Algebraic expressions and substitutionIB MYP Maths Extended: Revision notes

In algebra a variable is a letter standing for a number we do not know or that can change. $3a$ means $3\times a$, and $a^2$ means $a\times a$. An expression such as $4x+7$ has no equals sign. A formula such as $C=50+30d$ links quantities, and an equation such as $4x+7=19$ can be solved. In $4x+7$, the terms are $4x$ and $7$; the number in front of a letter is its coefficient (4 here); the number on its own is the constant.

Substitution means replacing each letter by its value. Put negative values in brackets and follow the order of operations: brackets, powers, multiplication and division, then addition and subtraction. For $a=3$ and $b=-2$: $a^2+b=9+(-2)=7$ and $3b^2=3\times(-2)^2=3\times4=12$. For a formula such as $C=50+30d$ with $d=6$: $C=50+30\times6=230$. To find $d$ from a known $C$, work backwards: $50+30d=410$, so $30d=360$ and $d=12$.

Like terms have exactly the same letters and powers, so they can be added or subtracted: $5x+3y-2x+4y=3x+7y$. Terms such as $x$, $x^2$ and $xy$ are unlike and cannot be combined. Keep the sign in front of each term with it: $4a-3b-a+5b=3a+2b$. Also $3x\times2x=6x^2$ and $a\times b=ab$.

To expand a bracket, multiply every term inside by the term outside: $3(2x-5)=6x-15$. A negative outside changes every sign inside: $-2(x-4)=-2x+8$. To expand and simplify, expand each bracket and then collect like terms: $2(x+3)+4(x-1)=2x+6+4x-4=6x+2$.

With two brackets, multiply every term in the first bracket by every term in the second (four products), then collect like terms. $(x+3)(x+5)=x^2+5x+3x+15=x^2+8x+15$. With negatives: $(x-4)(2x+3)=2x^2+3x-8x-12=2x^2-5x-12$. A squared bracket is two brackets: $(x+4)^2=(x+4)(x+4)=x^2+8x+16$. The special case $(x-3)(x+3)=x^2-9$ has no $x$ term.