Linear equation word problems: foundations

[Video Placeholder - Introduction to word problems]

Duration: 5:12

Note: Word problems test your ability to translate real-world situations into mathematical equations. On the SAT, you'll encounter various types of word problems that can be modeled with linear equations.

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How to solve linear equation word problems

Follow these steps to solve word problems:

  1. Read carefully - Identify what's being asked
  2. Define variables - Choose letters to represent unknowns
  3. Translate - Convert the words into an equation
  4. Solve - Find the value of the variable
  5. Check - Does your answer make sense in context?

Common word problem types

Example 1: Age problem

Sarah is 5 years older than twice her brother's age. The sum of their ages is 29. How old is Sarah's brother?

Step 1: Define variables

Let b = brother's age
Then Sarah's age = 2b + 5

Step 2: Set up equation

(brother's age) + (Sarah's age) = 29
b + (2b + 5) = 29

Step 3: Solve

3b + 5 = 29
3b = 24
b = 8

Answer: Sarah's brother is 8 years old.

Example 2: Money problem

A movie theater charges $10 for adults and $6 for children. For one showing, the theater sold 120 tickets and made $920. How many adult tickets were sold?

Step 1: Define variables

Let a = number of adult tickets
Then (120 - a) = number of child tickets

Step 2: Set up equation

(adult revenue) + (child revenue) = total revenue
10a + 6(120 - a) = 920

Step 3: Solve

10a + 720 - 6a = 920
4a + 720 = 920
4a = 200
a = 50

Answer: 50 adult tickets were sold.

Try it!

A car rental company charges a flat fee of $30 plus $0.25 per mile driven. If a customer is charged $75, how many miles did they drive?

Step 1: Define your variable

Let =

Step 2: Write the equation

Step 3: Solve for the variable

x =

The total cost equals the flat fee plus the per-mile charge times the number of miles.

More word problem types

Example 3: Distance problem

Two trains leave stations 240 miles apart at the same time and travel toward each other. One train travels at 70 mph while the other travels at 50 mph. How long will it take for them to meet?

Step 1: Define variables

Let t = time until they meet (in hours)

Step 2: Set up equation

(distance train 1 travels) + (distance train 2 travels) = total distance
70t + 50t = 240

Step 3: Solve

120t = 240
t = 2

Answer: They will meet in 2 hours.

Example 4: Mixture problem

A chemist needs to make 40 liters of a 25% acid solution. She has a 10% solution and a 50% solution available. How many liters of each should she mix?

Step 1: Define variables

Let x = liters of 10% solution
Then (40 - x) = liters of 50% solution

Step 2: Set up equation

(acid from 10%) + (acid from 50%) = total acid
0.10x + 0.50(40 - x) = 0.25(40)

Step 3: Solve

0.10x + 20 - 0.50x = 10
-0.40x + 20 = 10
-0.40x = -10
x = 25

Answer: 25 liters of 10% solution and 15 liters of 50% solution.

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