Show equation AX=B represents a linear system of two equations in two unknowns. Solve the system and substitute into the matrix equation to check results. I have no idea can you help me out
A=|1,2|
|-3,5|,
X=|x[1]x[2]|,
B=|-4 12|
The matrix equation AX=B results in the system of equations
x1 + 2x2 = -4
-3x1 + 5x2 = 12
I am sure you can solve that.
Sure! To solve the linear system represented by the equation AX = B, we can follow these steps:
Step 1: Write down the matrix equation AX = B:
|1 2| |x[1]| |-4 |
|-3 5| |x[2]| = |12|
Step 2: Multiply the matrix A by the vector X:
|1 * x[1] + 2 * x[2]| |-4|
|-3 * x[1] + 5 * x[2]| = |12|
Simplify:
1 * x[1] + 2 * x[2] = -4 --> Equation 1
-3 * x[1] + 5 * x[2] = 12 --> Equation 2
So we have the linear system:
Equation 1: x[1] + 2x[2] = -4
Equation 2: -3x[1] + 5x[2] = 12
Step 3: Solve the linear system.
To solve the system, we can choose a method like substitution, elimination, or matrix inversion. Let's use the substitution method:
From Equation 1: x[1] = -4 - 2x[2]
Substitute x[1] into Equation 2:
-3 * (-4 - 2x[2]) + 5x[2] = 12
Simplify:
12 + 6x[2] + 5x[2] = 12
11x[2] + 12 = 12
11x[2] = 0
x[2] = 0
Substitute x[2] = 0 into Equation 1:
x[1] + 2 * 0 = -4
x[1] = -4
So the solution to the system is x[1] = -4 and x[2] = 0.
Step 4: Substitute the solution into the matrix equation to check the results:
Substitute x[1] = -4 and x[2] = 0 into AX = B:
|1 2| |-4| |-4 |
|-3 5| | 0| = |12|
Multiply the matrices:
|-4 + 0| |-4|
|-3 * -4 + 0 * 5| = |12|
Simplify:
|-4| |-4|
|12| = |12|
The two matrices are equal, which confirms that our solution is correct.
Therefore, the solution to the linear system represented by the equation AX = B is x[1] = -4 and x[2] = 0.