The value of a certain two-digit number is eight times he sum of its digits. if the digits of the number are reversed the result is 45 less than the original number. find the original number.

**please show both equations and work... thanks =)

let the unit digit be x

let the tens digit be y

then 10y + x = 8(x+y)
10y + x = 8x + 8y
2y = 7x
y = 7x/2

the number reversed would be 10x+y

then 10y+x - (10x+y) = 45
9y -9x = 45
-x+y=5
sub in the 1st equation
-x + 7x/2 = 5
-2x + 7x = 10
x = 2
then y = 7

original number is 72

when a number is squared ,the result is five times the original number find the number

Let's assume the original two-digit number is represented as "10x + y," where x represents the tens digit and y represents the units digit.

According to the given information, the value of the number is eight times the sum of its digits. This can be represented as:

10x + y = 8(x + y)

Simplifying the equation:

10x + y = 8x + 8y

Rearranging the equation:

2x - 7y = 0 --(Equation 1)

Next, we know that if the digits of the number are reversed, the result is 45 less than the original number. This can be represented as:

10y + x = 10x + y - 45

Simplifying the equation:

9y - 9x = 45

Dividing both sides by 9:

y - x = 5 --(Equation 2)

Now we have a system of two equations (Equation 1 and Equation 2). We can solve this system to find the values of x and y.

Let's solve this system of equations using the elimination method:

Multiply Equation 2 by 2:

2(y - x) = 2(5)
2y - 2x = 10

Now, add this equation to Equation 1:

2x - 7y + 2y - 2x = 0 + 10
-5y = 10

Divide by -5:

y = -2

Substitute the value of y into Equation 2:

-2 - x = 5
-x = 7
x = -7

Therefore, the original number is 10x + y = 10(-7) + (-2) = -70 - 2 = -72.

However, since we are looking for a two-digit number, we disregard the negative sign. Therefore, the original number is 72.

To find the original number, let's first represent it as a two-digit number with the tens digit as "x" and the units digit as "y". Therefore, the original number can be written as 10x + y.

According to the given information:

1) "The value of the number is eight times the sum of its digits:"

So, the equation is: 10x + y = 8(x + y)

2) "If the digits of the number are reversed, the result is 45 less than the original number:"

When the digits are reversed, the new number becomes 10y + x. Therefore, we can write the equation as: 10x + y - 45 = 10y + x

Now, let's solve these two equations simultaneously to find the values of x and y:

Equation 1: 10x + y = 8(x + y)
Expanding: 10x + y = 8x + 8y
Simplifying: 2x - 7y = 0

Equation 2: 10x + y - 45 = 10y + x
Expanding: 9x - 9y = 45
Simplifying: x - y = 5

Now, we have a system of equations:
2x - 7y = 0 ----(A)
x - y = 5 ----(B)

To solve, we can multiply equation (B) by 2 to eliminate the x variable:
2(x - y) = 2(5)
2x - 2y = 10

Now, we can substitute this equation into equation (A):
2x - 7y = 0
(2x - 2y) - 5y = 0
10 - 5y = 0
-5y = -10
y = 2

By substituting the value of y into equation (B):
x - 2 = 5
x = 7

Therefore, the original two-digit number is 10x + y = 10(7) + 2 = 72.