Algebra and Equations

“Five less than $n$” means take 5 away from $n$, in that order: $n-5$. The words “less than” reverse the order you read them in.

Substitute, then use order of operations: $2\times 3 + 5 = 6 + 5 = 11$. Remember $2x$ means $2\times x$, not 23.

Substitute carefully: $2a = 2\times 5 = 10$ and $b^2 = (-3)^2 = 9$. So $2a - b^2 = 10 - 9 = 1$. (Writing $(-3)^2$ as $-9$ gives the wrong 19.)

Like terms have exactly the same variable to the same power. The coefficient does not have to match. $3x$ and $5x$ both have $x$ to the power 1, so they are like terms.

$3a$ means three lots of $a$ added together: $a + a + a$. ($a \times a \times a = a^3$, and $3 + a$ is a sum, not a product.)

Only like terms can be combined. $3a$ is a term in the variable $a$, but $4$ is a constant, so they are unlike and cannot be added together. The expression stays as $3a + 4$.

Group like terms: $3a+2a=5a$ and $4b-b=3b$, giving $5a+3b$. The minus sign stays attached to the $b$ in front of it.

$5x-8x=(5-8)x=-3x$. Taking away more than you started with gives a negative, and the $x$ stays.

Substitute $h = 5$ into the formula: $C = 8\times 5 + 15 = 40 + 15 = 55$. So it costs $55. (The $15 is a fixed fee added once, not for each hour.)

Substitute $x = 4$: $2x + 5 = 2\times 4 + 5 = 13$ and $x^2 = 4^2 = 16$. Since $16 > 13$, $x^2$ is larger. (Note $x^2$ means $x\times x$, not $x\times 2$.)

Distribute the 2 to both terms inside the brackets: $2\times x + 2\times 3 = 2x+6$.

Expand each bracket: $5(x+2)=5x+10$ and $-3(x-1)=-3x+3$ (the $-3$ multiplies both terms, so $-3\times-1=+3$). Combine: $5x-3x+10+3=2x+13$. (Forgetting to flip the sign of the $-1$ gives $2x+7$.)

The perimeter is the sum of the three sides: $x + (x+2) + (x+4) = 3x + 6$. Setting $3x + 6 = 24$ gives $3x = 18$, so $x = 6$. (The sides are then 6, 8 and 10 cm.)

Expand each bracket: $4(x+2)=4x+8$ and $3(x-1)=3x-3$. Collect like terms: $(4x+3x)+(8-3)=7x+5$.

Each new triangle adds 2 matches and the first triangle needs 3, so the rule is $2n+1$. Check: $n=1$ gives 3, $n=2$ gives 5, $n=3$ gives 7.

Set the rule equal to 27 and solve: $4n - 1 = 27$, so $4n = 28$ and $n = 7$. The 7th term is 27. (Check: $4\times 7 - 1 = 27$.)

Subtract 7 from both sides to undo the $+7$: $x = 12-7 = 5$.

Undo the $+5$ first: $2x = 13-5 = 8$. Then undo the $\times 2$: $x = \dfrac{8}{2} = 4$.

Collect the $x$ terms on one side: subtract $x$ from both sides to get $2x + 2 = 10$. Then $2x = 8$, so $x = 4$. (Check: $3\times 4 + 2 = 14$ and $4 + 10 = 14$.)

Subtract 2 from both sides: $\dfrac{x}{4} = 4$. Then multiply both sides by 4: $x = 16$. Undo adding and subtracting before multiplying and dividing.

Divide both sides by 2: $x+3 = 7$. Then subtract 3: $x = 4$. (Or expand first: $2x+6=14$, $2x=8$, $x=4$.)

Sam is $x$ and his sister is $x+3$. Their sum: $x + (x+3) = 2x + 3 = 25$. So $2x = 22$ and $x = 11$. (His sister is then 14; check $11 + 14 = 25$.)

Perimeter $=2(w+2w)=2(3w)=6w$. So $6w=30$, giving $w=5$ cm. (The length is then $2\times 5 = 10$ cm.) A rectangle has two of each side.

Let the smallest be $n$; the three even numbers are $n,\,n+2,\,n+4$, summing to $3n+6=48$. So $3n=42$ and $n=14$, giving 14, 16, 18. The largest is 18.

Adult $=x$, senior $=x-5$. Total: $3x+2(x-5)=55$. Expand: $3x+2x-10=55$, so $5x=65$ and $x=13$. The adult price is $13 (senior $8). Check: $3\times 13 + 2\times 8 = 39+16 = 55$.