5x, -3y, and 8.

Quick identification (underline coefficients; box constants; list terms)

7x + 4
Check
Coefficient: 7; Constant: 4; Terms: 7x, 4
-3y - 10
Check
Coefficient: -3; Constant: -10; Terms: -3y, -10
a + 2b + 2
Check
Coefficients: 1 (for a), 2 (for b); Constant: 2; Terms: a, 2b, 2

Common Pitfalls (and How to Avoid Them)

Pitfall 1: Confusing the letter “x” with the multiplication sign

In algebra, x is often a variable name. Multiplication can be shown in other ways to avoid confusion:

3 × 5 (times sign)
3·5 (dot)
3(5) (parentheses)
3*5 (asterisk, common in typing)

Example: 2x means 2 × x, not “2 times (the letter x symbol).” Here x stands for a number.

Quick check

Which one shows multiplication, not a variable? 4 × 6 or 4x?
Check
4 × 6 shows multiplication; 4x uses the variable x.

Pitfall 2: Treating a variable as a label instead of a number

A variable represents a number, so you can do arithmetic with it. If n = 5, then 3n means 3 × 5, which equals 15.

Wrong idea: Thinking 3n means “3 notebooks.” It only means that if you specifically defined n to represent “number of notebooks.” Even then, n is still a number (a count).

Step-by-step: substituting a value

Replace the variable with the given number.
Use parentheses if it helps you keep signs clear.
Compute.

Example: Evaluate 2x + 1 when x = -3.

2x + 1 = 2(-3) + 1 = -6 + 1 = -5

Quick substitution practice (with immediate checks)

Evaluate 5y when y = 4.
Check
5y = 5(4) = 20
Evaluate m - 7 when m = 2.
Check
m - 7 = 2 - 7 = -5
Evaluate -x + 6 when x = 6.
Check
-x + 6 = -6 + 6 = 0

Mixed Practice: Identify and Interpret

For each item: (1) circle the variable(s), (2) underline the coefficient(s), (3) box the constant(s), (4) list the terms.

Expression	Your work	Immediate check
`8x - 3`	Circle variable; underline coefficient; box constant; list terms	Check Variable: `x`; Coefficient: `8`; Constant: `-3`; Terms: `8x`, `-3`
`t + 5t + 1`	Circle variable; underline coefficients; box constant; list terms	Check Variable: `t`; Coefficients: `1` (in `t`), `5`; Constant: `1`; Terms: `t`, `5t`, `1`
`-2a + 4b - 9`	Circle variables; underline coefficients; box constant; list terms	Check Variables: `a`, `b`; Coefficients: `-2`, `4`; Constant: `-9`; Terms: `-2a`, `4b`, `-9`
`3x₁ + x₂ - 10`	Circle variables; underline coefficients; box constant; list terms	Check Variables: `x₁`, `x₂`; Coefficients: `3` (for `x₁`), `1` (for `x₂`); Constant: `-10`; Terms: `3x₁`, `x₂`, `-10`

Mixed Practice: Substitute and Compute

Compute each expression using the given value(s). Show substitution first, then simplify.

4n + 2 when n = 3
Check
```
4n + 2 = 4(3) + 2 = 12 + 2 = 14
```
7 - 2y when y = 5
Check
```
7 - 2y = 7 - 2(5) = 7 - 10 = -3
```
x₁ + x₂ when x₁ = 8 and x₂ = -1
Check
```
x₁ + x₂ = 8 + (-1) = 7
```
-3m + 4 when m = -2
Check
```
-3m + 4 = -3(-2) + 4 = 6 + 4 = 10
```

Now answer the exercise about the content:

In the expression 3x₁ + x₂ - 10, which statement correctly identifies the variables, coefficients, and constant?

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You missed! Try again.

Subscripts make x₁ and x₂ separate variable names. The numbers multiplying variables are coefficients: 3 for x₁ and an implied 1 for x₂. The fixed number term is the constant -10.