Word Problems and Applications

Apply linear systems to real-world word problems. Math 10-C Alberta mathematics curriculum.

Lesson 5.4 of Systems of Equations in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• Translating Words to Equations. Learn the key English-to-math translations needed to build equations from word problems. Practice converting phrases like 'exceeds by', 'less than', and 'twice' into algebraic expressions, and see common mistakes to avoid.
• Setting Up Systems. Practice writing two-equation systems from real-world scenarios involving perimeters, recycling totals, and text messages. Learn the count-and-value strategy for choosing which equation to write first.
• Full Solutions. Solve complete word problems using substitution and elimination: ski-rental break-even points, investment splitting, and digital download pricing. See every step from variable definition to verified answer.
• Harder Applications. Tackle trickier contexts including rectangle dimensions, acid mixtures, boats against currents, and road-trip speed problems. Learn the distance-rate-time framework and the mixture setup strategy.

Inside the lesson — a free preview

Common English-to-Math Translations

Why Translations Matter. Every word problem is really just an equation or system in disguise. The skill is reading the keywords that tell you which operation to use. Once you can translate the sentence, the algebra is straightforward.

Common Mistake — Order Matters!. "5 less than x" is written as x - 5, NOT 5 - x.

The phrase names the starting value first, then the amount subtracted. Always re-read carefully before committing to an order.

Tip on Units — Money Problems. When money is involved, pick one unit — either all cents or all dollars — and stick with it across the entire equation.

Mixing dollars and cents in the same equation is the number-one cause of wrong answers in money problems.

Worked example: Write an Equation for Each Statement

Part a) — The sum of two numbers is 15. Define variables:

Let x and y be the two numbers.

Translation:

Answer: x + y = 15

Part b) — A number exceeds another by 5. Define variables:

Let x be the bigger number and y the smaller.

Translation:

'Exceeds by' means the bigger number equals the smaller plus the excess.

Answer: x = y + 5

Part c) — The width of a rectangle is 3 m less than the length. Define variables:

Let l = length, w = width (in metres).

Translation:

'w is 3 m less than l' means start with l and subtract 3.

Answer: w = l - 3

Part d) — Daniel is 13 years older than Bobby. Define variables:

Let D = Daniel's age, B = Bobby's age.

Translation:

'Older than' means add the age difference to the younger person's age.

Answer: D = B + 13

Part e) — One number decreased by one half of another number equals 7. Define variables:

Let x and y be the two numbers.

Translation:

'Decreased by half of another' → subtract (1)/(2)y from x, set equal to 7.

Answer: x - (1)/(2)y = 7

Part f) — A collection of dimes and pennies has a value of 1.97. Define variables:

Let d = number of dimes, p = number of pennies.

Unit choice:

Convert \1.97 to 197 cents so every term is in cents.

Translation:

Each dime is worth 10 cents, each penny is worth 1 cent.

Answer: 10d + p = 197

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