Question

the difference between two numbers is 11. six times the larger number is 17 times the smaller number. write a system of equations describing the given conditions. then solve the system by the substitution method and find the two numbers.

1) let x represent the larger and y why represent the smaller number. type the equaton for "the difference between two numbers is 11" ?

2)type the equation for "six times the larger numer is 17 times the smaller number" ?

3) find the numbers
the value of the smaller number is?
the value of the larger nyumber is ?

Answer 1

"the difference between two numbers is 11"
"let x represent the larger and y why represent the smaller number."

Can you write the above as an equation?

Answer 2

What operation does "difference" mean?

Answer 3

subtract?

Answer 4

Correct.
That means that you have two numbers you are calling x and y.
The difference between them is 11.
That means, write an equation where you subtract one number from the other number, and it equals 11.
Notice that the difference in 11, a positive number. That means you must subtract the smaller number from the larger number.

Answer 5

21-10 is all i can think of

Answer 6

That is an good example using real numbers because you do get a difference of 11.

The problem is telling you about two numbers we don't yet know what they are.
We use variables (letters) to represent unknown numbers.
The problem specifically tells you to use x for the larger number and y for the smaller number.
Can you show in an equation using x and y that the difference between x and y is 11?

Answer 7

x-y=11?

Answer 8

Exactly.
You have answered part 1).

Note:
You wrote it correctly since we know that x is the larger number, and y is the smaller number, x - y will give a positive difference.
x - y = 11 is correct.
If you had written y - x = 11, that would have been incorrect because when you subtract a larger number from a smaller number, the difference is negative.

Answer 9

Ok, so far we have
1) x - y = 11

Answer 10

Now we need to do part 2)

Answer 11

Remember that the problem told us the larger number is x, and the smaller number is y.
Part 2)
"type the equation for "six times the larger number is 17 times the smaller number"

If the larger number is x, what is 6 times the larger number?

Answer 12

11?

Answer 13

No. We don't yet know what the numbers are.
The larger number could be 11, 21, 35, 101, or any other number.
For now, we are calling the larger number x.
How do you write 6 times the larger number using 6 and x?

Answer 14

x+6=17

Answer 15

x + 6 means "6 added to x", or "6 more than x", or "the sum of x and 6"
You are told 6 TIMES the larger number.
Which operation, addition, subtraction, multiplication, or division, does "times" mean?

Answer 16

multiplication?

Answer 17

Correct.
That means that 6 times the larger number is 6 times x.
6 times x is written as \(6 \times x\), or simply \(6x\) .

Answer 18

For 6 times the larger number we get 6x.

Now we need to write 17 times the smaller number.
The smaller number is y.
Times still means multiplication.
How do you write 17 times y in a similar way to 6x?

Answer 19

17y or it can also be written as 17 x Y

Answer 20

Correct.
17 times y = 17 * y = 17y

Answer 21

To finish part 2) we need to put together 6x and 17y.
The problem tells us that
"six times the larger number is 17 times the smaller number"

That means 6x is equal to 17y.
Use = and write the equation for 6x and 17y.

Answer 22

this is where i get lost

Answer 23

Why? All the hard work so far is done.
We know that 6 times the larger number is 6x.
We know that 17 times the smaller number is 17y.
To write that 6x is 17y means to simply replace "is" with "=".
"is" in the problem's sentence means "is the same as" or "is equal to"
The problem is stating "6 times the larger number "IS EQUAL TO" 17 times the smaller number.

Answer 24

soit would be 6x=17?

Answer 25

Close, but not just 17.
It's 17 times the smaller number, that we already know is 17y.

Answer 26

6x=17y

Answer 27

Exactly.
You have just answered part 2)

Answer 28

So far we have:
Part 1) x - y = 11
Part 2) 6x = 17y

Answer 29

this is very hard for me

Answer 30

its very challenging

Answer 31

We're doing it one step at a time.

Answer 32

Now we need to do part 3)

Answer 33

In part 3) we take both equations we wrote in parts 1) and 2) and we solve them as a system of equations using the substitution method.

Answer 34

how would we substitute?

Answer 35

To use the substitution method, we must first solve one equation for one variable.
Let's use the first equation,
x - y = 11
and solve it for x.

Answer 36

Solving for x means to isolate x.
We want x alone on the left side.

We have
x - y = 11

y is being subtracted from x.
In order to get rid of a subtraction, we must do the opposite operation to subtraction which is addition.

Answer 37

In order to undo the subtraction of y, we add y.
In an equation, we must do the same operation to both sides, so we add y to both sides of this equation.

\(x - y \color{red}{+ y} = 11 \color{red}{+ y}\)

Above, I am showing in red that we are adding the same to both sides, y.

Answer 38

We get

\(x = 11 + y\)

The first equation is now solved for x.

Now we are ready to do the substitution step.

Answer 39

Now we take the second equation,

\(6x = 17y\)

and wehre we see x, we \(\Large \color{red}{substitute} \) what x is equal to.

Answer 40

\(\Large 6 \color{red}{x} = 17y\)

\(\Large 6\color{red}{(11 + y)} = 17y\)

You see that we had x, but now we have 11 + y instead?
That was the substitution step of the substitution method.

Answer 41

Now we have a single equation with only one variable (unknown), the letter y.
We can solve for y.

Answer 42

We use the distributive property on the left side, to distribute the 6.

\(\Large 6(11 + y) = 17y\)

\(\Large 6 \times 11 + 6 \times y = 17y\)

\(\Large 66 + 6y = 17y\)

Now we want all y's together, so since 6y is being added to 66 on the left side, we subtract 6y from both sides.

\(\Large 66 + 6y - 6y = 17y - 6y\)

\(\Large 66 = 11y\)

We can simply switch sides to get:

\(\Large 11y = 66\)

We now divide both sides by 11 to get:

\(\Large \dfrac{11y}{11} = \dfrac{66}{11} \)

\(\Large y = 6\)

Since y stands for the smaller number, we now know that the smaller number is 6.

Answer 43

Now we use the first equation:

\(\Large x - y = 11\)

Since we know y is 6, we substitute 6 for y and solve for x.

\(\Large x - 6 = 11\)

We add 6 to both siodes:

\(\Large x - 6 + 6 = 11 + 6\)

\(\Large x = 17\)

Now we see that the larger number is 17.

The answer is:

The two numbers are 17 and 6.

Answer 44

so i would write my last 2 answers in number form?

Answer 45

Finally, we can check the answer to make sure we are correct.

First, the difference of the numbers must be 11:
17 - 6 = 11, so that works.

Second, 6x = 17y:
6(17) = 17(6)
This equation must be true because of the commutative property of multiplication.
This equation also checks, and we know for sure that our answer is correct.

Answer 46

1) x - y = 11
2) 6x = 17y
3) 17 and 6

Answer 47

Thew answer to part 3) is
17 and 6

Answer 48

so the value of the smaller number is 6? right?

Answer 49

thank you very very much for your help :-)

Answer 50

Yes, the smaller number is 6, and the larger number is 17.
You're welcome.