1)Cramer's Rule is used to solve the system of equations:3m-5n=12,4m+7n=-5
Which determinant represents the numerator for n?
answer=[12 -5]
[-5 7]
2)Cramer's Rule is used to solve the system of equations:3x-y+2z=17,4x+2y-3z=10,and 2x+5y-9z=-6
Which determinant represents the numerator for z?
answer=[3 -1 17]
[4 2 10]
[2 5 -6]
3)Which matrix would NOT be used to write a matrix equation for the system of equations:5m-2n=13 and -m+n=-2?
answer=[5 -2]
[-1 1]
4)Which product would be used to solve the matrix equation below by using inverse matrices?
[3 -4][a] [6]
[2 1] [b]= [7]
answer=1/11[1 4] [6]
[-2 3][7]
Thanks for the help:)
[3 12]
[4 -5]
is the answer for the 1
I am so stuck on these same exact problems on my Algebra 2 test.
You're welcome! Here are the step-by-step explanations for each question:
1) Cramer's Rule is used to solve the system of equations: 3m - 5n = 12 and 4m + 7n = -5. The determinant for finding the value of n's numerator is the determinant of the coefficients of n, which is represented by:
Answer: determinant = [12 -5]
[-5 7]
2) Cramer's Rule is used to solve the system of equations: 3x - y + 2z = 17, 4x + 2y - 3z = 10, and 2x + 5y - 9z = -6. The determinant for finding the value of z's numerator is the determinant of the coefficients of z, which is represented by:
Answer: determinant = [3 -1 17]
[4 2 10]
[2 5 -6]
3) To write a matrix equation for the system of equations 5m - 2n = 13 and -m + n = -2, we need to determine the matrix that represents the coefficients of m and n. The matrix that would NOT be used for this purpose is:
Answer: matrix = [5 -2]
[-1 1]
4) To solve the matrix equation [3 -4] [a] = [6] [2 1] [b] = [7] using inverse matrices, we need to find the product of the inverse of the coefficient matrix and the solution matrix. The product is given by:
Answer: product = (1/11) [1 4] [6]
[-2 3] [7]
I hope this helps! Let me know if you have any more questions.
1) To find the numerator for n using Cramer's Rule, you need to create a matrix involving the coefficients of the equations:
[3 -5]
[4 7]
Then, you calculate the determinant of this matrix:
Numerator for n = det([3 -5], [4 7]) = (3 * 7) - (4 * -5) = 21 + 20 = 41
So, the correct answer is the matrix [12 -5; -5 7].
2) To find the numerator for z using Cramer's Rule, you create a matrix using the coefficients of the equations:
[3 -1 17]
[4 2 10]
[2 5 -6]
Then, calculate the determinant of this matrix:
Numerator for z = det([3 -1 17], [4 2 10], [2 5 -6]) = 3(2(-6) - 10(5)) + (-1)(4(-6) - 10(2)) + 17(4(5) - 2(2))
= 3(-12 - 50) + (-1)(-24 - 20) + 17(20 - 4)
= 3(-62) + (1)(44) + 17(16)
= -186 + 44 + 272
= 130
So, the correct answer is the matrix [3 -1 17; 4 2 10; 2 5 -6].
3) To determine the matrix that would NOT be used to write a matrix equation for the given system of equations, you need to examine the coefficients of the equations:
5m - 2n = 13
-m + n = -2
The matrix form of this system of equations would be:
[5 -2]
[-1 1]
So, the correct answer is the matrix [5 -2; -1 1].
4) To solve the matrix equation [3 -4][a] = [6] and [2 1][b] = [7] using inverse matrices, you need to find the inverse of the coefficient matrix [3 -4; 2 1]. Let's call this matrix A.
First, calculate the determinant of A:
det(A) = (3 * 1) - (2 * -4) = 3 + 8 = 11
Next, find the inverse of A:
A^(-1) = (1/det(A)) * adj(A)
where adj(A) is the adjugate of matrix A.
adj(A) = [1 -4; -2 3]
So, A^(-1) = (1/11) * [1 -4; -2 3] = [1/11 -4/11; -2/11 3/11]
Finally, multiply the inverse matrix with the given vector [6; 7]:
[a] = A^(-1) * [6] = [1/11 -4/11] * [6] = (1/11)(1 * 6 + -4 * 7) = (1/11)(6 - 28) = (1/11)(-22) = -2
[b] = A^(-1) * [7] = [1/11 -4/11] * [7] = (1/11)(1 * 7 + -4 * 7) = (1/11)(7 - 28) = (1/11)(-21) = -3
So, the product used to solve the matrix equation is [a] = -2 and [b] = -3.