Algebra 1
Pupil Notes and worked examples
Key Skill 1: Add and subtract like terms
To simplify an expression, we collect like terms. Like terms involve letters and numbers.
Look at the expression:
4x + 5x – 2 – 2x + 7
The x terms go together to give 7x
The numbers go together to give 5
So 4x + 5x – 2 – 2x + 7 simplifies to 7x + 5.
Examples
Simplify the following:
(a)
7t + 6 + 3t + 4t + 5
(b)
= 14t + 11
(c)
x2 + 2y + 3x2 + 17 + 6x2 - y – 20 – 30x2
= - 20x2 + y – 3
9y + 15 – 4y + 4 + y - 3
= 6y + 16
(d)
- a – b – c - 2b - 3c - 4a - 10
= -5a - 3b – 4c -10
Written by John Donnelly
Key Skill 2: Perform basic calculations after substituting letters with numbers
In an algebraic expression letters can stand for numbers. When we substitute a specific value
for each letter, and then perform the operations, it’s called evaluating the expression.
Let’s evaluate the expression 5x + 10y when x = 6 and y = 3.
5x + 10y
= 5(6) + 10(3)
= 30 + 30
= 60
Examples
(a)
If a = 5, b = 7, c = 9 evaluate:
(i)
2a + 3b – c
(ii)
2ab
(iii)
= 2(5) + 3(7) - 9
= 2(5)(7)
= (9)2 – (7)2
= 10 + 21 – 9
= 70
= 81 – 49
= 22
(b)
c2 – b2
= 32
If x = - 3, y = - 5, z = 10 evaluate:
(i)
y2 + x2
(ii)
(z – x)2
(iii)
4y2
= (- 5)2 + (- 3)2
= (10 – (- 3))2
= 4(- 5)2
= 25 + 9
= (10 + 3)2
= 4(25)
= 34
= (13)2
= 100
= 169
Written by John Donnelly
Key Skill 3: Solve a variety of algebraic equations
In an equation letters stand for missing numbers. To solve an equation is to find the missing
value of the letters.
To keep balance we will do the same operation to both sides of the equation, the left hand side
and the right hand side.
To solve the equation 2x + 7 = 19 we need to find an x value which multiplies by 2, 7 is then
added, to give 19.
2x + 7 = 19
-7
-7
(We will take 7 away from both sides – the opposite of the add 7)
2x
2
(We will divide both sides by 2 – the opposite of the times by 2)
= 12
2
x=6
(So the value of 6 when times by 2, then 7 is added, gives 19)
The work in orange must be written always.
The work in black is not required but if it helps you to write it then do so.
Examples
Solve the following equations algebraically:
(a)
4x – 8 = 12
+8 +8
4x
4
(b)
= 20
4
5x + 32 = 22
- 32 -32
5x
5
x=5
(c)
= - 10
5
x=-2
7x – 3 = x + 21
+3
+3
(d)
2x + 9 = 5x + 18
-9
- 9
7x
-x
= x + 24
-x
2x
- 5x
= 5x + 9
- 5x
6x
6
=
-3x
 3
=
x=4
24
6
9
 3
x=-3
Written by John Donnelly
Key Skill 4: Read and interpret a problem in order to construct an algebraic equation
We need to be able to create an equation to represent a written problem, and then solve the equation.
Examples
(a)
I think of a number. I double this number and add ten. This is equal to thirty-two.
(i)
Create an equation to illustrate the above scenario.
Let n be the original number
(ii)
2n + 10 = 32
Solve this equation to find the number I originally thought of.
2n + 10 = 32
- 10 - 10
2n
2
= 22
2
n = 11
(b)
I think of a number. I multiply this number by six and take it away from thirty-nine.
This is equal to fifty-one.
(i)
Create an equation to illustrate the above scenario
Let n be the original number
(ii)
39 – 6n = 51
Solve this equation to find the number I originally thought of.
39 – 6n = 51
- 39
- 39
- 6n = 12
 6  6
n=-2
Written by John Donnelly