Equations enable us to describe complex problems in simple terms. They are built with numbers, pronumerals and an equal sign.It is clear that the number sentence
5 + 10 = 15
is an equation.
If the value of the pronumeral x is 5, then x can take the place of 5; and we can write this equation as
x + 10 = 15
This is an equation containing the pronumeral x. The value of the pronumeral x is 5.
Example 1
Describe each of the following equations in words:
Solution:
Example 2
Write an equation to represent each of the following statements:
a. When I add 7 to a number, the answer is 16.
b. When I subtract 8 from a number, the answer is 23.
c. When I multiply a number by 9, the answer is 27.
d. When I divide a number by 5, the answer is 29.
Solution:
Equations and a Pair of Scales
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Recall that:
An equation is a statement that contains an equal sign.
Consider the simple equation
x = 5
Visualise this equation as a balanced pair of scales with x and 5 measured in kilograms.
If we add 3 kg to the scale on the left-hand side, the scales will balance as long as we add 3 kg to the scale on the right-hand side. That is, x + 3 = 8.
Also, if we subtract the same weight, say 3 kg, from each side of the balance, the scales will remain balanced. That is, x – 3 = 2.
If we double the weight in the scale on the left-hand side, the scales will balance as long as we double the weight in the scale on the right-hand side. That is, 2x = 10.
Also, if we halve the weight in each scale of the balance, the scales will remain balanced.
Solving Equations
Solving an equation means to find the value of a pronumeral that makes a statement true.
In the preceding section, we observed that:
An equation behaves like a pair of balanced scales. The scales remain balanced as long as we do the same thing to both scales.
This suggests that to solve an equation, we can do the same thing to both sides of an equation. That is:
- The same number can be subtracted from both sides of an equation.
- The same number can be added to both sides of an equation.
- Both sides of an equation can be divided by the same number.
- Both sides of an equation can be multiplied by the same number.
We will now consider equations involving addition, subtraction, multiplication and division.
Operations such as +, –, × and ÷ are used to build an equation. To solve an equation, we use inverse (i.e. opposite) operations such that the pronumeral is the only term remaining on the left-hand side.
Equations Involving Addition
The inverse operation of + is –. So, to solve an equation involving addition, we undo the addition by subtracting the same number from both sides.
Example 3
Solve the equation x + 6 = 14.
Solution:
Note:
6 is added to x. So, we undo the addition by subtracting 6 from both sides.
Check:
Equations Involving Subtraction
The inverse operation of – is +. So, to solve an equation involving subtraction, we undo the subtraction by adding the same number to both sides.
Example 4
Solve the equation x – 9 = 17.
Solution:
Note:
9 is subtracted from x. So, we undo the subtraction by adding 9 to both sides.
Check:
Equations Involving Multiplication
The inverse operation of × is ÷. So, to solve an equation involving multiplication, we divide both sides of the equation by the same number.
Example 5
Solve the equation 8x = 72.
Solution:
Note:
x is multiplied by 8. So, we undo the multiplication by dividing both sides by 8.
Check:
Equations Involving Division
The inverse operation of ÷ is ×. So, to solve an equation involving division, we multiply both sides of the equation by the same number.
Example 6
Solution:
Note:
x is divided by 6. So, we undo the division by multiplying both sides by 6.
Check:
Linear equations help us to solve word problems. First, we assume that the number we are trying to find is represented by a pronumeral. Then the problem given in words is translated into an equation which is solved using the methods we have learned for solving equations. Finally, we write the answer in words.
Example 7
A number is added to 85 and the result is 172. Find the number.
Solution:
Let the number be x.
 So, the number is 87.
Check:
Equations Involving Two or More Operations
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To solve an equation involving two or more operations, start by carrying out an inverse operation on the number that is furthest away from x.
Example 8
Solve 7x + 4 = 25.
Solution:
The number 4 is furthest away from x. The inverse of + 4 is – 4. So, subtract 4 from both sides.
Problem Solving 2
Example 9
If three times a number, when diminished by 4, equals 17, what is the number?
Solution:
Let x be the number.
Three times x is 3x, and diminishing this by 4 gives 3x – 4, which we are told equals 17.
So, the number is 7.
Equations with the Pronumeral on Both Sides
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When a pronumeral is on both sides of an equation, remove the pronumeral term from the right-hand side of the equation by using inverse operations. Then continue to use inverse operations to solve the equation for the pronumeral.
Example 10
Solution:
Problem Solving 3
Example 11
If 7 less than three times a number is 9 more than the number, what is the number?
Solution:
Let x be the number.
Seven less than three times x is 3x – 7, and 9 more than x is x + 9.
So, the number is 8.
Check:
Equations Containing Brackets
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To solve the equation containing brackets, we may proceed as follows:
- Remove the brackets by using the Distributive Law.
- Collect the pronumeral terms on the left-hand side of the equation and the numerical terms on the right-hand side of the equation by doing the same thing to both sides of the equation.
Example 12
Solution:
Example 13
Solution:
Example 14
Find the width of a rectangular paddock whose length is 60 m and perimeter is 220 m.
Solution:
So, the width of the paddock is 50 m.
Equations Containing Fractions
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To solve equations containing fractions:
- Find the lowest common multiple of the denominators which is known as the lowest common denominator (LCD).
- Remove the fractions by multiplying both sides of the equation by the LCD.
- Solve the equation for the unknown pronumeral by performing the same operations to both sides of the equation.
Example 15
Solution:
Lowest common multiple of 3 and 1 is 3. So, we multiply both sides by 3 to obtain:
Example 16
Solution:
Lowest common multiple of 8 and 3 is 24. So, we multiply both sides by 24 to obtain:
Problem Solving 5
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Read the problem carefully and describe it by an equation. Then solve the equation and write the answer as a sentence.
Example 17
The sum of three consecutive numbers is 87. What are the numbers?
Solution:
Let the smallest number be n.
The word consecutive means one after the other. So, the next two consecutive numbers after n will be n + 1 and n + 2.
So, the three consecutive numbers are 28, 29 and 30.
Example 18
Use the information given in the following diagram to form an equation and then solve it to find the value of the pronumeral.
Solution:
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