Systems of Equations and Inequalities

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}3x-2y=4\\ 6x-4y=0\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}-x+2y=5\\ 4x-8y=6\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}2x-4y=-2\\ 3x+2y=3\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}3x+3y=3\\ 4x+2y=\frac{8}{3}\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}2x-3y=-1\\ 10x+10y=5\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}3x-2y=0\\ 5x+10y=4\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}2x+3y=6\\ x-y=\frac{1}{2}\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}\frac{1}{2}x+y=-2\\ x-2y=8\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}2x-y=-1\\ x+\frac{1}{2}y=\frac{3}{2}\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x+y-z=6\\ 3x-2y+z=-5\\ x+3y-2z=14\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x-y+z=-4\\ 2x-3y+4z=-15\\ 5x+y-2z=12\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x+2y-z=-3\\ 2x-4y+z=-7\\ -2x+2y-3z=4\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x+4y-3z=-8\\ 3x-y+3z=12\\ x+y+6z=1\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x-2y+3z=1\\ 3x+y-2z=0\\ 2x-4y+6z=2\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x-y+2z=5\\ 3x+2y=4\\ -2x+2y-4z=-10\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x+2y-z=0\\ 2x-4y+z=0\\ -2x+2y-3z=0\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x+4y-3z=0\\ 3x-y+3z=0\\ x+y+6z=0\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x-2y+3z=0\\ 3x+y-2z=0\\ 2x-4y+6z=0\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x-y+2z=0\\ 3x+2y=0\\ -2x+2y-4z=0\end{array}\right.$

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that $\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4$

$\left|\begin{array}{ccc}1& 2& 3\\ u& v& w\\ x& y& z\end{array}\right|$

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that $\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4$

$\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 2& 4& 6\end{array}\right|$

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that $\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4$

$\left|\begin{array}{ccc}x& y& z\\ -3& -6& -9\\ u& v& w\end{array}\right|$

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that

$\left|\begin{array}{llr}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4$

$\left|\begin{array}{ccc}1& 2& 3\\ x-u& y-v& z-w\\ u& v& w\end{array}\right|$

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that 

$$\begin{gathered} \left|\begin{array}{llr}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4 \\ \left|\begin{array}{ccc}1& 2& 3\\ x-3& y-6& z-9\\ 2u& 2v& 2w\end{array}\right| \end{gathered}$$

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that 

$$\begin{gathered} \left|\begin{array}{llr}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4 \\ \left|\begin{array}{ccc}x& y& z-x\\ u& v& w-u\\ 1& 2& 2\end{array}\right| \end{gathered}$$

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that 

$$\begin{gathered} \left|\begin{array}{llr}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4 \\ \left|\begin{array}{ccc}1& 2& 3\\ 2x& 2y& 2z\\ u-1& v-2& w-3\end{array}\right| \end{gathered}$$

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that 

$$\begin{gathered} \left|\begin{array}{llr}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4 \\ \left|\begin{array}{ccc}x+3& y+6& z+9\\ 3u-1& 3v-2& 3w-3\\ 1& 2& 3\end{array}\right| \end{gathered}$$

In Problems 51–56, solve for x,

$\left|\begin{array}{ll}x& x\\ 4& 3\end{array}\right|=5$

In Problems 51–56, solve for x, 

$\left|\begin{array}{ll}x& 1\\ 3& x\end{array}\right|=-2$

In Problems 51–56, solve for x, 

$\left|\begin{array}{rrr}x& 1& 1\\ 4& 3& 2\\ -1& 2& 5\end{array}\right|=2$

In Problems 51–56, solve for x, 

$\left|\begin{array}{rrr}3& 2& 4\\ 1& x& 5\\ 0& 1& -2\end{array}\right|=0$

In Problems 51–56, solve for x, 

$\left|\begin{array}{rrr}x& 2& 3\\ 1& x& 0\\ 6& 1& -2\end{array}\right|=7$

In Problems 51–56, solve for x, 

$\left|\begin{array}{lll}x& 1& 2\\ 1& x& 3\\ 0& 1& 2\end{array}\right|=-4x$

Geometry: Equation of a Line An equation of the line containing the two points $\left(x_1,y_1\right) \text{and } \left(x_2,y_2\right)$ may be expressed as the determinant

$\left|\begin{array}{lll}x& y& 1\\ x_1& y_1& 1\\ x_2& y_2& 1\end{array}\right|=0$

Prove this result by expanding the determinant and comparing the result to the two-point form of the equation of a line.

Geometry: Collinear Points Using the result obtained in Problem 57, show that three distinct points $\left(x_1,y_1\right), \left(x_2,y_2\right),\text{ and }\left(x_3,y_3\right)$ are collinear (lie on the same line) if and only if

$\left|\begin{array}{lll}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

Geometry: Area of a Triangle A triangle has vertices $\left(x_1,y_1\right),\left(x_2,y_2\right),\text{ and }\left(x_3,y_3\right)$. The area of the triangle is given by the absolute value of D, where

$D=\frac{1}{2}\left|\begin{array}{ccc}{x}_1& x_2& x_3\\ y_1& y_2& y_3\\ 1& 1& 1\end{array}\right|$

Use this formula to find the area of a triangle with vertices $(2,3),(5,2)\text{, and }(6,5)$.

Show that $\left|\begin{array}{lll}{x}^2& x& 1\\ y^2& y& 1\\ z^2& z& 1\end{array}\right|=(y-z)(x-y)(x-z)$

Complete the proof of Cramer’s Rule for two equations containing two variables.

Interchange columns $1$ and $3$ of a $3$ by $3$ determinant. Show that the value of the new determinant is $-1$ times the value of the original determinant.

Multiply each entry in row $2$ of a $3$ by $3$ determinant by the number $k$, $k\ne 0$. Show that the value of the new determinant is k times the value of the original determinant.

Prove that a $3$ by $3$ determinant in which the entries in column $1$ equal those in column $3$ has the value $0$.

Prove that, if row $2$ of a $3$ by $3$ determinant is multiplied by k, $k\ne 0$, and the result is added to the entries in row $1$, there is no change in the value of the determinant.

A matrix that has the same number of rows as columns is

called a(n)_______ matrix.

True or False 

Matrix addition is commutative.

To find the product AB of two matrices A and B, the number

of _______ in matrix A must equal the number of_______ in

matrix B.

True or False 

Matrix multiplication is commutative.

Suppose that A is a square n by n matrix that is nonsingular.

The matrix B such that AB = BA = In is called the__________

of the matrix A.

If a matrix A has no inverse, it is called .

True or False 

The identity matrix has properties similar to

those of the real number 1.

If AX = B represents a matrix equation where A is a

nonsingular matrix, then we can solve the equation using

X = ________.