Answer:
(a)![x_1=-2,x_2=1](https://tex.z-dn.net/?f=x_1%3D-2%2Cx_2%3D1)
(b)![x_1=\frac{3}{4} ,x_2=-\frac{1}{2} ,x_3=\frac{1}{4}](https://tex.z-dn.net/?f=x_1%3D%5Cfrac%7B3%7D%7B4%7D%20%2Cx_2%3D-%5Cfrac%7B1%7D%7B2%7D%20%2Cx_3%3D%5Cfrac%7B1%7D%7B4%7D)
Step-by-step explanation:
(a) For using Cramer's rule you need to find matrix
and the matrix
for each variable. The matrix
is formed with the coefficients of the variables in the system. The first step is to accommodate the equations, one under the other, to get
more easily.
![2x_1+5x_2=1\\x_1+4x_2=2](https://tex.z-dn.net/?f=2x_1%2B5x_2%3D1%5C%5Cx_1%2B4x_2%3D2)
![\therefore A=\left[\begin{array}{cc}2&5\\1&4\end{array}\right]](https://tex.z-dn.net/?f=%5Ctherefore%20A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D2%265%5C%5C1%264%5Cend%7Barray%7D%5Cright%5D)
To get
, replace in the matrix A the 1st column with the results of the equations:
![B_1=\left[\begin{array}{cc}1&5\\2&4\end{array}\right]](https://tex.z-dn.net/?f=B_1%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%265%5C%5C2%264%5Cend%7Barray%7D%5Cright%5D)
To get
, replace in the matrix A the 2nd column with the results of the equations:
![B_2=\left[\begin{array}{cc}2&1\\1&2\end{array}\right]](https://tex.z-dn.net/?f=B_2%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D2%261%5C%5C1%262%5Cend%7Barray%7D%5Cright%5D)
Apply the rule to solve
:
![x_1=\frac{det\left(\begin{array}{cc}1&5\\2&4\end{array}\right)}{det\left(\begin{array}{cc}2&5\\1&4\end{array}\right)} =\frac{(1)(4)-(2)(5)}{(2)(4)-(1)(5)} =\frac{4-10}{8-5}=\frac{-6}{3}=-2\\x_1=-2](https://tex.z-dn.net/?f=x_1%3D%5Cfrac%7Bdet%5Cleft%28%5Cbegin%7Barray%7D%7Bcc%7D1%265%5C%5C2%264%5Cend%7Barray%7D%5Cright%29%7D%7Bdet%5Cleft%28%5Cbegin%7Barray%7D%7Bcc%7D2%265%5C%5C1%264%5Cend%7Barray%7D%5Cright%29%7D%20%3D%5Cfrac%7B%281%29%284%29-%282%29%285%29%7D%7B%282%29%284%29-%281%29%285%29%7D%20%3D%5Cfrac%7B4-10%7D%7B8-5%7D%3D%5Cfrac%7B-6%7D%7B3%7D%3D-2%5C%5Cx_1%3D-2)
In the case of B2, the determinant is going to be zero. Instead of using the rule, substitute the values of the variable
in one of the equations and solve for
:
![2x_1+5x_2=1\\2(-2)+5x_2=1\\-4+5x_2=1\\5x_2=1+4\\ 5x_2=5\\x_2=1](https://tex.z-dn.net/?f=2x_1%2B5x_2%3D1%5C%5C2%28-2%29%2B5x_2%3D1%5C%5C-4%2B5x_2%3D1%5C%5C5x_2%3D1%2B4%5C%5C%205x_2%3D5%5C%5Cx_2%3D1)
(b) In this system, follow the same steps,ust remember
is formed by replacing the 3rd column of A with the results of the equations:
![2x_1+x_2 =1\\x_1+2x_2+x_3=0\\x_2+2x_3=0](https://tex.z-dn.net/?f=2x_1%2Bx_2%20%3D1%5C%5Cx_1%2B2x_2%2Bx_3%3D0%5C%5Cx_2%2B2x_3%3D0)
![\therefore A=\left[\begin{array}{ccc}2&1&0\\1&2&1\\0&1&2\end{array}\right]](https://tex.z-dn.net/?f=%5Ctherefore%20A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%261%260%5C%5C1%262%261%5C%5C0%261%262%5Cend%7Barray%7D%5Cright%5D)
![B_1=\left[\begin{array}{ccc}1&1&0\\0&2&1\\0&1&2\end{array}\right]](https://tex.z-dn.net/?f=B_1%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%261%260%5C%5C0%262%261%5C%5C0%261%262%5Cend%7Barray%7D%5Cright%5D)
![B_2=\left[\begin{array}{ccc}2&1&0\\1&0&1\\0&0&2\end{array}\right]](https://tex.z-dn.net/?f=B_2%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%261%260%5C%5C1%260%261%5C%5C0%260%262%5Cend%7Barray%7D%5Cright%5D)
![B_3=\left[\begin{array}{ccc}2&1&1\\1&2&0\\0&1&0\end{array}\right]](https://tex.z-dn.net/?f=B_3%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%261%261%5C%5C1%262%260%5C%5C0%261%260%5Cend%7Barray%7D%5Cright%5D)
![x_1=\frac{det\left(\begin{array}{ccc}1&1&0\\0&2&1\\0&1&2\end{array}\right)}{det\left(\begin{array}{ccc}2&1&0\\1&2&1\\0&1&2\end{array}\right)} =\frac{1(2)(2)+(0)(1)(0)+(0)(1)(1)-(1)(1)(1)-(0)(1)(2)-(0)(2)(0)}{(2)(2)(2)+(1)(1)(0)+(0)(1)(1)-(2)(1)(1)-(1)(1)(2)-(0)(2)(0)}\\ x_1=\frac{4+0+0-1-0-0}{8+0+0-2-2-0} =\frac{3}{4} \\x_1=\frac{3}{4}](https://tex.z-dn.net/?f=x_1%3D%5Cfrac%7Bdet%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D1%261%260%5C%5C0%262%261%5C%5C0%261%262%5Cend%7Barray%7D%5Cright%29%7D%7Bdet%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D2%261%260%5C%5C1%262%261%5C%5C0%261%262%5Cend%7Barray%7D%5Cright%29%7D%20%3D%5Cfrac%7B1%282%29%282%29%2B%280%29%281%29%280%29%2B%280%29%281%29%281%29-%281%29%281%29%281%29-%280%29%281%29%282%29-%280%29%282%29%280%29%7D%7B%282%29%282%29%282%29%2B%281%29%281%29%280%29%2B%280%29%281%29%281%29-%282%29%281%29%281%29-%281%29%281%29%282%29-%280%29%282%29%280%29%7D%5C%5C%20x_1%3D%5Cfrac%7B4%2B0%2B0-1-0-0%7D%7B8%2B0%2B0-2-2-0%7D%20%3D%5Cfrac%7B3%7D%7B4%7D%20%5C%5Cx_1%3D%5Cfrac%7B3%7D%7B4%7D)
![x_2=\frac{det\left(\begin{array}{ccc}2&1&0\\1&0&1\\0&0&2\end{array}\right)}{det\left(\begin{array}{ccc}2&1&0\\1&2&1\\0&1&2\end{array}\right)} =\frac{(2)(0)(2)+(1)(0)(0)+(0)(1)(1)-(2)(0)(1)-(1)(1)(2)-(0)(0)(0)}{4} \\x_2=\frac{0+0+0-0-2-0}{4}=\frac{-2}{4}=-\frac{1}{2}\\x_2=-\frac{1}{2}](https://tex.z-dn.net/?f=x_2%3D%5Cfrac%7Bdet%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D2%261%260%5C%5C1%260%261%5C%5C0%260%262%5Cend%7Barray%7D%5Cright%29%7D%7Bdet%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D2%261%260%5C%5C1%262%261%5C%5C0%261%262%5Cend%7Barray%7D%5Cright%29%7D%20%3D%5Cfrac%7B%282%29%280%29%282%29%2B%281%29%280%29%280%29%2B%280%29%281%29%281%29-%282%29%280%29%281%29-%281%29%281%29%282%29-%280%29%280%29%280%29%7D%7B4%7D%20%5C%5Cx_2%3D%5Cfrac%7B0%2B0%2B0-0-2-0%7D%7B4%7D%3D%5Cfrac%7B-2%7D%7B4%7D%3D-%5Cfrac%7B1%7D%7B2%7D%5C%5Cx_2%3D-%5Cfrac%7B1%7D%7B2%7D)
![x_3=\frac{det\left(\begin{array}{ccc}2&1&1\\1&2&0\\0&1&0\end{array}\right)}{det\left(\begin{array}{ccc}2&1&0\\1&2&1\\0&1&2\end{array}\right)}=\frac{(2)(2)(0)+(1)(1)(1)+(0)(1)(0)-(2)(1)(0)-(1)(1)(0)-(0)(2)(1)}{4} \\x_3=\frac{0+1+0-0-0-0}{4}=\frac{1}{4}\\x_3=\frac{1}{4}](https://tex.z-dn.net/?f=x_3%3D%5Cfrac%7Bdet%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D2%261%261%5C%5C1%262%260%5C%5C0%261%260%5Cend%7Barray%7D%5Cright%29%7D%7Bdet%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D2%261%260%5C%5C1%262%261%5C%5C0%261%262%5Cend%7Barray%7D%5Cright%29%7D%3D%5Cfrac%7B%282%29%282%29%280%29%2B%281%29%281%29%281%29%2B%280%29%281%29%280%29-%282%29%281%29%280%29-%281%29%281%29%280%29-%280%29%282%29%281%29%7D%7B4%7D%20%5C%5Cx_3%3D%5Cfrac%7B0%2B1%2B0-0-0-0%7D%7B4%7D%3D%5Cfrac%7B1%7D%7B4%7D%5C%5Cx_3%3D%5Cfrac%7B1%7D%7B4%7D)
Answer:
p = 0.263473
Step-by-step explanation:
264/1002 = 0.263473
Answer:
k = 4
Step-by-step explanation:
In order for the claim to be true, (x-3) must be a factor of each of the terms in the first expression. That is, you must have that expression factor as ...
(x -3)(2x +5) +(x -3)(x +7)
Factoring out (x -3), we get ...
= (x -3)(2x +5 +x +7) = (x -3)(3x +12)
= 3(x -3)(x +4)
The value of k must be 4.
Answer:
he is 1,305 feet below sea level
Step-by-step explanation:
805 + 500 = 1305 ft
he is 1,305 feet below sea level
Answer:
A and C are parallel
Step-by-step explanation:
They both are 70 degrees