Find the values of the variables.
matricesStart 2 by 2 Matrix 1st Row 1st Column 15 2nd Column 8 2nd Row 1st Column 4 x 2nd Column 0 End Matrix minus Start 2 by 2 Matrix 1st Row 1st Column 2 2nd Column 5 2nd Row 1st Column x 2nd Column 3 y plus 6 End Matrix equals Start 2 by 2 Matrix 1st Row 1st Column 13 2nd Column 3 2nd Row 1st Column 12 2nd Column 4 y plus 8 End Matrix
(1 point)
Responses
x = –4 and y = 2
x = –4 and y = 2
x = 5 and y = –1
x = 5 and y = –1
x = 4 and y = –2
x = 4 and y = –2
x = –5 and y = 1
x = –5 and y = 1.
To solve the given system of equations, we can set up two algebraic equations based on the given matrices:
Equation 1: 15 - 2 * 0 - (2 - 5 * 3 * y + 6) = 13 - 3 + 12
Equation 2: 8 - 4 * 0 - (0 + x * 3 * y + 6) = 3 + 4 * y + 8
Simplifying Equation 1:
15 - 2 * 0 - (2 - 15 * y + 6) = 22
15 - (2 - 15y + 6) = 22
15 - 2 + 15y - 6 = 22
7 + 15y = 22
15y = 22 - 7
15y = 15
y = 15/15
y = 1
Now substituting y = 1 into Equation 2:
8 - 4 * 0 - (0 + x * 3 * 1 + 6) = 3 + 4 * 1 + 8
8 - (0 + 3x + 6) = 3 + 4 + 8
8 - 3 - 3x - 6 = 15
-3x - 1 = 15
-3x = 15 + 1
-3x = 16
x = 16/-3
x = -16/3
x ≈ -5.333
Therefore, the values of the variables are:
x ≈ -5.333 and y = 1
To find the values of the variables x and y, we need to solve the given system of equations using matrix operations. Let's break down the steps:
Step 1: Write the system of equations in matrix form
We can express the given system of equations in matrix form as follows:
[15 8][4 x] - [2 5][x 3y + 6] = [13 3][12 4y + 8]
Step 2: Perform matrix operations
We can subtract the matrices on each side of the equation:
[15 - 2 8 - 5][4 - x] = [13 - 12 3 - 4y - 8]
[13 3][12 - 4y]y
Simplifying further:
[13 3][12 - 4y] = [11 -9]
[12 - 4y]
Step 3: Equate corresponding elements
We can equate the corresponding elements on each side of the equation:
13 = 11
3 = -9
12 = 12
-4y = -9
From the first two equations, we can see that 13 is not equal to 11 and 3 is not equal to -9. Therefore, there is no solution for this system of equations.
Hence, there are no values of x and y that satisfy the given system of equations.