Systems of Equations Word Problems

Learn to translate real-world situations into systems of equations and solve them step by step.

Advanced25 min15 problems
Practice Problems

Definition

A system of equations word problem asks you to find unknown quantities by:
  1. 1.Defining variables for the unknowns
  2. 2.Writing two equations that relate the variables
  3. 3.Solving the system using substitution or elimination
  4. 4.Checking your answer in the original problem
The key is translating words into mathematical relationships:
Word/PhraseMathematical Operation
sum, total, combined++ (addition)
difference, less than−- (subtraction)
times, product, each×\times (multiplication)
per, for each, ratetypically multiplication
is, equals, was== (equals sign)

Try it now

The sum of two numbers is 20 and their difference is 6. Which system of equations represents this situation?

Worked Examples

The sum of two numbers is 52. Their difference is 18. Find the two numbers.
1

Define variables

Let xx = the larger number Let yy = the smaller number → Two unknowns: xx and yy
2

Write equations from the words

Sum is 52: x+y=52x + y = 52 Difference is 18: x−y=18x - y = 18 → System: {x+y=52x−y=18\begin{cases} x + y = 52 \\ x - y = 18 \end{cases}
3

Solve using elimination

Add both equations: (x+y)+(x−y)=52+18(x + y) + (x - y) = 52 + 18 2x=702x = 70 → x=35x = 35
4

Find y

x+y=52x + y = 52 35+y=5235 + y = 52 y=17y = 17
5

Check

Sum: 35+17=5235 + 17 = 52 ✓ Difference: 35−17=1835 - 17 = 18 ✓ → Answer verified!

Common Mistakes

Using only one variable when two are needed

Why it's wrong: If you have two unknown quantities, you need two variables and two equations. Using one variable often leads to contradictions or overly complex expressions.

Correct: Always identify ALL unknown quantities first, then assign a different variable to each.

Mixing up which equation represents which condition

Why it's wrong: Writing 'cost equation' when you meant 'quantity equation' gives wrong answers even with correct algebra.

Correct: Label your equations clearly: 'Equation 1 (total quantity)' and 'Equation 2 (total value)'

Forgetting to check the answer in the original problem

Why it's wrong: Calculation errors happen. The answer might solve your equations but not match the word problem conditions.

Correct: Always substitute back into BOTH original conditions and verify in context (not just in your equations).

Setting up rate problems incorrectly

Why it's wrong: Downstream, the boat and current speeds ADD. Upstream, you SUBTRACT the current from boat speed.

Correct: Think about it: going with the current makes you faster (add), going against makes you slower (subtract).

Interactive Visual

Function graph

024681012141605101520xy(7, 8)

Tap or hover over the graph to read off values.

x + y = 1512x + 8y = 148
  • Intersection point: (7, 8)

Equation Solver

Solve for x,yx, y:

(1)x+y=52(2)x−y=18\begin{aligned} (1)\quad x + y &= 52 \\ (2)\quad x - y &= 18 \end{aligned}
1
(1)x+y=52(2)x−y=18\begin{aligned} (1)\quad x + y &= 52 \\ (2)\quad x - y &= 18 \end{aligned}
Add the two equationsThe yy terms cancel out

Tips:

  • • Whatever you do to one side, do to the other
  • • Undo operations in reverse order (PEMDAS backwards)
  • • Check your answer by substituting back

Follow along as we solve the equation step by step.

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Practice Problems

15 problems
Problem 1 of 15
Easy
The sum of two numbers is 20 and their difference is 6. Which system of equations represents this situation?

Why It Matters

Systems of equations word problems appear everywhere in real life:
  • Business: Finding the right mix of products to maximize profit
  • Chemistry: Mixing solutions to achieve a desired concentration
  • Finance: Calculating how much to invest in different accounts
  • Travel: Determining speeds when two objects are moving
  • Shopping: Finding individual prices when you know totals
The skill of translating real situations into equations is fundamental to using math in any career!

Real World Applications

Running a Business

Business owners use systems of equations to make pricing decisions and manage inventory.

Example:

A coffee shop sells small coffees for 3 euros and large coffees for 5 euros. Yesterday they sold 120 coffees totaling 480 euros. How many of each size did they sell?

1Try It Yourself

A bakery sells cookies for 2 euros and brownies for 3 euros. One day it sold 85 items for a total of 195 euros.

How many brownies did it sell?

Step 1: Write the mathematical expression

Let cc = cookies and bb = brownies: c+b=85c + b = 85 and 2c+3b=1952c + 3b = 195. Multiply the first equation by −2-2 and add it to the second, then write the calculation that gives bb:

Travel Planning

Planning trips involves calculating distances, speeds, and times that often require systems of equations.

Example:

A cyclist rode 50 km: 3 hours uphill and 2 hours downhill. Her downhill speed was 10 km/h higher than her uphill speed. What were the two speeds?

2Try It Yourself

A cyclist rides 50 km in 3.5 hours. Part of the time she rides uphill at 10 km/h, the rest of the time downhill at 25 km/h.

How many hours does she ride uphill?

Step 1: Write the mathematical expression

Let uu = hours uphill and dd = hours downhill: u+d=3.5u + d = 3.5 and 10u+25d=5010u + 25d = 50. Substitute d=3.5−ud = 3.5 - u, then write the calculation that gives uu:

Chemistry and Mixing

Chemists and pharmacists use systems to calculate how much of each substance to mix.

Example:

A chemist needs 100 mL of a 25% acid solution. She has 10% and 40% solutions available. How much of each should she mix?

3Try It Yourself

A juice company wants 500 mL of a 32% fruit juice blend. It mixes a 20% juice with a 50% juice.

How many mL of the 50% juice are needed?

Step 1: Write the mathematical expression

Let xx = mL of 20% juice and yy = mL of 50% juice: x+y=500x + y = 500 and 0.20x+0.50y=0.32×5000.20x + 0.50y = 0.32 \times 500. Multiply the first equation by −0.20-0.20 and add it to the second, then write the calculation that gives yy:

Key Takeaways

  • 1Define a variable for EACH unknown quantity in the problem
  • 2Identify TWO conditions that relate the variables and write an equation for each
  • 3Solve using substitution or elimination (choose the method that seems easier)
  • 4Always check your answer in the ORIGINAL word problem, not just the equations
  • 5Common problem types: sum/difference, cost/quantity, rate problems, mixture problems

Frequently Asked Questions

Look for phrases like 'how many', 'find the', or 'what is' - these indicate the unknowns. Each unknown quantity gets its own variable. Usually you'll have two unknowns that you're asked to find.
Look for phrases like 'how many', 'find the', or 'what is' - these indicate the unknowns. Each unknown quantity gets its own variable. Usually you'll have two unknowns that you're asked to find.
Use substitution when one equation is already solved for a variable (like y=2x+3y = 2x + 3) or can easily be solved. Use elimination when the coefficients match up nicely or when substitution would create messy fractions.
Check if it makes sense in context. You can't have -3 tickets or 2.7 people. But you CAN have negative temperatures, fractional hours, or decimal amounts of money. If the answer doesn't fit the context, recheck your setup.
Remember: distance = rate × time. For moving objects, think about whether they're moving in the same direction (subtract speeds to find relative speed) or opposite directions (add speeds). For current/wind problems, going WITH adds speed, going AGAINST subtracts.

Glossary

System of equations
Two or more equations with the same variables that must all be true simultaneously
Variable
A letter (like xx or yy) that represents an unknown quantity
Substitution
Solving one equation for a variable and plugging that expression into the other equation
Elimination
Adding or subtracting equations to cancel out one variable
Rate
A quantity measured per unit of something else (speed = distance per time, price = cost per item)

Formula Card

Setting Up Variables

Let xx = first unknown, yy = second unknown

Assign a variable to each unknown quantity

Distance Formula

d=r×td = r \times t

Distance equals rate times time

Total Value Formula

p1⋅q1+p2⋅q2=totalp_1 \cdot q_1 + p_2 \cdot q_2 = \text{total}

Sum of price times quantity for each item

Mixture Formula

c1⋅a1+c2⋅a2=cf⋅atc_1 \cdot a_1 + c_2 \cdot a_2 = c_f \cdot a_t

Concentration times amount equals final mixture

Today's challengeOne problem, three tries

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