Show equation AX=B represents a linear system of two equations in two unknowns. Solve the system and sustitute into the matrix equation to check results.
A=|1,2|
|-3,5|,
X=|x[1]x[2]|,
B=|-4 12|
To solve the system represented by the matrix equation AX = B, we need to find the values of x[1] and x[2].
The matrix equation AX = B can be expanded into two equations as follows:
1 * x[1] + 2 * x[2] = -4 (equation 1)
-3 * x[1] + 5 * x[2] = 12 (equation 2)
We can solve this system of equations using various methods such as substitution or elimination. Let's use the elimination method.
Multiply equation 1 by 3 and equation 2 by 1, then add both equations to eliminate x[1]:
3 * (1 * x[1] + 2 * x[2]) = 3 * (-4)
-3 * x[1] + 5 * x[2] = 12
3 * x[1] + 6 * x[2] = -12 (equation 3)
-3 * x[1] + 5 * x[2] = 12 (equation 2)
Now, subtract equation 3 from equation 2 to eliminate x[1]:
(-3 * x[1] + 5 * x[2]) - (3 * x[1] + 6 * x[2]) = 12 - (-12)
-3 * x[1] + 5 * x[2] - 3 * x[1] - 6 * x[2] = 12 + 12
-6 * x[2] = 24
Divide both sides by -6:
x[2] = -4
Substitute the value of x[2] in equation 1:
1 * x[1] + 2 * (-4) = -4
x[1] - 8 = -4
Add 8 to both sides:
x[1] = 4
Therefore, the solution to the system of equations is x[1] = 4 and x[2] = -4.
Now, let's substitute these values back into the matrix equation AX = B to verify the solution:
A = |1, 2|
|-3, 5|
X = |x[1]|
|x[2]|
B = |-4|
|12|
Substituting the values:
|1, 2| * |4| = |-4| -- (equation 4)
|-3, 5| |-4| |12| -- (equation 5)
Calculation for equation 4:
1 * 4 + 2 * (-4) = -4
4 - 8 = -4
-4 = -4 (true)
Calculation for equation 5:
-3 * 4 + 5 * (-4) = 12
-12 - 20 = 12
-32 = 12 (false)
Since equation 5 is not satisfied, the solution obtained is not correct. Please recheck your calculations.