Systems help please?

Question

A two-digit number is 3 times the sum of its digits. The number is also 45 less than the number formed by reversing the digits of t6he original number. What is the original number?
It says use a system of 2 equations with2 variables to solve the problem but i dont know how to use a system of equations for this.

Answer 1

if, x= the tens digit
y= the ones digit
equation:
10x+y=3(x+Y)
10x+y=10y+x-45

solution:
9(7x-2y)=(0)
2(9x-9y)=(-45)

63x-18y=0
-(18x-18y)=-90

45x=90
x=2
y=7
the original number is 27

Answer 2

Call the first digit x and the second digit y. Then, create equations that represent the statements.

What is the actual value of a 2 digit number xy where x is the 10's digit and y is the 1's digit? The value of the number represented is actually 10(x) + y.

The sentance says that the value of the 2 digit number is 3 times the sum of the digits. Well, the sum of the digits is x + y. So,

10x + y = 3(x + y) is the first equation.

The only hard part left is to figure out what the value of the number would be if you reversed the digits. Isn't that the same thing as putting the number that was in the 1's place into the 10's place? The value of yx is 10y + x.

The second equation then is:

10x + y = 10y + x - 45

Now you have the 2 equations and 2 unknowns:

10x + y = 3(x + y)
10x + y = 10y + x -45

Putting them in standard form:

7x - 2y = 0
9x - 9y = -45

Ouch, multiply the top one by 9 and the bottom one by 2.

63x-18y = 0
18x-18y=-90

Look what happens when you subtract the bottom one from the top one:

45x = 90

So, x = 2. Let's plug that value into 7x-2y=0

14-2y = 0
y = 7.

Let's check it to see if it works. Add the digits and you get 9.

Is 27 the same as 3 times 9? Yes, that one worked.

Is 27 less than 72 by exactly 45? Yes, it is. So, the original number must have been 27.

Answer 3

Sparrowhawk's answer is correct. Some elaboration on how to solve a problem like this:

Let t be the tens digit and u the units digit of the number tu. However, we cannot use tu in an equation because this means the product of t and u. So to write the number correctly, we say that it is equal to 10t+u.
So, 3 times the sum of the digits is 3(t+u). Now we can write our first equation:

3(t+u) = 10t+u (a)

Then we are told that by reversing the digits, the original number is 45 less. We first form the new number by reversing the digits: 10u + t.
Now since the old number is less, we need to add 45 to 10u + t, e.g. 10u+t+45

Now we are ready to write our next equation:

10u+t = 10t + u + 45 (b)

Now solve (a) and (b) simultaneously and you will find that t=2 and u=7 so that the original number tu = 27.

Check;
27 = 3(2+7)
72 = 27 + 45

Now let's see how we can create our own word problem:

2^3 - 1 = 7
2^2 + 3 = 7

The cube of a certain number less one is 7.
The sum of the square of this number and 3 is
also 7. Find the number.

Get the idea? You can form your own problems and then practice solving them. Enjoy!

Answer 4

let's assume that the 10th digit is t, and the first digit is n, so the number you want is tn.
because t is the tenth digit, it is put in equations as 10t
so, the first equation is:
10t +n = 3*(t+n)
10t +n = 3t +3n
7t -2n = 0
7t = 2n
t = 2n/7

if you reverse the digits then the tenth digit would be n, and the first is t. The original numer +45 = the reverse number
10t +n +45 = 10n +t
9t +45 = 9n
9( 2n/7) +45 = 9n
9 [ (2n/7) +5 = n ]
{ (2n/7) +5 = n }*7
2n +35 = 7n
5n = 35
so n = 7
t= 2n/7 =2*7/7 = 2

so the number is 27, and the reverse number is 72

Answer 5

A system of equations is a set of equations or in this case just a pair of equations. Other systems (sets) may have more equations.

In your case
let x be the 10's place digit and y be the one's place digit then

1) 10x + y = 3(x + y)
2) 10x + y = 10y + x - 45

Now it is up to you to solve these for x and y to find the number

Answer 6

the first equation would be 10x + y = 3(x+y)
the second is 10y + x -45 = 10x + y
if you solve these two by substitution/elimination/whatever, you get y = 2/7 * x + 45/7

the only integer solution to this, (so that 0<=x<=9 and 0<=y<=9) is (2,7) if you make x any higher so that y is an integer, y will be greater than 9, so the number is 27

Answer 7

Well, for these questions, you have to see what can be made into variables. In this case, it is a 2-digit number with information on the digits. Thus, each digit can be a variable!

If the two digit number is ab, where a and b are the digits, it can be represented by the equation 10a + b. Now use this to create your system of equations.

Try it yourself :-D

Answer 8

The two equations are:
3*(a+b)=a*10+b
a*10+b+45=b*10+a

Where the orginal number is ab. Note that when we write a number xyz its the same as x*100+y*10+z