the sum of two numbers is 51. Twice the first plus 4 times the second is 128. What are the numbers? Show your work.

let the first no be x

then the second no is 51-x

solve:
2x + 4(51-x) = 128

To solve this problem, we will set up a system of equations based on the given information. Let's call the first number x and the second number y.

The first equation is: "The sum of two numbers is 51."
So, we can translate this into the equation: x + y = 51. ----(Equation 1)

The second equation is: "Twice the first plus 4 times the second is 128."
We can translate this into the equation: 2x + 4y = 128. ----(Equation 2)

Now, we have a system of equations:
Equation 1: x + y = 51,
Equation 2: 2x + 4y = 128.

To solve this system of equations, we can use the method of substitution or elimination.

Let's solve by the method of substitution:
From Equation 1, we can isolate x: x = 51 - y.

Now, substitute this value of x into Equation 2:
2(51 - y) + 4y = 128.

Simplify and solve for y:
102 - 2y + 4y = 128,
102 + 2y = 128,
2y = 128 - 102,
2y = 26,
y = 26/2,
y = 13.

Now, substitute the value of y = 13 back into Equation 1 to find x:
x + 13 = 51,
x = 51 - 13,
x = 38.

Therefore, the two numbers are 38 and 13.

To summarize:
- We set up two equations from the given information.
- We solved the system of equations using the method of substitution.
- We found the values of x and y to be 38 and 13, respectively.

x=12