1) when using cramer's rule to solve the system of equations containing 5x - y = 5 and -3x + y = -9 the denominator for the x and y variables looks like which of these determinants ? a) |5 5 | b) |5 -1
|-3 9| |-3 1|
c) |5 -1|
|-9 1|
d) none of these
2) when using cramer's rule to solve the system of equations containing 3x - 6y = 9 and -4x + 2y = -6 the numerator for the x variable looks like which of these determinants ?
a) |3 -6| b) |9 -6| c) |-6 2|
|4 2| |-6 2| |3 4| d) none of these
can someone assist me with this? I'm having trouble
The denominator is composed of a determinant made up of the left-hand side of the equation.
The numerator of the variables is made up a determinant of the left-hand side, but with the corresponding column replaced with the vector of the right hand side.
Yes, I can assist you with this. Let's break down the process of using Cramer's rule to solve these systems of equations.
1) The denominator for the x and y variables in Cramer's rule involves the determinant of the coefficient matrix. Let's write down the coefficient matrix for the given system of equations:
| 5 -1 |
|-3 1 |
To find the determinant of this 2x2 matrix, we simply multiply the numbers on the main diagonal (5 * 1) and subtract the product of the numbers on the other diagonal (-1 * -3):
Determinant = (5 * 1) - (-1 * -3)
= (5 * 1) - (1 * 3)
= 5 - 3
= 2
So, the determinant of the coefficient matrix is 2. Now let's look at the given options:
a) |5 5 |
|-3 9|
Here, the determinant of the coefficient matrix is (5 * 9) - (5 * -3) = 45 - (-15) = 60. This option does not match the calculated determinant of 2.
b) |5 -1 |
|-3 1 |
Here, the determinant of the coefficient matrix is (5 * 1) - (-1 * -3) = 5 - 3 = 2. This option matches the calculated determinant of 2.
c) |5 -1 |
|-9 1 |
Here, the determinant of the coefficient matrix is (5 * 1) - (-9 * -1) = 5 - 9 = -4. This option does not match the calculated determinant of 2.
d) none of these
Therefore, the correct answer is (b) |5 -1 |.
2) Now let's move on to the numerator for the x variable in Cramer's rule. The numerator involves replacing the x column in the coefficient matrix with the column on the right side of the equations. Let's write down the modified matrix for the given system of equations:
| 9 -6 |
|-6 2 |
To find the determinant of this modified 2x2 matrix, we follow the same process as before:
Determinant = (9 * 2) - (-6 * -6)
= 18 - 36
= -18
Now let's look at the options:
a) |3 -6 |
|4 2 |
Here, the determinant of the modified matrix is (3 * 2) - (-6 * 4) = 6 - (-24) = 6 + 24 = 30. This option does not match the calculated determinant of -18.
b) |9 -6 |
|3 4 |
Here, the determinant of the modified matrix is (9 * 4) - (-6 * 3) = 36 - (-18) = 36 + 18 = 54. This option does not match the calculated determinant of -18.
c) |-6 2 |
|-6 2 |
Here, the determinant of the modified matrix is (-6 * 2) - (2 * -6) = -12 - (-12) = -12 + 12 = 0. This option does not match the calculated determinant of -18.
d) none of these
Therefore, none of the options provided match the calculated determinant of -18, so the correct answer is (d) none of these.