# Equations and word problems · Grade 8

> Equations and word problems, Grade 8: Linear equations, Special equations and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Equations and word problems (Grade 8) you practise 6 skills: Linear equations · Special equations · Word problems with equations · Equations with fractions · Speed, distance and time problems · Percentage problems with equations.

An equation is like a balanced scale: whatever you do to one side, you also do to the other, and the two sides stay equal. That is how you get the unknown on its own, one step at a time.

When an equation has fractions, multiply both sides by the common denominator. The fractions disappear, and you are left with an ordinary equation without fractions.

In a word problem, use a letter for what you are looking for, write an equation that describes the story, solve it, and check that the answer fits the story.

In speed problems, distance = speed × time. Cars driving towards each other cover the whole distance together, and a car that catches up with another covers the same distance in less time.

**Tip:** At the end, substitute the solution into the original equation and check that both sides are equal.

## What you learn

### Linear equations

- Example: Solve the equation: x + 11 = 20
- Answer: 9
- Solution: Subtract 11 from both sides: x = 20 − 11 = 9 The solution is 9.

### Special equations

- Example: How many solutions does the equation 3x + 5 = 5 + 3x have?
- Options: One solution · No solution · Infinitely many solutions
- Answer: Infinitely many solutions
- Solution: Simplifying each side: 3x + 5 = 3x + 5 The two sides are the same, so every number you substitute makes the equation true: it has infinitely many solutions.

### Word problems with equations

- Example: Sophie has 3 times as many coins as Oliver. Together they have 80 coins. How many coins does Oliver have?
- Answer: 20
- Solution: Let x be the number of coins Oliver has. Sophie has 3 times as many, so the equation is: x + 3x = 80 4x = 80 x = 80 ÷ 4 = 20 Oliver has 20 coins.

### Equations with fractions

- Example: Solve: x ÷ 3 − 8 = −7
- Answer: 3
- Solution: Add 8 to both sides: x ÷ 3 = 1 Multiply both sides by 3: x = 1 × 3 = 3

### Speed, distance and time problems

- Example: A car travels at 100 km/h for 2 hours. How far does it go?
- Answer: 200 km
- Solution: Distance is speed times time: 100 × 2 = 200 (A table: the speed, time and distance of each car.)

### Percentage problems with equations

- Example: The price of a cinema ticket went up by 50%, and a ticket now costs 72 shekels. What was the price before the increase?
- Answer: 48 shekels
- Solution: An increase of 50% means multiplying by 1.5. Call the old price x and write an equation: 1.5x = 72 Divide both sides by 1.5: x = 72 ÷ 1.5 = 48 The old price was 48 shekels.

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education's grade 8 transition maths curriculum (2026–27).

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