1. SOLVING SYSTEMS OF EQUATIONS
ALGEBRAICALLY
Word Problems
Holt Algebra 1

2. Example 1: Consumer Economics Application
Jenna is deciding between two cell-phone plans. The first plan has a $50 sign-up fee and costs $20 per month. The second plan has a $30 sign-up fee and costs $25 per month. After how many months will the total costs be the same? What will the costs be? If Jenna has to sign a one-year contract, which plan will be cheaper? Explain.
Write an equation for each option. Let y represent the total amount paid and x represent the number of months.

3. Example 1 Continued
Sign up fee
Total paid
Payment amount
for each
month.
plus is
Option 1 y =
$20 +
$50 x
Option 2 y =
$25 +
$30 x
Step 1 y = 20x+50
y = 25x + 30
Both equations are solved for y (y=mx + b)
Step 2
20x + 50 =
25x + 30
Set the 2 equations equal to each other which eliminates the y

4. Example 1 Continued
Step 3
20x + 50 =
25x + 30
Solve for x. Subtract 25x from both sides.
–25x – 25x
-5x + 50 = 30
Subtract 50 from both sides.
–50 –50
-5x = -20
Divide both sides by -5.
x = 4
-5x = -20
Step 4
y = 25x + 30
Write one of the original equations.
y = 25(4) + 30
Substitute 4 for x.
y =
y = 130
Simplify.

5. Example 1 Continued
Write the solution as an ordered pair.
Step 5
(4, 130)
In 4 months, the total cost for each option would be the same $130.
If Jenna has to sign a one-year contract, which plan will be cheaper? Explain.
Option 1: y = 20(12) + 50 = 290
Option 2: y = 25(12) + 30 = 330
Jenna should choose the first plan because it costs $290 for the year and the second plan costs $330.

6. Example 2
One cable television provider has a $60 setup fee and $80 per month, and the second has a $160 equipment fee and $70 per month.
a. In how many months will the cost be the same? What will that cost be.
Write an equation for each option. Let y represent the total amount paid and x represent the number of months.

7. Example 2 Continued
payment amount
for each
month.
Total paid
plus
fee is
Option 1 y =
$80 +
$60 x
Option 2 y =
$70 +
$160 x
Step 1 y = 80x + 60
y = 70x + 160
Both equations are solved for y (y=mx + b.
Step 2
80x + 60 =
70x + 160
Set both equations equal to each other to eliminate the y’s.

8. Example 2 Continued
Step 3
80x + 60 =
70x + 160
Solve for x. Subtract 70m from both sides.
–70x –70x
10x + 60 = 160
Subtract 60 from both sides.
–60 –60
10x = 100
Divide both sides by 10.
x = 10
Step 4
y = 70x + 160
Write one of the original equations.
y = 70(10) + 160
Substitute 10 for m.
y =
y = 860
Simplify.

9. Example 2 Continued
Step 5
(10, 860)
Write the solution as an ordered pair.
In 10 months, the total cost for each option would be the same, $860.
b. If you plan to move in 6 months, which is the cheaper option? Explain.
Option 1: y = (6) = 540
Option 2: y = (6) = 580
The first option is cheaper for the first six months.

10. Example 3
3. Plumber A charges $60 an hour. Plumber B charges $40 to visit your home plus $55 for each hour. For how many hours will the total cost for each plumber be the same? How much will that cost be? If a customer thinks they will need a plumber for 5 hours, which plumber should the customer hire? Explain.
8 hours; $480; plumber A: plumber A is cheaper for less than 8 hours.

11. Example 5
4. The Strauss family is deciding between two lawn-care services. Green Lawn charges a $49 start-up fee, plus $29 per month. Grass Team charges a $25 start-up fee, plus $37 per month.
a. In how many months will both lawn-care services cost the same?
b. If the family will use the service for only 6 months, which is the better option?
Explain.
3, $136; Green Lawn: for 6 months, Green Lawn’s service cost only $223 while Green Team’s cost $247

12. 5. ) Jack and Jason are saving for new scooters
5.) Jack and Jason are saving for new scooters. So far, Jack has saved $9 and can earn $6 per hour dog-sitting. Jason has saved $3 and can earn $9 per hour working as a lifeguard. After how many hours of work will Jack and Jason have saved the same amount? What will that amount be? Solve by substitution showing all work and your check.
(2, 21)

13. 6.) Angus runs 7 miles per week and increases his distance by 1 mile each week. Myles runs 4 miles per week and increases his distance by 2 miles each week. In how many weeks will Angus and Myles be running the same distance? What will that distance be?
(3, 10)

14. (15, 170) 7.) Derek is going to have a party catered.
Good Eats charges $120 plus $10 per
person. Food Fare charges $150
plus $8 per person. Find the # of
people for which the total cost
is the same for both catering
companies.
(15, 170)

15. Example 8
A jar contains x nickels and y dimes. There are 20 coins in the jar, and the total value of the coins is $ How many nickels and how many dimes are in the jar? (Hint: nickels are worth $0.05 and dimes are worth $0.10
Write an equation for number of coins in the jar. Let x represent the total number of nickels and y represent the number of dimes
Write an equation for the total money in the jar. Let x represent the total number of nickels and y represent the number of dimes

16. Solve 1st equation for “y” ; Slope intercept form (y=mx + b)
Example 8 Continued
nickels
plus
dimes is
Total in jar
Option 1 x = y + 20
Option 2
$0.05x =
$0.10y +
$1.40
Step 1 x + y= 20
-x x
y= -x+20
Solve 1st equation for “y” ;
Slope intercept form (y=mx + b)

17. Example 8 Continued
Step x + .10y= 1.40
-0.05x x
.10y= -.05x+1.40
Solve 2nd equation for “y” ;
Slope intercept form
(y=mx + b)
Step 2
Set the 2 equations equal to each other eliminating the y’s.

18. Example 8 Continued
Step 3
-x =
-½x + 14
Solve for x. Add ½ to
both sides.
+½x ½x
-.5x + 20 = 14
Subtract 20 from both sides.
–20
-.5x = -6
Divide both sides by -.5
x = 12
Step 4
y = -x + 20
Write one of the original equations.
y = -(12)+ 20
Substitute 12 for x.
y =
y = 8
Simplify.
(12,8) 12 nickels, 8 dimes