Algebra Foundations: recap

In a formula, letters stand for quantities. A letter whose value can change is a variable; a number that stays the same is a constant.

In $P = 2l + 2w$, $P$, $l$ and $w$ are variables and the 2s are constants.

In an equation such as $3x + 2 = 14$, the letter is an unknown: one particular number ($x = 4$) makes it true.

To substitute, replace each letter with its value.

A number next to a letter means multiply: $2l$ means $2 \times l$. Two letters next to each other also multiply: $lw$ means $l \times w$. Write the × signs in before working anything out.

Then work it out in order: brackets first, then × and ÷, then + and −. Give the units.

When you know a formula's answer but not one of its letters, substitute the values you know. What is left is an equation with one unknown.

Solve it by undoing: undo the last operation first (+ and − before × and ÷).

Then check: put your answer back into the formula. It should give the value you started with.

Words about a number tell you what to do to it. More than means add; less than means subtract; times, twice and lots of mean multiply; divided by and shared between mean divide.

Write the operations in the order the words say them. "3 more than twice n" is twice n (2n) and then 3 more: $2n + 3$.

"Less than" starts from the number: 7 less than n is $n - 7$, not $7 - n$.

Check an expression by trying a number you can work out in words.

Brackets group a part so it is worked out first.

If a sentence says to do something to the whole result of an earlier step, put that earlier step in brackets. "Add 3 to n, then double the result" is $2(n + 3)$.

If the later step is adding or subtracting, brackets are not needed: "double n, then add 3" is $2n + 3$.

The two are different. With $n = 4$: $2(n + 3) = 14$ and $2n + 3 = 11$.

To write an expression for a situation: (1) say what the letter stands for: the number that changes, such as the number of guests; (2) multiply it by the amount for each one; (3) add anything paid only once.

A hall costing 60 dollars plus 5 dollars per guest, with $g$ guests, is $5g + 60$.

Test the expression with one case you can check in words, then use it for any number.

The letter stands for a number, not for the objects: in $3a$, $a$ is how many apples.

An equation says two things are equal: $2x + 3 = 11$ says "two lots of something, plus 3, is 11".

The solution is the value that makes it true. To check a value, put it in place of the letter (this is called substitution) and work out each side. If both sides are the same, it is a solution.

Remember: $2x$ means $2 \times x$. If $x = 4$, then $2x$ is 8, not 24.

Backtracking means running an equation backwards. In $2x + 3 = 11$, $x$ is multiplied by 2, then 3 is added, and the answer is 11.

To find $x$, undo each step with its opposite, starting with the last one: take away 3 ($11 - 3 = 8$), then divide by 2 ($8 \div 2 = 4$).

Then check by running forwards: $2 \times 4 + 3 = 11$. If it matches, you have the solution.

An equation is like a balanced scale. If you do the same thing to both sides, it stays balanced, so the equation stays true.

To solve $2x + 3 = 11$: take 3 from both sides ($2x = 8$), then divide both sides by 2 ($x = 4$).

When $x$ is on both sides, take the same number of $x$ from both sides first.

Always check in the first equation: $2 \times 4 + 3 = 11$ ✓.

The number outside a bracket multiplies every term inside.

$3(2x + 4)$ is 3 lots of $2x + 4$: $3 \times 2x = 6x$ and $3 \times 4 = 12$, so $6x + 12$. Signs come too: $4(3x - 2) = 12x - 8$, and $-2(x - 5) = -2x + 10$.

After expanding, collect any like terms: $6x + 12 - x = 5x + 12$.

Factorising is expanding backwards. Find a number that goes into every term, divide every term by it, and put it outside the bracket.

$6x + 9$: 3 goes into 6 and 9, so $6x + 9 = 3(2x + 3)$. Take out the highest common factor: $12x - 8 = 4(3x - 2)$, not $2(6x - 4)$, which still shares a 2. A letter can be common too: $2x^2 + 4x = 2x(x + 2)$.

Check by expanding: you should get back to where you started.

To change the subject, undo what was done to the letter you want, last step first, doing the same to the whole of both sides.

$y = 3x + 5$: $x$ was multiplied by 3, then 5 was added. Take 5 away: $y - 5 = 3x$. Divide by 3: $x = \dfrac{y - 5}{3}$.

Check by putting in numbers: $x = 2$ gives $y = 11$, and $\dfrac{11 - 5}{3} = 2$.

To solve an equation, do the same to both sides until $x$ is on its own:

1. Expand any brackets.
2. Take the smaller $x$ term from both sides, so $x$ is on one side only.
3. Undo the numbers with the inverse operation (add to undo $-$, subtract to undo $+$).
4. Divide (or multiply) to get $x$.

$3x + 7 = 5x - 9 \to 7 = 2x - 9 \to 16 = 2x \to x = 8$. Check: $3(8) + 7 = 31$ and $5(8) - 9 = 31$. On a graph, $x = 8$ is where the two sides meet.

Solve an inequality like an equation, doing the same to both sides. Adding, subtracting, and multiplying or dividing by a positive keep the inequality sign. Multiplying or dividing by a negative reverses it: $-2x > 6$ gives $x < -3$ (because $2 < 5$ but $-2 > -5$).

On a number line: an open circle for $<$ and $>$ (the end is not included), closed for $\le$ and $\ge$, shaded towards the solutions. Check by trying a value: $x = -4$ gives $8 > 6$. ✓