Algebra · Junior Cert

Algebra Basics

Junior Cert Higher · Letters, evaluating & graphs · Tap NEXT to begin

Section 1 of 6

What Algebra Is

Algebra just means using letters instead of numbers. The letter stands for a value that can change.

You already meet it everywhere: $V=\pi r^2 h$ for the volume of a cylinder, time measured in minutes and seconds, money, and scores in sport.

Section 2 of 6

Evaluating

To evaluate an expression, put the given number in place of the letter and work it out.

Worked example — when $x = 3$

Find the value of $5x$.

$5x = 5(3)$

$= 15$

Worked example — $2x+1$

Find the value of $2x+1$ when $x=3$.

$2(3)+1$

$= 7$

Worked example — $x^2+3x+1$

Find the value of $x^2+3x+1$ when $x=3$.

$3^2 + 3(3) + 1$

$= 9+9+1 = 19$

You try

Find the value of $x^2-2$ when $x=3$.

Pen and paper out — try it before you reveal.

$3^2 - 2$

$= 9-2 = 7$

Worked example — the same letter, different values

Find $2x$ when $x=1$ and when $x=1.5$.

$x=1:\ \ 2x = 2(1) = 2$

$x=1.5:\ \ 2x = 2(1.5) = 3$

Worked example — when $x = 5$

Find the value of $7x$.

$7(5)$

$= 35$

Worked example — $2x+3$

Find $2x+3$ when $x=5$.

$2(5)+3$

$= 13$

You try

Find the value of $x^2-2x+7$ when $x=5$.

Pen and paper out — try it before you reveal.

$5^2 - 2(5) + 7$

$25 - 10 + 7$

$= 22$

Section 3 of 6

Tables & Graphs

A rule like $y=2x+1$ links two letters. Choose values of $x$, work out $y$, then plot the pairs $(x,y)$.

Worked example — $y = 2x+1$

Find $y$ when $x = 1,\,2,\,3,\,4$.

$x=1:\ y = 2(1)+1 = 3$

$x=2:\ y = 2(2)+1 = 5$

$x=3:\ y = 2(3)+1 = 7$

$x=4:\ y = 2(4)+1 = 9$

Points: $(1,3)\ (2,5)\ (3,7)\ (4,9)$

Section 4 of 6

The Words We Use

An expression like $3x^2+7x+9$ is also called a polynomial or a function.

In $3x^2$: the 3 is the coefficient, the $x$ is the variable (the number that changes), and the 2 is the power (also called the index or exponent).

$3x^2+7x+9$ has 3 terms. The $9$ on its own is the constant.

The same words apply to $2x^3+5x^2+7x+9$: coefficient $2$, variable $x$, power $3$ — and it has 4 terms with constant $9$.

Section 5 of 6

Adding & Subtracting

You can only add or subtract like terms — terms with exactly the same letter and power. Remember $x^2 = x\cdot x$.

Worked example

Simplify $2x+3+5x+6$.

$7x+9$

Worked example

Simplify $5x+2+7x+8$.

$12x+10$

Worked example

Simplify $x^2+3x+2x+6$.

$x^2+5x+6$

Worked example

Simplify $2x^2+5x+6+x^2+7x+3$.

$3x^2+12x+9$

Worked example

Simplify $2x+3+5x+7$.

$7x+10$

You try

Simplify $5a+7b+3a+22b$.

Pen and paper out — try it before you reveal.

Add the $a$’s: $5a+3a = 8a$

Add the $b$’s: $7b+22b = 29b$

$8a+29b$

You try

Simplify $x^2+7x+3x+10$.

Pen and paper out — try it before you reveal.

Add the $x$ terms: $7x+3x = 10x$

$x^2+10x+10$

Section 6 of 6

Real-Life Graphs

In a real problem the independent quantity (like time) is $x$, and the dependent quantity (like money) is $y$.

Worked example — a plumber’s bill

A plumber charges a €20 call-out fee and €10 per hour. Show the fees in a table and a graph, and form an equation.

Time $0\to$ €$20$ (the start)

$1\to 10(1)+20 = 30$

$2\to 10(2)+20 = 40$

$3\to 10(3)+20 = 50$

$4\to 10(4)+20 = 60$

$y = 10x+20$

You try

I have €10 and save €5 every day. Show the savings in a table and form an equation.

Pen and paper out — try it before you reveal.

$0\to 10$ (the start)

$1\to 5(1)+10 = 15$

$2\to 5(2)+10 = 20$

$3\to 5(3)+10 = 25$

$y = 5x+10$

That’s the basics.

Evaluating, the words we use, like terms and real-life graphs — the ground floor of algebra.