College Algebra Solutions for Your Success

The reduced row echelon form of the augmented matrix of a system of three linear equations in three variables must be of the form

$\left[\begin{array}{cccc}1& 0& 0& a\\ 0& 1& 0& b\\ 0& 0& 1& c\end{array}\right]$

where a, b, c, are real numbers. Justify your answer.

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent.

$\left[\begin{array}{ccccc}1& 0& 0& |& -1\\ 0& 1& 0& |& 3\\ 0& 0& 1& |& 4\\ 0& 0& 0& |& 0\end{array}\right]$

find an equation of the form $y=a{x}^2+bx+c$, whose graph passess through the points (2, 3), (-2, 7) and (1, -2). You must use a system of linear equations in three variables for this problem.

If a, b are elements of a ring and m, $n\in Z$, show that $(na)(mb)=(mn)(ab)$

The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.
${\displaystyle {\lambda }_1=1\Rightarrow \left\{\left[\begin{array}{c}2\\ -1\end{array}\right]\right\},{\lambda }_2=3\Rightarrow \left\{\left[\begin{array}{c}3\\ 1\end{array}\right]\right\}}$

Solve the system of linear equations using the substitution method.
$x+y-z=4$

$3x+2y+4z=17$

$-x+5y+z=8$

The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.
${\displaystyle \lambda 1=-1\to \left\{\begin{array}{cc}1& 1\end{array}\right\},\lambda 2=2\to \left\{\begin{array}{cc}1& -1\end{array}\right\}}$

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent. $\left[\begin{array}{ccccc}1& 0& -1& |& 1\\ 0& 1& 2& |& 1\end{array}\right]$

Describe the transformations that must be applied to $y=x^2$ to create the graph of each of the following functions.
a) ${\displaystyle y=\frac{1}{4}{(x-3)}^2+9}$
b) ${\displaystyle y={\left(\left(\frac{1}{2}\right)x\right)}^2-7}$

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $x_1,x_2\dots$ from left to right. $[1,2,0,2,-1,3]$

A 30-cm-diameter wheel is turning at a speed of 3 radians per second. What is the linear speed, in meters per minute, of a point on its rim?

if x, y belong to $R^p$, than is it true that the relation norm $(x+y)=norm(x)+norm(y)$ holds if and only if $x=cyory=cxwithc>0$

${\displaystyle f\left(x\right)=2x}$
${\displaystyle \frac{f(x+\mathrm{△}x)-f\left(x\right)}{\mathrm{△}}x}$

For the matrix A below, find a nonzero vector in the null space of A and a nonzero vector in the column space of A
$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$ 
Find a vector in the null space of A that is not the zero vector 
$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$