Applications of Linear Equations
A word problem turns into an equation once you name the unknown and translate each sentence into math. Distance, work and mixture problems each run on one simple relationship, so once you spot which one you're in, the setup writes itself.
What the symbols mean
- Distance
Distance equals rate times time.
Explanation
An application is simply a real situation, described in words, that turns into a linear equation once you translate it. The formulas up top are three everyday facts: (distance is speed times time), work rate (if a job takes 3 hours, you do a third of it per hour) and amount concentration volume (2 liters of 30% juice holds 0.6 liters of juice). Almost every problem in this lesson is one of those three in disguise. The math is the easy part; the translating takes a little practice.
A method:
- Read carefully; decide what you're solving for and name it.
- Write other quantities in terms of that variable.
- Translate the relationship into an equation.
- Solve, then check that the answer makes sense (no negative times or lengths).
Work problems. Times don't add, but rates do. If one pipe fills a tank in 3 hours and another in 6, then each hour they fill of the tank together, so the whole job takes 2 hours.
Mixtures. When you pour two solutions together, the total amount of the pure ingredient doesn't change. So write "pure stuff before = pure stuff after," and that's your equation.
A common mistake. With the pipes, it's tempting to average the times and say hours, or add them and say . But two pipes working together have to be faster than either one alone, so the answer must be under 3 hours. Adding rates is what gets that right.
See it
Worked examples
Two cars leave the same town in opposite directions at 55 mph and 65 mph. When are they 300 miles apart?
- 1
They're moving apart, so the gap between them grows at mph.
Practice
Type your answer and check it. Solve them all to complete the lesson.
Two cars leave in opposite directions at 55 and 65 mph. After how many hours are they 300 miles apart?
One pipe fills a tank in 3 hours, another in 6. How long together?
How many liters of 40% acid must be added to 10 L of 10% acid to get 20% acid?
A rectangle's length is 3 more than twice its width, and its perimeter is 36. Find the width.
After a 20% discount a jacket costs $64. What was the original price?
A plane flies 600 km with the wind in 2 hours and back against the wind in 2.5 hours. What is the wind speed, in km/h?
How many liters of pure antifreeze must be added to 10 L of 30% antifreeze to make a 50% mixture?
The hardest one here. Get it, and you've got this lesson.
Two trains 300 km apart head toward each other at 70 and 80 km/h. A bird flies back and forth between them at 120 km/h until they meet. How far does the bird fly in all?
Where it's used
Pharmacists dilute medications, logistics planners schedule shipments and contractors estimate job times with exactly these patterns.