ARTICLE CONTENTSOn this page 9 sections
Simple Equations
Performance Objectives
By the end of the lesson, students should be able to:
- translate word expressions or statements into mathematical equations;
- use mathematical equations to represent worded sentences or statements;
- solve simple equations and verify the solutions.
Introduction to Algebraic Symbols and Notations
Algebra is a branch of mathematics that uses letters and symbols to represent numbers in equations and expressions. In algebra, letters are used to stand for unknown values that can be calculated.
A variable is a letter or symbol used to represent an unknown number.
Common variables used in algebra include $x$, $y$, $z$, $a$, $b$ and $m$. For example, in the expression $x + 3$, the letter $x$ is the variable.
A constant is a fixed number that does not change its value.
In the expression $x + 3$, the number $3$ is a constant because its value is always three.
An equation is a mathematical statement showing that two expressions are equal.
An equation always contains an equals sign ($=$). The expression on the left side of the equals sign is called the Left-Hand Side (LHS), and the expression on the right side is called the Right-Hand Side (RHS). For example, $x + 3 = 10$ is an equation.
Translating Word Statements into Mathematical Equations
To translate word statements into mathematical equations, key words must be identified and replaced with mathematical symbols. Certain words always point to specific arithmetic operations.
The following table shows common words and their corresponding mathematical operations:
| Word or Phrase | Mathematical Operation | Example Phrase | Algebraic Expression |
|---|---|---|---|
| Sum of, added to, increased by, more than | Addition ($+$) | $5$ more than $x$ | $x + 5$ |
| Difference, subtracted from, decreased by, less than | Subtraction ($-$) | $3$ subtracted from $y$ | $y – 3$ |
| Product of, times, multiplied by, twice | Multiplication ($\times$) | Twice a number $p$ | $2p$ |
| Quotient, divided by, shared among | Division ($\div$ or fraction) | $k$ divided by $4$ | $\frac{k}{4}$ |
| Is, equals, results in, gives, is the same as | Equality ($=$) | $x$ plus $2$ is $7$ | $x + 2 = 7$ |
Observe the diagram below to see how a word sentence is mapped directly to algebraic symbols.
Worked Examples: Translating Words to Equations
The following examples demonstrate how to translate word statements into mathematical equations:
Example 1: Translate the statement “The sum of a certain number and $8$ is $15$” into an equation.
Let the unknown number be represented by the variable $y$.
The term “sum of $y$ and $8$” is written as $y + 8$.
The term “is $15$” is written as $= 15$.
Therefore, the equation is:
$$y + 8 = 15$$
Example 2: Translate the statement “When $3$ is subtracted from twice a number, the result is $11$” into an equation.
Let the unknown number be represented by $n$.
The term “twice a number” is written as $2n$.
The term “$3$ is subtracted from twice a number” is written as $2n – 3$.
The term “the result is $11$” is written as $= 11$.
Therefore, the equation is:
$$2n – 3 = 11$$
Example 3: Translate the statement “A teacher shares $24$ pencils equally among $x$ students, and each student gets $6$ pencils” into an equation.
The term “$24$ pencils shared equally among $x$ students” is written as $\frac{24}{x}$.
The term “each gets $6$” is written as $= 6$.
Therefore, the equation is:
$$\frac{24}{x} = 6$$
Translating Mathematical Equations into Word Sentences
Translating mathematical equations back into word sentences requires interpreting the symbols and operations in reverse. A single equation can often be written in several different ways in words.
Worked Examples: Translating Equations to Words
The following examples demonstrate how to translate mathematical equations into word sentences:
Example 1: Translate the equation $x – 7 = 10$ into a word sentence.
The equation can be written in words as:
“When $7$ is subtracted from a certain number, the result is $10$.”
Alternatively, it can be written as:
“The difference between a number and $7$ is $10$.”
Example 2: Translate the equation $3y + 4 = 19$ into a word sentence.
The equation can be written in words as:
“Three times a number increased by $4$ is equal to $19$.”
Alternatively, it can be written as:
“The sum of thrice a number and $4$ is $19$.”
Example 3: Translate the equation $\frac{m}{5} = 6$ into a word sentence.
The equation can be written in words as:
“A certain number divided by $5$ gives $6$.”
Alternatively, it can be written as:
“The quotient of a number and $5$ is $6$.”
Solving Simple Equations
A simple equation is an algebraic equation in which the highest power of the variable is one.
To solve a simple equation means to find the numerical value of the variable that makes the equation true. This value is called the solution or root of the equation.
The fundamental rule for solving equations is the balance method. An equation is like a balance scale. To keep the scale balanced, whatever operation is performed on one side of the equation must also be performed on the other side.
Look at the balance scale diagram below to understand this concept visually.
The four basic operations used to solve simple equations are:
- Subtracting the same number from both sides.
- Adding the same number to both sides.
- Dividing both sides by the same non-zero number.
- Multiplying both sides by the same non-zero number.
Type 1: Equations Involving Addition ($x + a = b$)
To solve equations where a number is added to the variable, subtract that same number from both sides of the equation.
Example 1: Solve $x + 6 = 15$.
To isolate $x$, subtract $6$ from both sides of the equation:
$$
\begin{aligned}
x + 6 – 6 &= 15 – 6 \\
x &= 9
\end{aligned}
$$
Example 2: Solve $y + 12 = 20$.
To isolate $y$, subtract $12$ from both sides of the equation:
$$
\begin{aligned}
y + 12 – 12 &= 20 – 12 \\
y &= 8
\end{aligned}
$$
Type 2: Equations Involving Subtraction ($x – a = b$)
To solve equations where a number is subtracted from the variable, add that same number to both sides of the equation.
Example 1: Solve $m – 8 = 4$.
To isolate $m$, add $8$ to both sides of the equation:
$$
\begin{aligned}
m – 8 + 8 &= 4 + 8 \\
m &= 12
\end{aligned}
$$
Example 2: Solve $p – 15 = 3$.
To isolate $p$, add $15$ to both sides of the equation:
$$
\begin{aligned}
p – 15 + 15 &= 3 + 15 \\
p &= 18
\end{aligned}
$$
Type 3: Equations Involving Multiplication ($ax = b$)
To solve equations where the variable is multiplied by a number, divide both sides of the equation by that same number.
Example 1: Solve $4x = 24$.
To isolate $x$, divide both sides of the equation by $4$:
$$
\begin{aligned}
\frac{4x}{4} &= \frac{24}{4} \\
x &= 6
\end{aligned}
$$
Example 2: Solve $7k = 35$.
To isolate $k$, divide both sides of the equation by $7$:
$$
\begin{aligned}
\frac{7k}{7} &= \frac{35}{7} \\
k &= 5
\end{aligned}
$$
Type 4: Equations Involving Division ($\frac{x}{a} = b$)
To solve equations where the variable is divided by a number, multiply both sides of the equation by that same number.
Example 1: Solve $\frac{a}{3} = 8$.
To isolate $a$, multiply both sides of the equation by $3$:
$$
\begin{aligned}
\frac{a}{3} \times 3 &= 8 \times 3 \\
a &= 24
\end{aligned}
$$
Example 2: Solve $\frac{w}{5} = 12$.
To isolate $w$, multiply both sides of the equation by $5$:
$$
\begin{aligned}
\frac{w}{5} \times 5 &= 12 \times 5 \\
w &= 60
\end{aligned}
$$
Verifying Solutions
Verifying a solution means checking if the calculated value of the variable is correct. This is done by substituting the calculated value back into the original equation. If the Left-Hand Side (LHS) equals the Right-Hand Side (RHS), the solution is correct.
Worked Examples: Verifying Solutions
The following examples demonstrate how to verify solutions to simple equations:
Example 1: Solve $x + 9 = 17$ and verify the solution.
First, solve the equation by subtracting $9$ from both sides:
$$
\begin{aligned}
x + 9 – 9 &= 17 – 9 \\
x &= 8
\end{aligned}
$$
To verify, substitute $x = 8$ into the Left-Hand Side (LHS) of the original equation:
$$
\begin{aligned}
\text{LHS} &= x + 9 \\
&= 8 + 9 \\
&= 17
\end{aligned}
$$
The Right-Hand Side (RHS) is $17$. Since $\text{LHS} = \text{RHS}$, the solution $x = 8$ is verified as correct.
Example 2: Solve $5y = 30$ and verify the solution.
First, solve the equation by dividing both sides by $5$:
$$
\begin{aligned}
\frac{5y}{5} &= \frac{30}{5} \\
y &= 6
\end{aligned}
$$
To verify, substitute $y = 6$ into the Left-Hand Side (LHS) of the original equation:
$$
\begin{aligned}
\text{LHS} &= 5y \\
&= 5 \times 6 \\
&= 30
\end{aligned}
$$
The Right-Hand Side (RHS) is $30$. Since $\text{LHS} = \text{RHS}$, the solution $y = 6$ is verified as correct.
Example 3: Solve $\frac{m}{4} – 2 = 3$ and verify the solution.
First, solve the equation by adding $2$ to both sides:
$$
\begin{aligned}
\frac{m}{4} – 2 + 2 &= 3 + 2 \\
\frac{m}{4} &= 5
\end{aligned}
$$
Next, multiply both sides by $4$:
$$
\begin{aligned}
\frac{m}{4} \times 4 &= 5 \times 4 \\
m &= 20
\end{aligned}
$$
To verify, substitute $m = 20$ into the Left-Hand Side (LHS) of the original equation:
$$
\begin{aligned}
\text{LHS} &= \frac{m}{4} – 2 \\
&= \frac{20}{4} – 2 \\
&= 5 – 2 \\
&= 3
\end{aligned}
$$
The Right-Hand Side (RHS) is $3$. Since $\text{LHS} = \text{RHS}$, the solution $m = 20$ is verified as correct.
Summary
The key points covered in this unit are:
- Variables are letters representing unknown numbers, while constants are fixed numbers.
- An equation is a mathematical statement showing that two expressions are equal, separated by an equals sign ($=$).
- Word statements can be translated into algebraic equations by identifying key operational words.
- Simple equations are solved using the balance method, where the same operation is performed on both sides of the equation to isolate the variable.
- Solutions are verified by substituting the value of the variable back into the original equation to check if the Left-Hand Side equals the Right-Hand Side.
Evaluation Exercises
The following questions are designed to test the understanding of translating and solving simple equations:
Section A: Translate Word Expressions into Mathematical Equations
Translate each of the following statements into a mathematical equation using a suitable variable:
- The sum of a certain number and $12$ is $25$.
- When $9$ is subtracted from a number, the result is $14$.
- Four times a certain number gives $32$.
- A number divided by $6$ is equal to $7$.
- When $5$ is added to twice a number, the result is $19$.
Section B: Translate Mathematical Equations into Word Sentences
Translate each of the following equations into a word sentence:
- $x + 7 = 15$
- $y – 10 = 3$
- $5m = 40$
- $\frac{a}{8} = 4$
- $2k – 3 = 11$
Section C: Solve and Verify Simple Equations
Solve each of the following equations and verify the solutions:
- $x + 8 = 20$
- $y – 11 = 5$
- $6m = 42$
- $\frac{p}{3} = 9$
- $2a + 5 = 17$
- $3x – 4 = 14$
- $\frac{y}{5} + 2 = 8$
- $10 + k = 25$
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