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 first: $3\times(-3)+7$. A positive times a negative is negative, so $3\times(-3)=-9$, then $-9+7=-2$. Keep the negative sign on $x$.

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.

Multiply the coefficients ($2\times 3=6$) and combine the variables ($a\times b=ab$): $6ab$. Neither variable is squared, since each appears once.

Divide the coefficient: $\dfrac{12}{4}=3$. The variable $a$ comes along unchanged: $3a$.

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

Distribute the 3 to both terms: $3\times 2x - 3\times 4 = 6x-12$. The minus sign carries through.

Distribute the $-2$ to both terms: $(-2)\times x + (-2)\times 3 = -2x-6$. The negative multiplies everything inside.

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.

Substitute $n=7$ into the rule: $3\times 7 + 5 = 21 + 5 = 26$.

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

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

Add 4 to both sides: $3x = 14+4 = 18$. Then divide both sides by 3: $x = \dfrac{18}{3} = 6$.

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 $x+3$. The sum is $x+(x+3)=25$, which simplifies to $2x+3=25$.

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 integers be $n,\,n+1,\,n+2$. Their sum is $3n+3=78$, so $3n=75$ and $n=25$. The integers are 25, 26, 27, so the largest is 27.

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: $3\times 13 + 2\times 8 = 39+16 = 55$.