Age Problem

Maria is 3 years older than twice her brother's age. If Maria is 17 years old, how old is her brother?

Define the variable: Let $x$ = brother's age = $x$ = unknown

Translate to equation: "3 years older than twice" means $2x + 3$ "Maria is 17" means equals 17 = $2x + 3 = 17$

Subtract 3 from both sides: $2x + 3 - 3 = 17 - 3$ = $2x = 14$

Divide both sides by 2: $\frac{2x}{2} = \frac{14}{2}$ = $x = 7$

Check the answer: Twice 7 is 14, plus 3 is 17 ✓ = Brother is 7 years old

Answer: Maria's brother is $7$ years old.

Shopping Problem

A shirt costs 15 dollars more than a pair of socks. Together they cost 47 dollars. How much does each item cost?

Define the variable: Let $x$ = price of socks (the smaller unknown) = $x$ = socks price

Express the other unknown: Shirt is "15 dollars more" so shirt = $x + 15$ = Shirt = $x + 15$

Write the equation: Together they cost 47: socks + shirt = 47 = $x + (x + 15) = 47$

Combine like terms: $x + x + 15 = 47$ = $2x + 15 = 47$

Solve for x: $2x = 47 - 15 = 32$ $x = 16$ = Socks = 16 dollars

Find shirt price: Shirt = $x + 15 = 16 + 15$ = Shirt = 31 dollars

Answer: The socks cost $16$ dollars and the shirt costs $31$ dollars.

Distance Problem

Two cyclists start from the same point and ride in opposite directions. One rides at 12 km/h and the other at 18 km/h. After how many hours will they be 90 km apart?

Define the variable: Let $t$ = time in hours = $t$ = hours

Write expressions for distances: Distance = speed $\times$ time Cyclist 1: $12t$ km Cyclist 2: $18t$ km = Distances: $12t$ and $18t$

Set up the equation: Total distance apart = 90 km $12t + 18t = 90$ = $30t = 90$

Solve for t: $t = \frac{90}{30}$ = $t = 3$

Check: After 3 hours: $12(3) + 18(3) = 36 + 54 = 90$ ✓ = 3 hours

Answer: The cyclists will be 90 km apart after $3$ hours.

Mistake: Defining the wrong quantity as the variable

Why: Choosing the larger or more complex quantity as $x$ makes the equation harder to solve.

Correct: Define $x$ as the simpler or smaller unknown. Express other quantities in terms of $x$.

Mistake: Forgetting to check if the answer makes sense

Why: Mathematical solutions can be correct but not meaningful (e.g., negative ages).

Correct: Always verify: Does the answer fit the context? Is it positive when it should be? Does it satisfy the original problem?

Mistake: Translating "less than" incorrectly

Why: "5 less than x" is $x - 5$, NOT $5 - x$. Order matters with subtraction!

Correct: "A less than B" always means $B - A$. The number that comes after "than" goes first.

Event Planning

Event planners use equations to budget for venues, catering, and guest counts.

A venue charges 200 euros base fee plus 35 euros per guest. With a budget of 900 euros, how many guests can you invite?

Sports Statistics

Athletes and coaches use equations to track performance and set goals.

A basketball player has scored 156 points in 12 games. How many points per game must she average in the next 8 games to reach 300 total points?

Q: How do I know which quantity to call x?

A: Choose the simplest unknown as $x$. If the problem says 'one thing is 5 more than another,' let $x$ be the smaller one, then express the other as $x + 5$.

Q: What if my answer is negative but that doesn't make sense?

A: Re-read the problem! A negative age or negative count of items means you made an error setting up the equation. Check your translation of words to symbols.

Q: How do I translate 'twice as much' or 'three times as many'?

A: 'Twice as much as $x$' means $2x$. 'Three times as many as $x$' means $3x$. The multiplier goes in front of the variable.