3.1 Word Problem Systems
Name: ____________________________
Solve the following solutions using two unknowns. Then solve the equations by the
method of choice—graphing, substitution, elimination.
1. The sum of two numbers is 41. The first number is 1 less than twice the second. Find the numbers.
2. The sum of two numbers is 53. Three times the first number is equal to 19 more than the second
number. What are the numbers?
3. The sum of two numbers is 12. The difference of the first number and the second number is –4. Find
the numbers.
4. Twice the first number minus a second number is –1. Twice the second number added to three times
the first number is 9. Find the two numbers.
5. Ezekiel has some coins in his pocket consisting of dimes, nickels, and pennies. He has two more
nickels than dimes and three times as many pennies as nickels. How many of each kind of coin does
he has 18 coins total?
6. Mrs. Crane received a total of 180 coins each day for her cash drawer at the restaurant where she was
a cashier. She had the same number of nickels as quarters and the same number of pennies as dimes.
She had 10 more dimes than nickels. How many of each did she have?
7. A man is four times as old as his son. In 3 years the father will be three times as old as his son. How
old is each now?
8. Benita’s age is three times Ana’s. If 20 is added to Ana’s age and 20 is subtracted from Benita’s, their
ages will be equal. How old is each now?
9. The length of a rectangle is 5 feet more than twice the width. The perimeter is 28 feet. Find the
length and width of the rectangle.
10. The width of a rectangle is 7 feet less than the length. The perimeter is 74 feet. Find the length and
width of the rectangle.
11. The second angle of a triangle is 20o greater than the first angle. The third angle is twice the second.
Find the three angles.
12. The third angle is three times the second angle. The first angle is half of the second angle. Find the
three angles.
13. The sum of the digits of a three-digit number is 6. The hundreds digit is twice the units digit. The tens
digit equals the sum of the other two digits. What is the number?
14. The tens digit of a two-digit number is 5 more than the ones digit. The sum of the digits is 9. Find the
two-digit number.
15. Last year the volleyball team paid $5 per pair for socks and $17 per pair for shorts on a total purchase
of $315. This year they spent $342 to buy the same number of pairs of socks and shorts because the
socks now cost $6 a pair and the shorts cost $18. Write a system of two equations that represents the
number of pairs of socks and shorts bought each year. How many pairs of socks and shorts did the
team buy each year?
16. At a store, toothbrushes cost 𝑥 dollars and bars of soap cost 𝑦 dollars. One customer bought 2
toothbrushes and 1 bar of soap for $11. Another customer bought 6 toothbrushes and 5 bars of soap
for $38. Write and solve a system of equations for 𝑥 and 𝑦.
3.1 𝑊𝑜𝑟𝑑 𝑃𝑟𝑜𝑏𝑙𝑒𝑚 𝑆𝑦𝑠𝑡𝑒𝑚𝑠 𝑜𝑓 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠
Name: ____________________________
Solve the following solutions using two unknowns. Then solve the equations by the
method of choice—graphing, substitution, elimination.
1. The sum of two numbers is 41. The first number is 1 less than twice the second. Find the numbers.
2. The sum of two numbers is 53. Three times the first number is equal to 19 more than the second
number. What are the numbers?
3. The sum of two numbers is 12. The difference of the first number and the second number is –4. Find
the numbers.
4. Twice the first number minus a second number is –1. Twice the second number added to three times
the first number is 9. Find the two numbers.
5. Ezekiel has some coins in his pocket consisting of dimes, nickels, and pennies. He has two more
nickels than dimes and three times as many pennies as nickels. How many of each kind of coin does
he has 18 coins total?
6. Mrs. Crane received a total of 180 coins each day for her cash drawer at the restaurant where she was
a cashier. She had the same number of nickels as quarters and the same number of pennies as dimes.
She had 10 more dimes than nickels. How many of each did she have?
7. A man is four times as old as his son. In 3 years the father will be three times as old as his son. How
old is each now?
8. Benita’s age is three times Ana’s. If 20 is added to Ana’s age and 20 is subtracted from Benita’s, their
ages will be equal. How old is each now?
9. The length of a rectangle is 5 feet more than twice the width. The perimeter is 28 feet. Find the
length and width of the rectangle.
10. The width of a rectangle is 7 feet less than the length. The perimeter is 74 feet. Find the length and
width of the rectangle.
11. The second angle of a triangle is 20o greater than the first angle. The third angle is twice the second.
Find the three angles.
12. The third angle is three times the second angle. The first angle is half of the second angle. Find the
three angles.
13. The sum of the digits of a three-digit number is 6. The hundreds digit is twice the units digit. The tens
digit equals the sum of the other two digits. What is the number?
14. The tens digit of a two-digit number is 5 more than the ones digit. The sum of the digits is 9. Find the
two-digit number.
15. Last year the volleyball team paid $5 per pair for socks and $17 per pair for shorts on a total purchase
of $315. This year they spent $342 to buy the same number of pairs of socks and shorts because the
socks now cost $6 a pair and the shorts cost $18. Write a system of two equations that represents the
number of pairs of socks and shorts bought each year. How many pairs of socks and shorts did the
team buy each year?
16. At a store, toothbrushes cost 𝑥 dollars and bars of soap cost 𝑦 dollars. One customer bought 2
toothbrushes and 1 bar of soap for $11. Another customer bought 6 toothbrushes and 5 bars of soap
for $38. Write and solve a system of equations for 𝑥 and 𝑦.
Hint: Remember you are looking for money—round the price to two decimal places.