a piece of wire that is 58cm long is cut into three pieces. the longest piece is 6cm shorter than the sum of the lengths of the other two pieces. five times the lengths of the shortest piece is 10cm less than the sum of the lengths of the other two pieces. determine the lengths of the three pieces of wire.
someone please help me?!
What?!?!?!!? I just can’t do it. It doesn’t make any sense to me. 😠🤯
represent the unknowns using variables:
let x = shortest piece
let y = 2nd longest piece
let (x+y) - 6 = longest piece *according to the 2nd statement*
set up equations:
(1) x + y + (x+y) - 6 = 58
*total length must be 58*
(2) 5x = y + (x+y) - 6 - 10
*according to 3rd statement*
these equations are simplified to:
(1) y = 32 - x
(2) y = 2x + 8
we then substitute the first equation to the second, o that everything will be in terms of x,, or you can do the other way around:
32 - x = 2x + 8
-3x = -24
x = 8 cm [length of shortest]
*using the simplifies equation (1), we get y:
y = 32-x = 24 cm [length of 2nd longest]
(x+y)-6 = 26 cm [length of longest]
**to check, see if their sum is equal to 58 cm,,
so there,, :)
Wow. Good job 👏
To solve this problem, we can break it down into steps:
Step 1: Assign variables
Let's assign variables to the lengths of the three pieces of wire. We'll call the lengths of the three pieces x, y, and z, where x is the shortest piece, y is the medium-length piece, and z is the longest piece.
Step 2: Set up equations
Based on the information given, we can set up two equations:
Equation 1: "the longest piece is 6cm shorter than the sum of the lengths of the other two pieces"
z = x + y - 6
Equation 2: "five times the lengths of the shortest piece is 10cm less than the sum of the lengths of the other two pieces"
5x = y + z - 10
Step 3: Solve the system of equations
We now have a system of two equations with three variables. To solve this system, we can use substitution.
Substitute Equation 1 into Equation 2:
5x = y + (x + y - 6) - 10
Simplify:
5x = 2y + x - 16
Combine like terms:
4x - 2y = -16
Step 4: Continue with the system of equations
Now we have a new equation derived from the system:
4x - 2y = -16
Combine Equation 1 and Equation 2:
z = x + y - 6 (Equation 1)
5x = y + z - 10 (Equation 2)
Rearrange Equation 2:
y = 5x - z + 10
Substitute y into the first equation:
z = x + (5x - z + 10) - 6
Combine like terms:
z = 6x + 4
Now we have two equations:
4x - 2y = -16
z = 6x + 4
Step 5: Solve for x
We will solve the first equation for y in terms of x, and then substitute it into the second equation:
4x - 2(5x - z + 10) = -16
4x - 10x + 2z - 20 = -16
-6x + 2z = 4
Multiply the equation by 3 to eliminate the coefficients:
-18x + 6z = 12
Combine this equation with the second equation:
-18x + 6z = 12
z = 6x + 4
Rearrange the second equation:
18x - 6z = -12
Add both equations:
0 = 0
Step 6: Analyze the result
After solving the system of equations, we find that the resulting equation is always true, indicating that x, y, and z can take on any real values, satisfying the given conditions. This means that there is no unique solution for the lengths of the three pieces of wire.