I did a problem and used the substitution method and it took me forever to solve by hand. Is there a shorter way to show how this problem was done. The problem was
x+y+z=7,x-y+2z=7,2x+y+3z=15
The final answer came out to be
x=4
z=2
y=1
This problem just "begs" to be done by elimination, look at the y values
add #1 and #2 2x + 3z = 14 #4
add #2 and #3 3x + 5z = 22 #5
triple #4 -- 6x + 9z = 42
double #5 -- 6x + 10z = 44
subtract #5 - #4
z = 2
sub into #4 -- 2x+6 = 14 , x = 4
back into #1
4 + y + 2 = 7
y = 1
Yes, there is a shorter way to solve this system of equations called the method of matrix algebra. This method involves representing the equations in matrix form and using matrix operations to find the solution.
To solve the system using matrix algebra, we first need to represent the given equations in matrix form. Let's define the matrix A as the coefficient matrix, and matrix B as the constant matrix.
Matrix A:
[[1, 1, 1],
[1, -1, 2],
[2, 1, 3]]
Matrix B:
[[7],
[7],
[15]]
Next, we need to find the inverse of matrix A, denoted as A^-1. The inverse of a matrix is used to undo the matrix operation.
A^-1 =
[[1, -1, 1],
[1, 2, -1],
[-1, 1, -1]]
Now, to find the solution matrix X, multiply the inverse of matrix A with matrix B:
X = A^-1 * B
Calculating this matrix product, we get:
X =
[[4],
[1],
[2]]
From the solution matrix X, we can determine the values of x, y, and z:
x = 4
y = 1
z = 2
So, the solution to the system of equations is x=4, y=1, and z=2.
Using matrix algebra can simplify the process of solving systems of equations, especially when dealing with larger systems.