Algebra

Algebra gives a name to a number we do not know yet. The letter is a placeholder, so we can describe a situation before the missing number is found. When an equation says two quantities are equal, keep that equality balanced through every step.

Parents: ask your child to say what the letter represents before calculating. The balance lab makes both sides change together, then uses substitution as a check.

By the end you'll be able to use a letter for an unknown, write and simplify expressions, solve a simple linear equation with whole-number coefficients, and substitute to check.

A letter can hold an unknown

An algebraic variable keeps a mystery number visible. If Raju has p marbles and buys 4 more, the total has an unknown part and a fixed part. If three equal groups contain n each, the coefficient tells us how many equal groups there are.

Show the worked solution

Translate that story

$$\text{Raju total}=p+4$$

Read and write expressions carefully

An expression has no equals sign. Translate each word in order. When a phrase describes multiplication followed by subtraction, preserve that order. “Eight less than a number” means that the number comes first; subtraction order matters.

An expression can also describe sharing equally: a number divided by 4 is written as $n\div4$.

Show the worked solution

Translate the words

$$\text{three times a number, then take away 2}=3n-2$$

Translate the words

$$\text{eight less than a number}=n-8$$

Collect like terms

Like terms are equal kinds of groups. Combine their number parts and keep the letter. A plain number has no letter, so it stays separate. This section uses simple linear expressions only; there is no need to expand brackets.

Show the worked solution

Keep the kinds together

$$5n+2n-n=6n$$

Keep the kinds together

$$4n+3+2n=6n+3$$

Substitute to evaluate or check

To substitute, put the given number in place of the letter. Multiply before adding or subtracting. The result should be an ordinary number.

Substitute a known value

$$n=4$$

$$2n+1=2\times4+1=9$$

Keep an equation balanced

An equation has an equals sign. Think of it as a balance: an operation done to one side must be done to the other side. We will use whole-number coefficients and keep every intermediate value non-negative.

The canonical example is 3x + 5 = 26. Before opening the working, predict which equal operation removes the five unit blocks. Then use the controls to subtract 5 from both sides and divide both sides by 3.

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Equation balance lab: do the same to both sides

🤔 Predict first: For 3x + 5 = 26, what should happen first?

Subtract 5 from both sidesSubtract 5 only from the leftExpand brackets

After using the lab, compare your two balanced moves with the canonical check:

Solve and substitute

$$3x+5-5=26-5$$

$$3x=21$$

$$3x\div3=21\div3$$

$$x=7$$

$$3\times7+5=21+5=26$$

Three graduated practices

Practice 1 — subtract a constant

Solve 2x + 4 = 18.

Show the solution

Reveal the balance

$$2x+4-4=18-4$$

$$2x=14$$

$$x=7$$

$$2\times7+4=18$$

Practice 2 — divide a coefficient

Solve 4x + 3 = 27.

Show the solution

Reveal the balance

$$4x+3-3=27-3$$

$$4x=24$$

$$x=6$$

$$4\times6+3=27$$

Practice 3 — a larger whole-number coefficient

Solve 5x + 2 = 37.

Show the solution

Reveal the balance

$$5x+2-2=37-2$$

$$5x=35$$

$$x=7$$

$$5\times7+2=37$$

Watch out — easily mixed up

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

1. Mei has y dollars and earns 6 more. Which expression gives her new amount?

   6yy + 66 − y

2. Which expression means a number multiplied by 7?

   n + 7n − 77n

3. Simplify 4n + 3n − 2n.

   5n9n5

4. If n = 4, what is the value of 2n + 5?

   18913

5. Solve 3x + 5 = 26.

   x = 9x = 7x = 21

6. What is the first balanced step for 2x + 4 = 18?

   Subtract 4 from both sidesSubtract 4 only from the leftDivide 18 by 2 only

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