PLEASE help with the below.
Solve the system of equations by first expressing it in matrix form as and then evaluating.
a). 3x-2y=5
4x-y =-10
b). 3x -2y =-2
4x -y = 3
Thank you.
Sure! To solve the system of equations using matrix form, we first need to express the system in the form of AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
a) For the system:
3x - 2y = 5
4x - y = -10
We can write it as:
| 3 -2 | | x | | 5 |
| 4 -1 | x | y | = | -10 |
The coefficient matrix A is:
A = | 3 -2 |
| 4 -1 |
The variable matrix X is:
X = | x |
| y |
And the constant matrix B is:
B = | 5 |
|-10 |
Now, let's evaluate the system of equations by solving for X.
To do this, we use the equation X = A^(-1) * B, where A^(-1) is the inverse of matrix A.
First, let's find the inverse of matrix A:
A = | 3 -2 |
| 4 -1 |
To find the inverse, we calculate the determinant of matrix A:
det(A) = (3 * -1) - (-2 * 4) = -3 + 8 = 5
Next, we calculate the adjugate of A:
adj(A) = | -1 2 |
| -4 3 |
Finally, we calculate the inverse of A:
A^(-1) = (1 / det(A)) * adj(A)
= (1 / 5) * | -1 2 |
| -4 3 |
= | -1/5 2/5 |
| -4/5 3/5 |
Now, we can evaluate X:
X = A^(-1) * B
= | -1/5 2/5 | * | 5 |
| -10 |
= | (-1/5 * 5) + (2/5 * -10) |
| (-4/5 * 5) + (3/5 * -10) |
= | (-1) + (-4) |
| (-4) + (-6) |
= | -5 |
|-10 |
Therefore, the solution to the system of equations is:
x = -5
y = -10
b) For the system:
3x - 2y = -2
4x - y = 3
We can write it as:
| 3 -2 | | x | | -2 |
| 4 -1 | x | y | = | 3 |
The coefficient matrix A is:
A = | 3 -2 |
| 4 -1 |
The variable matrix X is:
X = | x |
| y |
And the constant matrix B is:
B = | -2 |
| 3 |
Using the same steps as above, we can calculate the inverse of matrix A:
det(A) = (3 * -1) - (-2 * 4) = -3 + 8 = 5
adj(A) = | -1 2 |
| -4 3 |
A^(-1) = (1 / det(A)) * adj(A)
= (1 / 5) * | -1 2 |
| -4 3 |
= | -1/5 2/5 |
| -4/5 3/5 |
Now, we can evaluate X:
X = A^(-1) * B
= | -1/5 2/5 | * | -2 |
| 3 |
= | (-1/5 * -2) + (2/5 * 3) |
| (-4/5 * -2) + (3/5 * 3) |
= | (2/5) + (6/5) |
| (8/5) + (9/5) |
= | 8/5 |
| 17/5 |
Therefore, the solution to the system of equations is:
x = 8/5
y = 17/5
To solve this system of equations using matrix form, we need to represent the coefficients of the variables (x and y) and the constants on both sides of the equations in matrix form. Here's how you can do it:
a) 3x - 2y = 5
4x - y = -10
To represent the coefficients and the constants in matrix form:
| 3 -2 | | x | = | 5 |
| 4 -1 | | y | | -10 |
The left-hand side of the equation represents the coefficients of the variables (x and y) in the system of equations, while the right-hand side represents the constants.
To solve the equation, we need to evaluate the matrix equation in the form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
The equation would be: AX = B
Evaluating the equation for the given system of equations:
| 3 -2 | | x | = | 5 |
| 4 -1 | | y | | -10 |
Now, we need to find the inverse of matrix A. Let's call it A inverse.
A * A inverse = I (Identity matrix)
Now, multiply both sides of the equation by A inverse.
(A inverse) * A * X = (A inverse) * B
Since (A inverse) * A gives the identity matrix, we have:
I * X = (A inverse) * B
This simplifies to:
X = (A inverse) * B
We can use matrix multiplication to find the answer X.
b) 3x - 2y = -2
4x - y = 3
To represent the coefficients and the constants in matrix form:
| 3 -2 | | x | = | -2 |
| 4 -1 | | y | | 3 |
Follow the same steps as mentioned above to find the solution by evaluating the equation AX = B.
I hope this helps! Let me know if you have any further questions.
uggghhh, matrices for such simple systems ???
anyway ....
1st one:
3 -2 5
4 -1 -10
1 1 -15 ---> #2 - #1
3 -2 5 ----> #1
1 1 -15 ---> #1
0 5 -50 ---> 3x#1 - #2
1 1 -15 ---->#1
0 1 -10 ---> #2 ÷ 5
1 0 -5 ---#1 - #2
0 1 -10 ---> #2
so x = -5 and y = -10
---------------------------
same thing using substitution
form #2, y = 4x+10
sub into #1
3x - 2(4x+10) = 5
3x - 8x - 20 = 5
-5x = 25
x = -5 , then y = -20+10 = -10
You try the second one, remember in the matrix method there is no one correct way,
what I did above just seemed like a way to me, you might have taken a different path.