Algebra · Second Year
Algebra
Second Year · Letters, evaluating, like terms & brackets · Tap NEXT to begin
Section 1 of 5
Using Letters
Algebra uses a letter (like $x$) to stand for a number we don’t know yet. Turn words into expressions.
Worked example — write it in maths
For a number $x$: (i) add $3$; (ii) subtract $50$; (iii) $8$ times the number.
(i) $x + 3$
(ii) $x - 50$
(iii) $8 \times x = 8x$
$x+3$, $x-50$, $8x$
You try
Write in maths: (i) a number $x$ increased by $7$; (ii) double a number; (iii) $5$ less than a number.
“Increased by” = add, “double” = times 2.
(i) $x + 7$
(ii) $2x$
(iii) $x - 5$
$x+7,\ 2x,\ x-5$
Section 2 of 5
Evaluating Expressions
To evaluate, put the given numbers in place of the letters. Watch the signs.
Worked example — substitute the values
With $x = 3$ and $y = 2$, find (i) $5x$; (ii) $x+y$; (iii) $5x+3y$; (iv) $7x-2y$.
(i) $5(3) = 15$
(ii) $3 + 2 = 5$
(iii) $5(3) + 3(2) = 15 + 6 = 21$
(iv) $7(3) - 2(2) = 21 - 4$
$15,\ 5,\ 21,\ 17$
Worked example — with a negative
With $p = -3$ and $q = 5$, find (i) $2p + 6q$; (ii) $5p + q^2$.
(i) $2(-3) + 6(5) = -6 + 30 = 24$
(ii) $5(-3) + 5^2 = -15 + 25 = 10$
$24$ and $10$
You try
With $x = 3$ and $y = 2$, find $x^2 + y^2$.
Square each, then add.
$3^2 + 2^2 = 9 + 4$
$13$
You try
With $p = -3$ and $q = 5$, find $q^2 - 2p$.
$q^2 = 25$; $-2p = -2(-3) = +6$.
$25 - (-6) = 25 + 6$
$31$
Section 3 of 5
Collecting Like Terms
Like terms
Only add or subtract terms with the same letter and power: $3p$ and $2p$ are like terms; $5p^2$ and $3p$ are not.
Worked example — simplify
Simplify $3x + 5 + 6x + 2$.
$3x + 6x = 9x$; $5 + 2 = 7$
$9x + 7$
Worked example — two letters
Simplify $5g + 6p + 2g + 7p$.
$5g + 2g = 7g$; $6p + 7p = 13p$
$7g + 13p$
Worked example — with a power
Simplify $5p^2 + 3p + 2p + 8$.
$3p + 2p = 5p$
$5p^2 + 5p + 8$
Worked example — with two powers
Simplify $6x^2 - 3x + 2x^2 - 7x$.
$6x^2 + 2x^2 = 8x^2$; $-3x - 7x = -10x$
$8x^2 - 10x$
You try
Simplify $5y + 3 + 7y - 9$.
Collect the $y$’s, then the numbers.
$5y + 7y = 12y$; $3 - 9 = -6$
$12y - 6$
You try
Simplify $3a + 2b + 5a + 7b$.
$a$’s together, $b$’s together.
$3a + 5a = 8a$; $2b + 7b = 9b$
$8a + 9b$
You try
Simplify $3x^2 - 5x - 6x - 7$.
$-5x - 6x = -11x$.
$3x^2 - 11x - 7$
$3x^2 - 11x - 7$
Section 4 of 5
Multiplying Terms
Multiply the numbers, then the letters. Remember $x \times x = x^2$.
Worked example — number times term
Multiply (i) $7(6e)$; (ii) $5(4p)$.
(i) $7 \times 6 = 42 \Rightarrow 42e$
(ii) $5 \times 4 = 20 \Rightarrow 20p$
$42e$ and $20p$
Worked example — term times term
Multiply (i) $5x(3x)$; (ii) $3y(7y)$.
(i) $5 \times 3 = 15$, $x \times x = x^2 \Rightarrow 15x^2$
(ii) $21y^2$
$15x^2$ and $21y^2$
You try
Multiply $4a(6a)$.
$4 \times 6 = 24$, $a \times a = a^2$.
$24a^2$
$24a^2$
Section 5 of 5
Expanding Brackets
Multiply everything inside
$a(b + c) = ab + ac$. Multiply the outside term by each term inside.
Worked example — one bracket
Expand (i) $5(2a + 6b)$; (ii) $6(3x + 9)$.
(i) $10a + 30b$
(ii) $18x + 54$
$10a+30b$; $18x+54$
Worked example — a term outside
Expand (i) $2x(3x + 5)$; (ii) $6p(7p + 3)$.
(i) $6x^2 + 10x$
(ii) $42p^2 + 18p$
$6x^2+10x$; $42p^2+18p$
Worked example — two brackets, then simplify
Expand and simplify $3(2a + 5b) + 7(5a + 3b)$.
$6a + 15b + 35a + 21b$
$6a + 35a = 41a$; $15b + 21b = 36b$
$41a + 36b$
You try
Expand $4(3x + 2)$.
$4 \times 3x$ and $4 \times 2$.
$12x + 8$
$12x + 8$
You try
Expand $3x(6x + 7) + 2(6x + 7)$.
Expand each, then collect like terms.
$18x^2 + 21x + 12x + 14$
$21x + 12x = 33x$
$18x^2 + 33x + 14$
You try
Expand and simplify $3p(6p - 5) - 4(6p - 5)$.
Mind the minus on the second bracket.
$18p^2 - 15p - 24p + 20$
$-15p - 24p = -39p$
$18p^2 - 39p + 20$
That’s Algebra.
Words to letters, evaluating, collecting like terms, multiplying terms and expanding brackets — the start of algebra.