# 7.8 Solving systems with cramer's rule  (Page 5/11)

 Page 5 / 11

## Verbal

Explain why we can always evaluate the determinant of a square matrix.

A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product—even if it does end up being 0.

Examining Cramer’s Rule, explain why there is no unique solution to the system when the determinant of your matrix is 0. For simplicity, use a $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ matrix.

Explain what it means in terms of an inverse for a matrix to have a 0 determinant.

The inverse does not exist.

The determinant of $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is 3. If you switch the rows and multiply the first row by 6 and the second row by 2, explain how to find the determinant and provide the answer.

## Algebraic

For the following exercises, find the determinant.

$|\begin{array}{cc}1& 2\\ 3& 4\end{array}|$

$-2$

$|\begin{array}{rr}\hfill -1& \hfill 2\\ \hfill 3& \hfill -4\end{array}|$

$|\begin{array}{rr}\hfill 2& \hfill -5\\ \hfill -1& \hfill 6\end{array}|$

$7$

$|\begin{array}{cc}-8& 4\\ -1& 5\end{array}|$

$|\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 3& \hfill -4\end{array}|$

$-4$

$|\begin{array}{rr}\hfill 10& \hfill 20\\ \hfill 0& \hfill -10\end{array}|$

$|\begin{array}{cc}10& 0.2\\ 5& 0.1\end{array}|$

$0$

$|\begin{array}{rr}\hfill 6& \hfill -3\\ \hfill 8& \hfill 4\end{array}|$

$|\begin{array}{rr}\hfill -2& \hfill -3\\ \hfill 3.1& \hfill 4,000\end{array}|$

$-7,990.7$

$|\begin{array}{rr}\hfill -1.1& \hfill 0.6\\ \hfill 7.2& \hfill -0.5\end{array}|$

$|\begin{array}{rrr}\hfill -1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill -3\end{array}|$

$3$

$|\begin{array}{rrr}\hfill -1& \hfill 4& \hfill 0\\ \hfill 0& \hfill 2& \hfill 3\\ \hfill 0& \hfill 0& \hfill -3\end{array}|$

$|\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}|$

$-1$

$|\begin{array}{rrr}\hfill 2& \hfill -3& \hfill 1\\ \hfill 3& \hfill -4& \hfill 1\\ \hfill -5& \hfill 6& \hfill 1\end{array}|$

$|\begin{array}{rrr}\hfill -2& \hfill 1& \hfill 4\\ \hfill -4& \hfill 2& \hfill -8\\ \hfill 2& \hfill -8& \hfill -3\end{array}|$

$224$

$|\begin{array}{rrr}\hfill 6& \hfill -1& \hfill 2\\ \hfill -4& \hfill -3& \hfill 5\\ \hfill 1& \hfill 9& \hfill -1\end{array}|$

$|\begin{array}{rrr}\hfill 5& \hfill 1& \hfill -1\\ \hfill 2& \hfill 3& \hfill 1\\ \hfill 3& \hfill -6& \hfill -3\end{array}|$

$15$

$|\begin{array}{rrr}\hfill 1.1& \hfill 2& \hfill -1\\ \hfill -4& \hfill 0& \hfill 0\\ \hfill 4.1& \hfill -0.4& \hfill 2.5\end{array}|$

$|\begin{array}{rrr}\hfill 2& \hfill -1.6& \hfill 3.1\\ \hfill 1.1& \hfill 3& \hfill -8\\ \hfill -9.3& \hfill 0& \hfill 2\end{array}|$

$-17.03$

$|\begin{array}{ccc}-\frac{1}{2}& \frac{1}{3}& \frac{1}{4}\\ \frac{1}{5}& -\frac{1}{6}& \frac{1}{7}\\ 0& 0& \frac{1}{8}\end{array}|$

For the following exercises, solve the system of linear equations using Cramer’s Rule.

$\begin{array}{l}2x-3y=-1\\ 4x+5y=9\end{array}$

$\left(1,1\right)$

$\begin{array}{r}5x-4y=2\\ -4x+7y=6\end{array}$

$\left(\frac{1}{2},\frac{1}{3}\right)$

$\begin{array}{l}2x+6y=12\\ 5x-2y=13\end{array}$

$\left(2,5\right)$

$\begin{array}{l}10x-6y=2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\hfill \\ -5x+8y=-1\hfill \end{array}$

$\begin{array}{l}4x-3y=-3\\ 2x+6y=-4\end{array}$

$\left(-1,-\frac{1}{3}\right)$

$\begin{array}{r}4x-5y=7\\ -3x+9y=0\end{array}$

$\begin{array}{l}4x+10y=180\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\hfill \\ -3x-5y=-105\hfill \end{array}$

$\left(15,12\right)$

For the following exercises, solve the system of linear equations using Cramer’s Rule.

$\left(1,3,2\right)$

$\left(-1,0,3\right)$

$\begin{array}{r}4x-3y+4z=10\\ 5x-2z=-2\\ 3x+2y-5z=-9\end{array}$

$\left(\frac{1}{2},1,2\right)$

$\begin{array}{r}\hfill 5x+2y-z=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill -7x-8y+3z=1.5\\ \hfill 6x-12y+z=7\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$

$\left(2,1,4\right)$

Infinite solutions

$\begin{array}{r}\hfill 4x-6y+8z=10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill -2x+3y-4z=-5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill 12x+18y-24z=-30\end{array}$

## Technology

For the following exercises, use the determinant function on a graphing utility.

$|\begin{array}{rrrr}\hfill 1& \hfill 0& \hfill 8& \hfill 9\\ \hfill 0& \hfill 2& \hfill 1& \hfill 0\\ \hfill 1& \hfill 0& \hfill 3& \hfill 0\\ \hfill 0& \hfill 2& \hfill 4& \hfill 3\end{array}|$

$24$

$|\begin{array}{rrrr}\hfill 1& \hfill 0& \hfill 2& \hfill 1\\ \hfill 0& \hfill -9& \hfill 1& \hfill 3\\ \hfill 3& \hfill 0& \hfill -2& \hfill -1\\ \hfill 0& \hfill 1& \hfill 1& \hfill -2\end{array}|$

$|\begin{array}{rrrr}\hfill \frac{1}{2}& \hfill 1& \hfill 7& \hfill 4\\ \hfill 0& \hfill \frac{1}{2}& \hfill 100& \hfill 5\\ \hfill 0& \hfill 0& \hfill 2& \hfill 2,000\\ \hfill 0& \hfill 0& \hfill 0& \hfill 2\end{array}|$

$1$

$|\begin{array}{rrrr}\hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 2& \hfill 3& \hfill 0& \hfill 0\\ \hfill 4& \hfill 5& \hfill 6& \hfill 0\\ \hfill 7& \hfill 8& \hfill 9& \hfill 0\end{array}|$

## Real-world applications

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.

Two numbers add up to 56. One number is 20 less than the other.

Yes; 18, 38

Two numbers add up to 104. If you add two times the first number plus two times the second number, your total is 208

Three numbers add up to 106. The first number is 3 less than the second number. The third number is 4 more than the first number.

Yes; 33, 36, 37

Three numbers add to 216. The sum of the first two numbers is 112. The third number is 8 less than the first two numbers combined.

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.

You invest $10,000 into two accounts, which receive 8% interest and 5% interest. At the end of a year, you had$10,710 in your combined accounts. How much was invested in each account?

$7,000 in first account,$3,000 in second account.

given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
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Abhi
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salma
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salma
Commplementary angles
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Sherica
im all ears I need to learn
Sherica
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Tamia
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Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice