4. The sum of the digits of a two-digit number is 11. If 45 is added to the number, the digits will be reversed.
a. Write a system of equations that you can use to find the digits of the number. Tell what each variable represents.
b. Solve the system of equations to find the two-digit number. Show your work.
A:
a. ?
b. ?
a. Let's assume that the tens digit of the two-digit number is represented by the variable 'x' and the units digit is represented by the variable 'y'.
From the given statement, "The sum of the digits of a two-digit number is 11", we can write the equation:
x + y = 11
Also, from the given statement, "If 45 is added to the number, the digits will be reversed", we get:
10y + x + 45 = 10x + y
b. To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution.
First, we can rearrange the first equation to solve for 'x':
x = 11 - y
Substituting this value of 'x' into the second equation, we get:
10y + (11 - y) + 45 = 10(11 - y) + y
10y + 11 - y + 45 = 110 - 10y + y
9y + 56 = 110 - 9y
18y = 54
y = 3
Substituting the value of 'y' back into the first equation, we can find 'x':
x + 3 = 11
x = 11 - 3
x = 8
Therefore, the digits of the two-digit number are 8 and 3. The number itself is 83.
a. To solve this problem, we can use a system of equations. Let's assume the tens digit of the number is represented by 'x' and the ones digit is represented by 'y'.
1. The sum of the digits is 11, so the first equation can be written as: x + y = 11.
2. If 45 is added to the number, the digits will be reversed. This means that the new number can be written as 10y + x (since the ones digit now becomes the tens digit and vice versa). So, the second equation can be written as: 10y + x = (10x + y) + 45.
b. Now, let's solve the system of equations to find the two-digit number.
Substituting the value of y from the first equation into the second equation, we get:
10(11 - x) + x = (10x + (11 - x)) + 45.
Simplifying the equation, we have:
110 - 10x + x = 10x + 11 + 45.
Combining like terms, we get:
-x + 110 = 10x + 56.
Adding x to both sides of the equation, we have:
110 = 11x + 56.
Subtracting 56 from both sides of the equation, we get:
54 = 11x.
Dividing both sides of the equation by 11, we get:
x = 54/11.
Since x represents the tens digit, and it must be a whole number, we find that x = 4.
Substituting x = 4 into the first equation, we find:
4 + y = 11.
Subtracting 4 from both sides of the equation, we get:
y = 11 - 4.
Therefore, y = 7.
So the two-digit number is 47.