Write down the first 3 equations making x, y, and z as their subjects.
x = 4 - 3w
y = (1 + 2w) / 3 = (1/3) + (2w/3)
z = 2y - w = 2[(1/3) + (2w/3)] - w = (2/3) + (4w/3) - w = (2/3) + (w/3)
Then, substitute these equations as values of x, y, and z in the 4th equation. This way you will get an equation which is only in terms of 'w'. Now, you can make 'w' the subject and get its value. Like this:
You can solve this multiple ways. The most difficult method for solving this would probably be through using Row-Echelon form and the simplest, the method that I used, is by using a graphing calculator and plugging the numbers into two matrices. Your first matrix is going to be a 4x4 matrix b/c there are four columns and four rows (x,y,z,w) and your second matrix is going to be a 4x1 and it is all the numbers on the right side of the equal sign (4,0,1,5).
So matrix A will look like this:
[1 0 0 3]
[0 2 -1 -1]
[0 3 0 -2]
[2 -1 0 4]
And matrix B will look like this:
[4]
[0]
[1]
[5]
After entering each number into its correct spot in its respective matrix you take the first matrix, matrix A and multiply it by its inverse by using the x^-1 button on your calculator and then multiply that answer by matrix B. On the screen it ends up looking like: [A]^-1 [B]. And that is how you find your answer(s).
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Verified answer
Hi Hannah,
Write down the first 3 equations making x, y, and z as their subjects.
x = 4 - 3w
y = (1 + 2w) / 3 = (1/3) + (2w/3)
z = 2y - w = 2[(1/3) + (2w/3)] - w = (2/3) + (4w/3) - w = (2/3) + (w/3)
Then, substitute these equations as values of x, y, and z in the 4th equation. This way you will get an equation which is only in terms of 'w'. Now, you can make 'w' the subject and get its value. Like this:
Substituting values in equation 4:
2x - y + 4z = 5
2(4 - 3w) - [(1/3) + (2w/3)] + 4[(2/3) + (w/3)] = 5
8 - 6w - (1/3) - (2w/3) + (8/3) + (4w/3) = 5
(31/3) - (16w/3) = 5
31 - 16w = 15
-16w = -16
w = -16/-16 = 1 >>>>> value of 'w' found
Now, use the value of 'w' to find the values of x, y, and z from the initial equations that we made.
x = 4 - 3w = 4 - 3(1) = 1
y = (1/3) + (2w/3) = 1/3 + 2/3 = 3/3 = 1
z = (2/3) + (w/3) = 2/3 + 1/3 = 3/3 = 1
This is the actual way to solve the question. You can always use graphical calculators etc. but I prefer this method.
You can see how the matrices are made from Lucas's answer. tc:)
The answer is: x = 1; y = 1; z = 1; w = 1
You can solve this multiple ways. The most difficult method for solving this would probably be through using Row-Echelon form and the simplest, the method that I used, is by using a graphing calculator and plugging the numbers into two matrices. Your first matrix is going to be a 4x4 matrix b/c there are four columns and four rows (x,y,z,w) and your second matrix is going to be a 4x1 and it is all the numbers on the right side of the equal sign (4,0,1,5).
So matrix A will look like this:
[1 0 0 3]
[0 2 -1 -1]
[0 3 0 -2]
[2 -1 0 4]
And matrix B will look like this:
[4]
[0]
[1]
[5]
After entering each number into its correct spot in its respective matrix you take the first matrix, matrix A and multiply it by its inverse by using the x^-1 button on your calculator and then multiply that answer by matrix B. On the screen it ends up looking like: [A]^-1 [B]. And that is how you find your answer(s).