Grade 9: Algebraic Equations
April 2013
Grade 9
Algebraic Equations
Goals:
□ Set up equations to describe problem situations
□ Analyse and interpret equations that describe a particular situation
□ Solve equations by
o Inspection
o Operation Inverses
o Laws of Exponents
□ Determine the numerical value of an expression or equation by substitution
Terminology




Equation
Variable
Additive Inverse
Multiplicative Inverse
Note: We’ll return to algebraic equations in term 3 to discuss solving equations which
involve tables, factorisation, and products of factors.
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Grade 9: Algebraic Equations
Revision
Key concepts which will be addressed in exercise 1 include algebraic expressions,
substitution, the distributive law, and combining like terms. Exercise 1 will end with solving
simple algebraic equations.
Exercise 1
1. Write the following expressions using variables:
1.1. 4 times a number 𝑥.
1.2. 7 is subtracted from a number 𝑡.
1.3. A number 𝑤 is split into 4 equal groups. How much is in each group?
1.4. A number 𝑞 times itself.
1.5. Juan had 𝑐 bananas, but then Amy came and stole 𝑏 of them. How many does he
have left?
1.6. Nacho thought he had 𝑥 only minutes of homework, but then he realized that he
also has 𝑧 minutes of science homework he forgot to include. How much
homework does he have?
1.7. If Jessica is 𝑦 years old today, how old will she be in 3 years?
1.8. If Phila has 𝑎 marbles. If she loses half of them, how many will she have?
1.9. 6 friends have 𝑛 pencils each. How many pencils do they have all together?
1.10. The square root of a number 𝑞.
2. Inverse Operations:
2.1. What is the inverse of −2?
2.2. What is the inverse of × 17?
2.3. What is the inverse of ÷ 100?
2.4. What is the inverse of +3,5?
3. Substitution:
3.1. If 𝑦 = 2𝑥 + 1, calculate 𝑦 when 𝑥 = 2.
3.2. If 𝑦 = 𝑥 2 , calculate 𝑦 when 𝑥 = 5.
3.3. If 𝑦 = 2𝑥 − 10, calculate 𝑦 when 𝑥 = 3.
3.4. If 𝑦 = 𝑥 + 1, calculate 𝑦 when 𝑥 = 2.
4. Distribution: Simplify each of the following expressions using the distributive property.
Remember, the distributive property is 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐.
Example: −2(𝑥 + 1) = (−2)(𝑥) + (−2)(1) = −2𝑥 − 2
4.1.
4.2.
4.3.
2(𝑥 + 2)
3(𝑥 − 2)
2(𝑚 + 𝑛)
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Grade 9: Algebraic Equations
4.4.
4.5.
4.6.
4.7.
4.8.
5(𝑥 + 5)
𝑥(𝑥 + 1)
𝑦(𝑥 − 2)
−1(𝑥 + 3)
−2(𝑥 − 2
5. Combine the following like terms:
5.1. 3𝑥 + 4𝑥
5.2. 3𝑧 − 4𝑧
5.3. 3 + 2𝑎 + 4𝑎
5.4. 4𝑥 2 + 9𝑥 2
5.5. 5𝑐 − 9𝑐 + 4
5.6. 3𝑏 3 − 𝑏 3
5.7. 𝑦 + 𝑦 + 𝑦
5.8. 4𝑎 − 5𝑎 + 2
6. Solve the following equations for 𝑥 by inspection:
6.1.
6.2.
6.3.
6.4.
6.5.
6.6.
𝑥 = 1+8
𝑥+2=4
7=𝑥−2
𝑥 ÷ 2 = 10
2𝑥 = 8
𝑥 + 3 = −2
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Grade 9: Algebraic Equations
Methods for Solving Equations
All of the questions in the following exercise are the same: “Solve for __.” In other words,
find the variable value which makes the statement true (you can and should check your
work by substitution). The following are techniques you will need to use to solve equations:
Inverse Operations. To separate 𝑥, use the inverse operation of whatever number is “in the
way” of 𝑥 being by itself
e.g.
𝑥−3=6
𝑥 = 6+3
(Add 3 to both sides)
𝑥=9
Combine Like Terms. Description
𝑥 = −3𝑥 + 12
e.g.
𝑥 + 3𝑥 = 12
4𝑥 = 12
𝑥 = 12/4
(Add 3𝑥 to both sides)
(Combine like terms)
(Divide both sides by 4)
𝑥=3
Distributive Law.
e.g.
−2(𝑥 + 2) = 𝑥 + 17
−2𝑥 − 4 = 𝑥 + 17
−2𝑥 − 𝑥 − 4 = 17
(Distributive Law)
(Subtract 𝑥 from both sides)
−2𝑥 − 𝑥 = 17 + 4
(Add 4 to both sides)
−3𝑥 = 21
(Combine like terms)
𝑥 = 21/(−3)
(Divide both sides by −3)
𝑥=3
BODMAS. For more complicated equations, work backwards carefully and use BODMAS.
e.g.
𝑥+3
−6 ( 4 ) = −36
𝑥+3
=6
4
𝑥+3
4 ( 4 ) = 4(6)
𝑥 + 3 = 24
𝑥 + 3 − 3 = 24 − 3
(Divide both sides by −6)
(Multiply both sides by 4)
(Simplify)
(Subtract 3 from both sides)
𝑥 = 21
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Grade 9: Algebraic Equations
Exercise 2
1. Inverse Operations: Solve for 𝑎. Show working!
1.1. 𝑎 + 3 = 10
1.2. 2𝑎 = 26
1.3. 4𝑎 = −16
1.4. −20 = −𝑎
1.5. 16 = 6 + 2𝑎
𝑎
1.6. 10 = 3
1.7.
1.8.
1
4
𝑎
=8
20 = −10 − 10𝑎
2. Combine Like Terms: Solve for 𝑏. Show working!
2.1. 𝑏 + 𝑏 = 6
2.2. 3𝑏 − 2𝑏 = 10
2.3. 6 + 𝑏 = −𝑏
2.4. 2𝑏 = 3𝑏 + 7
2.5. 17𝑏 = 10𝑏 − 21
2.6. −2𝑏 + 29 = 4𝑏 + 5
3. Distributive Law: Solve for 𝑐. Show working!
3.1. 2(𝑐 + 1) = 4
3.2. 3(𝑐 − 2) = 0
3.3. 2𝑐 − (𝑐 + 1) = 8
3.4. 2(1 − 𝑐) = 22
3.5. 4 = 3(𝑐 + 1) + 1
3.6. 7𝑐 = 5(2𝑐 + 3)
4. BODMAS: Solve for 𝑥. Show working!
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.
4.7.
4.8.
𝑥+2
3
𝑥−7
2
=1
=2
2(2𝑥 − 3) = 𝑥 + 9
𝑥
+ 2 = −1
3
𝑥+2
3
= −1
2𝑥 + 2 = 0
2(𝑥 + 2) = 0
𝑥
=𝑥+2
3
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Grade 9: Algebraic Equations
Revision and Problem Solving
Exercise 3
1. Solve the following equations for 𝑥:
1.1. 𝑥 2 = 4
1.2. 𝑥 2 = 42
1.3. 𝑥 2 = 10
1.4. −2 = 𝑥 2
2. Nathaniel has 17 coins. He wants to figure out how many more coins he needs in order
to have 50 coins.
2.1. Write an equation to express the above situation, using 𝑐 to represent the
number of coins Nathaniel needs.
2.2. Solve the equation. How many coins does Nathaniel need?
3. Samantha owes her parents R100. She wants buy a CD which costs R90, but first she will
have to pay back her parents.
3.1. Write an equation to express the above situation, using 𝑚 to represent the
amount of money Samantha needs to earn.
3.2. Solve the equation. How much money does Samantha need to earn in order to
buy her CD?
4. Musa earns R25 per hour for working in his parents’ garden. He wants to save R150 to
pay for a trip to uShaka.
4.1. Write an equation to express the above situation, using ℎ to represent the number
of hours Musa works.
4.2. Solve the above equation. How many hours does Musa need to work before he can
go to uShaka?
5. Anna is 6 years older than Thandi. In 3 years, Anna will be twice as old as Anna. Write
two equations to represent the above statements, then find out how old Anna and
Thandi are right now.
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