It is important to grasp the potential of algebra for solving problems that appear to be baffling. It’s not just about manipulating numbers; it’s about finding a pathway to clarity where there was once confusion. This capacity of algebra to transform complex situations into manageable equations is pivotal in various fields—from engineering to economics.

Seeing a problem through the lens of algebra is like having a map: you might not know the terrain yet, but you’ve got a guide to navigate through it. The real-world problems we encounter often aren’t clear-cut; they’re messy and undefined. Algebra provides the tools to shape these problems into clear and solvable queries.

The cornerstone of applying algebra effectively lies in a methodical approach. It begins with a simple, yet profound step: reading the problem carefully. This may sound trivial, but it’s where the journey to the solution begins. You need to become a detective, studying the text for clues, understanding the context, and most importantly, distinguishing between the necessary and extraneous information.

RESOURCES

• Algebra: A Complete Introduction: The Easy Way to Learn Algebra (Teach Yourself) (https://amzn.to/4gIiXSU)
• KS3 Maths – Algebra for Beginners: With Answers (Step-By-Step Answer Key) | KS3 / KS4 Maths Workbook for Ages 12-15 (Years 7-10) | 100 Days of … Algebra Problems, Equations & Inequalities (https://amzn.to/4gFBeAj)
• Algebra Essentials Practice Workbook with Answers: Linear & Quadratic Equations, Cross Multiplying, and Systems of Equations: Improve Your Math Fluency Series: Volume 12 (https://amzn.to/3PnKPje)
• Algebra I Workbook For Dummies, 3rd Edition (https://amzn.to/4gZLYtd)
• Algebra II All-in-One For Dummies (https://amzn.to/3ZVMxgw)
• Everything You Need to Ace Pre-Algebra and Algebra I in One Big Fat Notebook (Big Fat Notebooks) (https://amzn.to/3DUss2M)
• KS2 Algebra Workbook: supporting Maths mastery for ages 10-11 (Year 6) (SATs Made Simple) (https://amzn.to/4iVmVcq)

These are affiliate links. If you click a link and buy the product, then the blogger gets a percentage of the sale or some other type of compensation. Prices are not different if you use these affiliate links. You will not pay more by clicking through to a link.

EXAMPLE 1

Linda was selling tickets for the school play.  She sold 10 more adult tickets than child tickets and she sold twice as many senior tickets as child tickets. Adult tickets cost £5, child tickets cost £2, and senior tickets cost £3. Linda made £700. Find the number of each type of ticket that was sold.

Step 1: Let x represent the number of child tickets sold

This is really important information, because we are going to let x = the number of child tickets. Every other expression will be based on x or the number of child tickets.

Step 2: Write an expression to represent the number of adult tickets sold

Since she sold 10 more adult tickets than child tickets, the expression is:

x + 10

Step 3: Write an expression to represent the number of senior tickets sold

Since she sold twice as many senior tickets as child tickets, the expression is:

2x

Step 4: Write an equation to represent the total ticket sales

Since adult tickets cost £5, child tickets cost £2, senior tickets cost £3, and the total amount of money made from the ticket sales was £700, the equation is:

5(x + 10) + 2(x) + 3(2x) = 700

Expand the brackets to make the equation look like this:

5x + 50 + 2x + 6x = 700

Collect the like terms to make the equation look like this:

13x + 50 = 700

Step 5: Solve the equation

Subtract 50 from both sides of the equation to get:

13x = 650

Divide both sides by 13 to get:

x = 50

Step 6: Substitute into the original expressions

Child tickets: x = 50 which means that 50 tickets were sold

Adult tickets: x + 10 = 50 + 10 = 60 which means that 60 tickets were sold

Senior tickets: 2x = 2(50) = 100 which means that 100 tickets were sold

EXAMPLE 2

Some people go to the cinema. 4 adults and 2 children pay £47 for their tickets. 1 adult and 3 children pay £25.50 for their tickets. Work out the costs of an adult ticket and a child ticket.

Step 1: Put the words into equations

Let a = adult ticket price and c = child ticket price in order to generate the following simultaneous equations:

4a + 2c = 47 (1)

a + 3c = 25.5 (2)

Step 2: Find the solution for a

Multiply everything in (1) by 3 to make the equation look like this:

12a + 6c = 141 (3)

Multiply everything in (2) by 2 to make the equation look like this:

2a + 6c = 51 (4)

Subtract (4) from (3) to cancel out the c variable:

10a = 90

Divide both sides by 10 to get:

a = 9

Step 3: Use the solution for a to find the solution for c

Substitute the value of a into (2) to make the equation look like this:

9 + 3c = 25.5

Subtract 9 from both sides to get:

3c = 16.5

Divide both sides by 3 to get:

c = 5.5

Adult ticket price = £9

Child ticket price = £5.50