Simple Equations and Word Problems

Introduction to Simple Equations

An equation is a mathematical statement that shows that two expressions are equal. Every equation has an "equal to" sign (=) separating the left-hand side (LHS) from the right-hand side (RHS). A simple equation (also called a linear equation in one variable) is an equation that contains only one unknown value, usually represented by a letter such as x, y, n or a, and the highest power of the unknown is 1. For example, in the equation x + 5 = 12, the letter x stands for a number we do not yet know, and our task is to find that number. Learning to solve simple equations is one of the most important skills in Junior Secondary School Mathematics because it forms the foundation for algebra, geometry, and everyday problem solving involving money, distance, age, and quantities.

Basic Terms Used in Equations

Before we solve equations, we must understand a few key terms. A variable (or unknown) is a letter that represents a number we are trying to find, such as x or y. A constant is a fixed number that does not change, such as 5, 12, or -3. A coefficient is the number multiplied by the variable; in the term 4x, the coefficient is 4. A term is a single number, variable, or the product of numbers and variables, such as 3x, 7, or -2y. An expression is a group of terms connected by plus or minus signs, for example 3x + 7. When an expression is set equal to another expression using the equal sign, it becomes an equation, such as 3x + 7 = 22.

The Golden Rule of Equations

The most important rule when solving equations is: whatever operation you perform on one side of the equation, you must perform the same operation on the other side. This keeps the equation balanced, just like a see-saw or a weighing balance. If you add a number to the left side, you must add the same number to the right side. If you divide the left side by a number, you must divide the right side by the same number. This rule ensures that the equation remains true throughout the solving process.

Solving Simple Equations by Addition and Subtraction

Consider the equation x + 8 = 20. To find x, we need to remove the 8 from the left-hand side. Since 8 is being added, we subtract 8 from both sides:

x + 8 - 8 = 20 - 8
x = 12

We can check this answer by substituting x = 12 back into the original equation: 12 + 8 = 20, which is true. This process of checking is very important and should never be skipped.

Now consider y - 6 = 15. Since 6 is being subtracted, we add 6 to both sides:

y - 6 + 6 = 15 + 6
y = 21

Check: 21 - 6 = 15. Correct.

Solving Simple Equations by Multiplication and Division

Consider 4x = 24. Here, x is being multiplied by 4. To isolate x, we divide both sides by 4:

4x ÷ 4 = 24 ÷ 4
x = 6

Check: 4 × 6 = 24. Correct.

Now consider n/3 = 9. Here, n is being divided by 3. To isolate n, we multiply both sides by 3:

n/3 × 3 = 9 × 3
n = 27

Check: 27/3 = 9. Correct.

Solving Equations with More Than One Operation

Many equations require more than one step. Consider 3x + 5 = 20. We solve this in two stages. First, remove the constant term by subtracting 5 from both sides:

3x + 5 - 5 = 20 - 5
3x = 15

Next, divide both sides by 3:

3x ÷ 3 = 15 ÷ 3
x = 5

Check: 3(5) + 5 = 15 + 5 = 20. Correct. The general approach is to always deal with addition or subtraction first, and then deal with multiplication or division, working backwards from the order of operations.

Equations with the Variable on Both Sides

Sometimes the unknown appears on both sides of the equation, such as 5x - 4 = 2x + 11. To solve this, we collect all the x-terms on one side and all the numbers on the other side. Subtract 2x from both sides:

5x - 2x - 4 = 2x - 2x + 11
3x - 4 = 11

Add 4 to both sides:

3x = 15

Divide both sides by 3:

x = 5

Equations Involving Brackets

When an equation contains brackets, we first expand (open) the brackets before solving. Consider 2(x + 3) = 16. Expanding the bracket gives:

2x + 6 = 16

Subtract 6 from both sides: 2x = 10. Divide by 2: x = 5.

Translating Word Problems into Equations

One of the most useful applications of simple equations is solving real-life word problems. The key skill is translating the words of the problem into a correct mathematical equation. Certain words give clues about which operation to use:

"Sum", "total", "added to", "more than" usually suggest addition. "Difference", "less than", "reduced by", "subtracted from" usually suggest subtraction. "Product", "times", "twice", "multiplied by" usually suggest multiplication. "Quotient", "divided by", "shared equally among" usually suggest division. "Is", "equals", "gives", "results in" usually represent the equal sign.

Worked Word Problems

Example 1: The sum of a number and 9 is 23. Find the number. Let the number be x. Then x + 9 = 23, so x = 23 - 9 = 14.

Example 2: Musa is x years old. In 5 years, he will be 18 years old. How old is Musa now? We write x + 5 = 18, so x = 13. Musa is 13 years old now.

Example 3: A number is multiplied by 4 and then 7 is added to the result to give 39. Find the number. We write 4x + 7 = 39. Subtracting 7 gives 4x = 32, and dividing by 4 gives x = 8.

Example 4: Three times a certain number, decreased by 6, equals 21. Find the number. We write 3x - 6 = 21. Adding 6 gives 3x = 27, and dividing by 3 gives x = 9.

Example 5: Amaka bought some oranges at ₦20 each and spent a total of ₦140. How many oranges did she buy? Let the number of oranges be n. Then 20n = 140, so n = 140 ÷ 20 = 7 oranges.

Example 6 (Age problem): The present age of a father is 4 times the age of his son. If the son is x years old and the father is 40, find the son's age. We write 4x = 40, so x = 10. The son is 10 years old.

Example 7 (Perimeter problem): The length of a rectangle is twice its width. If the perimeter is 30 cm, find the width. Let the width be w, so the length is 2w. Perimeter = 2(length + width) = 2(2w + w) = 6w. So 6w = 30, giving w = 5 cm.

Common Mistakes to Avoid

Students often make the following mistakes when solving equations: forgetting to perform the same operation on both sides of the equation; making sign errors when moving terms across the equal sign (remember that when a term crosses the equal sign, its sign changes — addition becomes subtraction and multiplication becomes division); failing to expand brackets correctly; and not checking the final answer by substitution. Always take time to verify your solution by putting it back into the original equation.

Why Simple Equations Matter

Simple equations are used in everyday life far beyond the classroom. Traders use them to calculate profit and loss, engineers use them to determine unknown measurements, and even simple household budgeting involves solving for unknown amounts. Mastering simple equations in JSS2 prepares you for more advanced topics such as simultaneous equations, quadratic equations, and inequalities that you will meet in later classes.

Summary

An equation states that two expressions are equal. To solve a simple equation, isolate the variable by performing inverse operations equally on both sides: use subtraction to undo addition, addition to undo subtraction, division to undo multiplication, and multiplication to undo division. Always simplify brackets first, collect like terms, and check your answer by substitution. Word problems can be solved by carefully translating the given information into an equation before solving it step by step.