The ratio of the tens digit to the units digit is 1:4. If the digits are reversed, the sum of the new number and the original number is 110. Find the original number by using a systems of equations with elimination
well, with that ratio it has to be 14 or 28
I agree.
To solve the problem using a system of equations with elimination, we can let the tens digit be x and the units digit be y.
From the given information, we have two equations:
1) x/y = 1/4 (ratio of the tens digit to the units digit is 1:4)
2) (10y + x) + (10x + y) = 110 (sum of the original number and the new number is 110)
Let's solve this system of equations using the elimination method.
Multiplying equation 1 by 4 on both sides, we get:
4(x/y) = 4(1/4)
4x/y = 1
Now, we can eliminate the fraction by multiplying both sides of equation 2 by y:
y[(10y + x) + (10x + y)] = y(110)
10y^2 + xy + 10xy + y^2 = 110y
10y^2 + 11xy + y^2 - 110y = 0
11xy + 11y^2 - 110y = 0
xy + y^2 - 10y = 0
Now we have two equations in terms of xy and y:
4x = y (equation 3)
xy + y^2 - 10y = 0 (equation 4)
To solve the system of equations, we can substitute equation 3 into equation 4:
(4x)y + y^2 - 10y = 0
4xy + y^2 - 10y = 0
Combining like terms:
y^2 + 4xy - 10y = 0 (equation 5)
Now we have two equations:
4x = y (equation 3)
y^2 + 4xy - 10y = 0 (equation 5)
To solve this system of equations, we'll use the elimination method. Multiply equation 3 by 4:
16x = 4y (equation 6)
Now we'll subtract equation 6 from equation 5:
(y^2 + 4xy - 10y) - (16x - 4y) = 0
y^2 + 4xy - 10y - 16x + 4y = 0
y^2 + 4xy + 4y - 10y - 16x = 0
y^2 + 4xy - 6y - 16x = 0
Now we have the equation:
y^2 + 4xy - 6y - 16x = 0 (equation 7)
We can rearrange equation 7 to express x in terms of y:
x = (6y - y^2) / 4
Substituting this expression for x into equation 3, we get:
4(6y - y^2) / 4 = y
6y - y^2 = y
6y - y^2 - y = 0
y(6 - y - 1) = 0
y(5 - y) = 0
Therefore, y = 0 or y = 5.
Let's consider y = 0:
If y = 0, then x = (6(0) - (0)^2) / 4 = 0
So, one possible solution is x = 0 and y = 0, which represents the number 00. However, this is not a valid solution as there is no tens digit of 0.
Now, let's consider y = 5:
If y = 5, then x = (6(5) - (5)^2) / 4 = 15/4 = 3.75
Since x is not a whole number, this solution is also not valid.
Therefore, there are no valid solutions for the original number that satisfies the given conditions.