Not everything goes up. Temperatures fall, points get deducted, hay bales get eaten, budgets shrink. When something is decreasing at a constant rate, equations with negative coefficients describe what's happening. The balance rule works exactly the same — we just divide both sides by a negative number, which is perfectly valid. The only thing to keep in mind: dividing both sides by $(-2)$ gives a negative answer if the right side is positive, and vice versa. Let the arithmetic decide the sign.
Key rule for equations: dividing both sides by a negative number is legal — the equation stays balanced. The sign of $x$ comes from the arithmetic, not from a special rule.
Example: $-2x = 8 \;\mid\; \div(-2)$ → $x = -4$. Straightforward.
Worked Examples
Worked Example 1
In a dressage test, Copper starts with a base score of 80 points. Each mistake costs 3 points. After the test, Copper's score is 62. How many mistakes did he make?
Let $x$ = number of mistakes. Each mistake subtracts 3 points:
The temperature in the stable overnight. At midnight the thermometer read $T$ °C. By 6 am it had dropped 4 °C and the reading was $-1$ °C. What was the midnight temperature?
Let $x$ = midnight temperature (°C):
$$x - 4 = -1 \quad \mid +4$$
$$x = 3$$
It was 3 °C at midnight.
Worked Example 3 — negative coefficient of x
Mia's feed budget. She starts the week with $50 and spends the same amount each day on feed. After 4 days she has $18 left. How much does she spend per day?
Coming soon — inequalities: dividing by a negative in an equation is fine. But for inequalities (like $-3x > 12$), dividing by a negative flips the direction: $x < -4$. We'll cover this in Lesson 07. For now, notice the difference exists — equations are safe.
Practice Problems
Problems 1–4: the equation is given. Solve on paper.
Hay supply. Mia has 40 kg of hay. Copper eats 3 kg per day. After how many days will she have 22 kg left?
Equation: $40 - 3x = 22$
Point deduction. At the county show, a rider starts with 100 points. Knocking a pole deducts 5 points. After the round, the rider has 75 points. How many poles were knocked?
Equation: $100 - 5x = 75$
Solve this equation. Equation: $-2x + 14 = 4$
Solve this equation. Equation: $7 - 4x = -1$
From here — write the equation yourself, then solve.
Weight watch. Copper weighed 480 kg in spring. The vet says he is losing 2 kg per week on his new diet. After how many weeks will he weigh 460 kg?
Cold night. The temperature started at 5 °C and dropped by the same amount each hour. After 4 hours it was $-3$ °C. By how many degrees did it drop per hour?
Savings countdown. Mia has 90 € in her horse fund. She withdraws 12 € each week for feed. After how many weeks will she have 42 € left?
Show penalties. At a dressage competition, each rider starts with 200 points. Mia's score was 158. Each fault costs the same number of points. She had 6 faults. How many points does each fault cost?
Copper on the vet scale. Copper weighed $x$ kg in January. By August he had lost 3 kg per month. After 5 months his vet weight was 455 kg. What was his January weight?
Challenge — x on both sides, negative. At a schooling show, Mia's horse earns 5 points per completed jump. The judge deducts 3 points per refusal. Mia completed 8 jumps and had $r$ refusals. Her final score was the same as a rider who completed 6 jumps and had no refusals. Write and solve an equation for $r$.
Hint: both sides represent final scores.
What's next → You can now solve equations with negative coefficients, and you've seen a glimpse of what changes with inequalities. Lesson 05 is a full story set at a horse competition — it pulls together everything from Lessons 01–04 in one realistic scenario. After that, Lesson 06 opens the world of inequalities for real.
Show answers
$40 - 3x = 22$ → $x = 6$ days
$100 - 5x = 75$ → $x = 5$ poles
$-2x + 14 = 4$ → $x = 5$
$7 - 4x = -1$ → $x = 2$
$480 - 2x = 460$ → $x = 10$ weeks
$5 - 4x = -3$ → $x = 2$ °C per hour
$90 - 12x = 42$ → $x = 4$ weeks
$200 - 6x = 158$ → $x = 7$ points per fault
$x - 5 \times 3 = 455$ i.e. $x - 15 = 455$ → $x = 470$ kg
$8(5) - 3r = 6(5)$ i.e. $40 - 3r = 30$ → $r = \frac{10}{3}$. Since refusals must be whole numbers, this situation is impossible — a good discussion point!