s olving a s ystems of e quations w ord p roblem

SOLVING A SYSTEMS OF EQUATIONS WORD PROBLEM

SYSTEMS WORD PROBLEMS

Example 1: The sum of two numbers is 18 and their difference is 2. What are the two numbers?

We are always looking to get equations out of word problems.

Let’s read this problem carefully to find any equations written in it

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SYSTEMS WORD PROBLEMS

Remember: Read the question first! This question wants to know 2 different

numbers

THE SUM OF TWO NUMBERS IS 18 AND THEIR

DIFFERENCE IS 2. WHAT ARE THE TWO NUMBERS?

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SYSTEMS WORD PROBLEMS

If we don’t know something what do we call it? x!

So one number is x, what should we call the second number since we don’t know it? y!

THE SUM OF TWO NUMBERS IS 18 AND THEIR

DIFFERENCE IS 2. WHAT ARE THE TWO NUMBERS?

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SYSTEMS WORD PROBLEMS

Now read the first part of the question What does the word “sum” mean?

To Add! So we are adding x and y together

THE SUM OF TWO NUMBERS IS 18 AND THEIR

DIFFERENCE IS 2. WHAT ARE THE TWO NUMBERS?

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SYSTEMS WORD PROBLEMS

“Is” means equal, so x + y = 18

The second part says their “difference”. What does that mean?

Minus So x minus y equals 2

THE SUM OF TWO NUMBERS IS 18 AND THEIR

DIFFERENCE IS 2. WHAT ARE THE TWO NUMBERS?

x + yx + y = 18x – y = 2

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SYSTEMS WORD PROBLEMS

We now have a system of equations we can solve using addition!

Add the equations together

Don’t forget that x + x = 2x!

Divide by 2

Now that we know x we have to find y

x + y = 18x – y = 2+__________

2x = 20__ __2 2x = 10

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SYSTEMS WORD PROBLEMS

Plug x into one of the equations

Solve for y Subtract 10 from both

sides

So we know the two numbers are:

x + y = 1810 + y = 18-10 -10

y = 8

10 and 816.

SYSTEMS WORD PROBLEMS

Does our answer makes sense?

Yes, 10 and 8 add up to 18 and subtract to 2!

THE SUM OF TWO NUMBERS IS 18 AND THEIR

DIFFERENCE IS 2. WHAT ARE THE TWO NUMBERS?

10 and 8

SYSTEMS WORD PROBLEMS

1. The sum of two numbers is 12 and the difference is 2. What are the two numbers?

2. The sum of two numbers is -3 and their difference is 15. What are the two numbers?

Your Turn!

SYSTEMS WORD PROBLEMS

1. The sum of two numbers is 12 and the difference is 2. What are the two numbers?

2. The sum of two numbers is -3 and their difference is 15. What are the two numbers?

7 and 5

6 and -9

Your Turn!

SYSTEMS WORD PROBLEMS

Example 2: One number plus two times another number is 13. Three times the first number minus the second number is 4. What are the two numbers?

Again, always read the question first!

This problem is again asking for two numbers.

Since we don’t know what they are we call them x and y

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SYSTEMS WORD PROBLEMS

Start reading the first sentence

“One number plus” means: “two times another number”

means: “is 13” means:

ONE NUMBER PLUS TWO TIMES ANOTHER NUMBER IS 13. THREE TIMES THE FIRST

NUMBER MINUS THE SECOND NUMBER IS 4. WHAT ARE THE

TWO NUMBERS?

x +2y= 1318.

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SYSTEMS WORD PROBLEMS

The second sentence says, “three times the first number” which means:

“minus the second number”: “is 4”

ONE NUMBER PLUS TWO TIMES ANOTHER NUMBER IS 13. THREE TIMES THE FIRST

NUMBER MINUS THE SECOND NUMBER IS 4. WHAT ARE THE

TWO NUMBERS?

3x- y= 421. 22.

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SYSTEMS WORD PROBLEMS

Now we have two equations we can use to solve the problem with.

Multiply the second equation by 2 so we can eliminate the y’s

x + 2y = 133x – y = 4(3x – y = 4)2

SYSTEMS WORD PROBLEMS

Add the equations

Divide by 7

One number is 3, so plug it into an equation to find the other number

x + 2y = 136x – 2y = 8+____________

7x = 21 __ __7 7x = 3

SYSTEMS WORD PROBLEMS

Plug 3 in as x

Solve for y Subtract 3 from

both sides

Divide by 2

The other number is 5

x + 2y = 133 + 2y = 13-3 -3

2y = 10__ __2 2y = 5

SYSTEMS WORD PROBLEMS

The answer is: Does it make sense? 3 plus 2(5) is 13 3(3) minus 5 is 4 Yes!

ONE NUMBER PLUS TWO TIMES ANOTHER NUMBER IS 13. THREE TIMES THE FIRST

NUMBER MINUS THE SECOND NUMBER IS 4. WHAT ARE THE

TWO NUMBERS?

3 and 526.

SYSTEMS WORD PROBLEMS

1. Three times one number plus another number is 12. The first number minus two times the second number is negative 10. What are the two numbers?

2. Two times a number plus three times a second number is 9. Three times the first number minus four times the second is negative 29. What are the numbers?

Practice

SYSTEMS WORD PROBLEMS

1. Three times one number plus another number is 12. The first number minus two times the second number is negative 10. What are the two numbers?

2. Two times a number plus three times a second number is 9. Three times the first number minus four times the second is negative 29. What are the numbers?

2 and 6

-3 and 5

Practice

SYSTEMS WORD PROBLEMS

Example 3: Juan has 10 nickels and dimes in his pocket. He has 70¢ in change. How many dimes does Juan have?

What’s the first thing you do when you see a word problem?

Read the question! 27.

SYSTEMS WORD PROBLEMS

Example 3: Juan has 10 nickels and dimes in his pocket. He has 70¢ in change. How many dimes does Juan have?

We’re looking for dimes, but we don’t know how many dimes or nickels he has

Instead of using x and y, we will use d for dimes and n for nickels

We do this to make sure we know what we’re actually solving for

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SYSTEMS WORD PROBLEMS

Example 3: Juan has 10 nickels and dimes in his pocket. He has 70¢ in change. How many dimes does Juan have?

Next, let’s look at the numbers 10 nickels and dimes means that 10 is a total Totals always go on the other side of the

equals sign!

= 10

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SYSTEMS WORD PROBLEMS

Example 3: Juan has 10 nickels and dimes in his pocket. He has 70¢ in change. How many dimes does Juan have?

So, how did we get to the total of 10?

We can count how many dimes and nickels we have

= 10

SYSTEMS WORD PROBLEMS

Example 3: Juan has 10 nickels and dimes in his pocket. He has 70¢ in change. How many dimes does Juan have?

In one hand we have the nickels

In the other hand we have dimes

To get the total number we add them together!

= 10n d+

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SYSTEMS WORD PROBLEMS

So one equation is:

Now we need to look at the other number

n + d = 10

SYSTEMS WORD PROBLEMS

Example 3: Juan has 10 nickels and dimes in his pocket. He has 70¢ in change. How many dimes does Juan have?

The other number is 70¢

This is a total number, so it goes on the other side of the equals sign

Remember, we have to write money as a decimal, so this is written as .70

= .70

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SYSTEMS WORD PROBLEMS

Example 3: Juan has 10 nickels and dimes in his pocket. He has 70¢ in change. How many dimes does Juan have?

In one hand we have the nickels

Since our total is in money, we have to use the value of nickels

So we have $.o5 for each nickel

Multiply that by the number of nickels (n)

.05n= .70

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SYSTEMS WORD PROBLEMS

Example 3: Juan has 10 nickels and dimes in his pocket. He has 70¢ in change. How many dimes does Juan have?

In the other hand we have dimes

Each dime is $.10 To get the total price,

we add how much we have in nickels with how much we have in dimes

.05n .10d+= .70

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SYSTEMS WORD PROBLEMS

Now we have our second equation! .05n + .10d = .70

The question asks how many dimes, so we should eliminate the nickels (n) and solve for dimes (d)

Multiply the first equation by -.o5

n + d = 10-.05

-.05n - .05d = .536.

SYSTEMS WORD PROBLEMS

We can now add the equations together

Divide by .05

d = 4 means Juan has 4 dimes

-.05n - .05d = .5.05n + .10d = .70+ ________________

.05d = .2____ __.05 .05

d = 4

SYSTEMS WORD PROBLEMS

Example 3: Juan has 10 nickels and dimes in his pocket. He has 70¢ in change. How many dimes does Juan have?

d = 4 Does our answer make sense?

We don’t need to solve for the number of nickels because we are only asked for dimes.

We can plug in 4 for d and see if it makes sense

n + 4 = 10

n = 6-4 -438.

SYSTEMS WORD PROBLEMS

Example 3: Juan has 10 nickels and dimes in his pocket. He has 70¢ in change. How many dimes does Juan have?

d = 4 4 dimes equals $.40 6 nickels equals $.30 So in total Juan has $.70

Yes, our answer makes sense!

n = 6

SYSTEMS WORD PROBLEMS

1. Randall has 9 nickels and dimes in his pocket. He has a total of $0.75. How many nickels does Randall have?

2. Alex has $1.55 in change. He has 11 dimes and quarters. How many dimes does Alex have?

3. Demarcus has 34 dimes and quarters in his change jar. He has $6.55 in total. How many quarters does Demarcus have in his jar?

Your Turn

SYSTEMS WORD PROBLEMS

1. Randall has 9 nickels and dimes in his pocket. He has a total of $0.75. How many nickels does Randall have?

2. Alex has $1.55 in change. He has 11 dimes and quarters. How many dimes does Alex have?

3. Demarcus has 34 dimes and quarters in his change jar. He has $6.55 in total. How many quarters does Demarcus have in his jar?

3 Nickels

8 Dimes

21 Quarters

Your Turn

SYSTEMS WORD PROBLEMS

Example 4: Rush Airlines has three times as many flights out of Hicksville as Crash Airlines. Together they have 16 flights daily out of Hicksville Airport. How many flights does each have?

First read the question

It’s asking for how many flights for each airline

Since we have two things we don’t know we have two variables

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SYSTEMS WORD PROBLEMS

Example 4: Rush Airlines has three times as many flights out of Hicksville as Crash Airlines. Together they have 16 flights daily out of Hicksville Airport. How many flights does each have?

C = Crash AirR = Rush Air

Let C = Crash flights Let R = Rush flights

SYSTEMS WORD PROBLEMS

Example 4: Rush Airlines has three times as many flights out of Hicksville as Crash Airlines. Together they have 16 flights daily out of Hicksville Airport. How many flights does each have?

Read the first sentence for and equation Cross out any words that don’t matter Now it say Crash has 3 times what Rush

has Therefore 3C is equal to R 3C = R

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SYSTEMS WORD PROBLEMS

Example 4: Rush Airlines has three times as many flights out of Hicksville as Crash Airlines. Together they have 16 flights daily out of Hicksville Airport. How many flights does each have?

Read the second sentence for an equation Cross out any words that don’t matter Together they have 16 flight That means if we add the number of Crash

flights with Rush flights we get a total of 16 So C + R = 16

C + R = 16

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SYSTEMS WORD PROBLEMS

We can now solve the system of equations

3C = RC + R = 16

What’s the best way to solve this problem?

Use substitution Plug 3C in for R Combine like terms

Divide by 4

Crash Airlines has 4 flights each day

C + 3C = 164C = 16___ __4 4C = 4

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SYSTEMS WORD PROBLEMS

Now that we know Crash Airlines has 4 flights we can find how many Rush Airlines has by substituting 4 in for C in either equations

Plug in 4 for C

Rush Airlines has 12 flights

So our answer is Crash Airlines has 4 flights and Rush airlines has 12

3C = R3(4) = R

12 = R

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SYSTEMS WORD PROBLEMS

Example 5: The junior class sold 70 spaghetti dinners and 90 chicken dinners and made $730. The senior class sold 110 spaghetti dinners and 120 chicken dinners and made $1040. How much was each chicken dinner?

Remember, read the question first! It’s asking for the price of a chicken

dinner We also don’t know the price of spaghetti

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SYSTEMS WORD PROBLEMS

Example 5: The junior class sold 70 spaghetti dinners and 90 chicken dinners and made $730. The senior class sold 110 spaghetti dinners and 120 chicken dinners and made $1040. How much was each chicken dinner?

We will use: s = spaghettic = chicken

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SYSTEMS WORD PROBLEMS

Example 5: The junior class sold 70 spaghetti dinners and 90 chicken dinners and made $730. The senior class sold 110 spaghetti dinners and 120 chicken dinners and made $1040. How much was each chicken dinner?

Read the first sentence $730 is a total, so it’s on the other side of

the equal sign Cross out anything that doesn’t matter It now says 70s plus 90c equals $730

= 730 70s + 90c = 73054.

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SYSTEMS WORD PROBLEMS

Example 5: The junior class sold 70 spaghetti dinners and 90 chicken dinners and made $730. The senior class sold 110 spaghetti dinners and 120 chicken dinners and made $1040. How much was each chicken dinner?

Read the second sentence $1040 is a total Cross out anything that doesn’t matter 110s and 120c totals 1040

= 1040110s + 120c = 1040

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SYSTEMS WORD PROBLEMS

We now have to solve this system Unfortunately we have to multiply

both equations to solve making the numbers bigger, but I know you can do it!

Make sure to multiply one equation by a negative so we can cancel a variable!

Multiply the first equation by -110 and the second by 70

110s + 120c = 1040 70s + 90c =

730-11070

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SYSTEMS WORD PROBLEMS

Add the equations Divide by -1500 c = 5 means the

chicken dinner was $5

-7700s – 9900c = -803007700s + 8400c = 72800

+______________________-1500c = -7500_______ ______-1500 -1500

c = 560.

SYSTEMS WORD PROBLEMS

We don’t have to, but once you know the price of a chicken dinner you can plug it into an equation to find the price of a spaghetti dinner

70s + 90c = 730 70s + 90(5) =

730 70s + 450 = 730-450 -450

SYSTEMS WORD PROBLEMS

s = 4 means spaghetti dinners were $4

So the answer is $4 for spaghetti and $5 for chicken

70s = 280

____ ___70 70

s = 4

SYSTEMS WORD PROBLEMS

1. Cheap Rental Company rented vans for $50 and cars for $30. If they rented 12 vehicles for $420, how many cars did they rent?

2. Adult tickets cost $5 and student tickets cost $2 for the game. There were 230 tickets sold and the total was $721. How many of each type of ticket was sold?

Your Try

SYSTEMS WORD PROBLEMS

1. Cheap Rental Company rented vans for $50 and cars for $30. If they rented 12 vehicles for $420, how many cars did they rent?

2. Adult tickets cost $5 and student tickets cost $2 for the game. There were 230 tickets sold and the total was $721. How many of each type of ticket was sold?

9 Cars

143 Students, 87 Adults

Your Try