An Algebraic Equation is a mathematical statement that shows two algebraic expressions are equal. It contains an equals sign (=) and may include one or more variables whose values are unknown.
The main objective of solving an algebraic equation is to find the value(s) of the unknown variable that make both sides of the equation equal.
Algebraic equations are widely used in mathematics, science, engineering, economics, finance, business, computer programming, and many real-life situations involving unknown quantities.
A Variable is a letter or symbol that represents an unknown value in an equation.
(i) In 3x + 5 = 20, the variable is x.
(ii) In 4y − 7 = 9, the variable is y.
(iii) In 2a + 8 = 18, the variable is a.
A Constant is a fixed numerical value that does not change.
(i) In 5x + 9 = 24, the constants are 9 and 24.
(ii) In 7y − 6 = 15, the constants are −6 and 15.
(iii) In 4a + 12 = 32, the constants are 12 and 32.
A Coefficient is the numerical value that multiplies a variable.
If no number appears before the variable, the coefficient is understood to be 1.
(i) In 8x = 40, the coefficient of x is 8.
(ii) In −3y + 5 = 11, the coefficient of y is −3.
(iii) In x + 6 = 12, the coefficient of x is 1.
Every algebraic equation consists of two sides separated by the equals sign (=).
The expression on the left of the equals sign is called the Left-Hand Side (LHS), while the expression on the right is called the Right-Hand Side (RHS).
Consider the equation:
4x + 7 = 23
Left-Hand Side (LHS): 4x + 7
Right-Hand Side (RHS): 23
A Solution (also called a Root) of an algebraic equation is the value of the variable that makes the equation true.
Consider the equation:
x + 6 = 14
The solution is:
x = 8
Verification:
8 + 6 = 14 ✔
Since both sides are equal, x = 8 is the correct solution.
A Linear Equation is an equation in which the highest power of the variable is 1.
(i) 3x + 5 = 20
(ii) 7y − 9 = 12
(iii) 4a + 8 = 32
A Quadratic Equation is an equation in which the highest power of the variable is 2.
(i) x² + 5x + 6 = 0
(ii) 2y² − 7y + 3 = 0
(iii) 4a² + a − 5 = 0
A Cubic Equation is an equation in which the highest power of the variable is 3.
(i) x³ + 2x² − x − 6 = 0
(ii) y³ − 4y + 1 = 0
(iii) 2a³ + 5a² − 8 = 0
Simultaneous Equations are two or more equations containing the same variables and having a common solution.
(i) 2x + y = 8
x − y = 1
(ii) 3a + b = 12
2a − b = 3
To solve an algebraic equation, we perform the same mathematical operation on both sides of the equation until the variable is isolated.
This ensures that both sides of the equation remain equal throughout the solution.
Solve x + 9 = 18.
Subtract 9 from both sides:
x + 9 − 9 = 18 − 9
x = 9
The four basic operations commonly used when solving algebraic equations are:
Solve x + 8 = 19.
Step 1: Identify the operation attached to the variable.
The variable x has 8 added to it.
Step 2: Remove the added number by subtracting 8 from both sides.
x + 8 − 8 = 19 − 8
Step 3: Simplify both sides.
x = 11
Final Answer: x = 11
Solve x − 15 = 24.
Step 1: Identify the operation attached to the variable.
The variable x has 15 subtracted from it.
Step 2: Remove the subtraction by adding 15 to both sides.
x − 15 + 15 = 24 + 15
Step 3: Simplify both sides.
x = 39
Final Answer: x = 39
Solve 7x = 63.
Step 1: Identify the coefficient of the variable.
The coefficient of x is 7.
Step 2: Divide both sides by 7.
7x ÷ 7 = 63 ÷ 7
Step 3: Simplify both sides.
x = 9
Final Answer: x = 9
Solve 4x + 9 = 37.
Step 1: Remove the constant term.
Subtract 9 from both sides.
4x + 9 − 9 = 37 − 9
4x = 28
Step 2: Remove the coefficient of x.
Divide both sides by 4.
4x ÷ 4 = 28 ÷ 4
Step 3: Simplify both sides.
x = 7
Final Answer: x = 7
A taxi company charges a fixed booking fee plus $6 for every kilometre travelled. If the total fare for a trip is $54 and the booking fee is $12, determine the number of kilometres travelled.
Step 1: Form the algebraic equation.
Let the number of kilometres travelled be x.
The fare equation is:
6x + 12 = 54
Step 2: Remove the constant term.
Subtract 12 from both sides.
6x + 12 − 12 = 54 − 12
6x = 42
Step 3: Remove the coefficient of x.
Divide both sides by 6.
6x ÷ 6 = 42 ÷ 6
x = 7
Step 4: State the answer.
The taxi travelled 7 kilometres.
Final Answer: 7 km
Attempt the following Questions:
(i) Solve the equation 7x − 9 = 40.
(ii) A stationery shop sold 6 exercise books to each student. After selling an additional 8 exercise books, the total number of exercise books sold was 56. If the number of students is represented by x, form an algebraic equation and hence find the value of x.
(iii) Solve the equation 5(2x − 3) = 35.
(iv) The perimeter of a rectangular garden is 54 m. Its length is (2x + 5) metres and its width is (x + 2) metres. Using the perimeter formula P = 2(L + W), form an equation and find the value of x.
(v) Two brothers share $156 such that the older brother receives $24 more than the younger brother. Let the younger brother's share be x. Form an equation and determine how much each brother receives.
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