4/25/25\n7. find the determinant of the matrix $\begin{bmatrix}1&1\\ - 3&4end{bmatrix}$\n8. find the…

7. Find the determinant of the matrix $\begin{bmatrix}1&1\ - 3&4\end{bmatrix}$

Explanation:

Step1: Recall determinant formula

For a $2\times2$ matrix $\begin{bmatrix}a&b\c&d\end{bmatrix}$, the determinant is $ad - bc$. Here $a = 1$, $b = 1$, $c=-3$, $d = 4$. $1\times4-1\times(-3)$

Step2: Calculate the result

$4 + 3=7$

Answer:

$7$

8. Find the determinant of the matrix $\begin{bmatrix}10&-1&3\5&2&5\-2&1&2\end{bmatrix}$

Explanation:

Step1: Expand the determinant

[ \begin{align*} &10\times\begin{vmatrix}2&5\1&2\end{vmatrix}-(-1)\times\begin{vmatrix}5&5\-2&2\end{vmatrix}+3\times\begin{vmatrix}5&2\-2&1\end{vmatrix}\ \end{align*} ]

Step2: Calculate $2\times2$ determinants

For $\begin{vmatrix}2&5\1&2\end{vmatrix}=2\times2 - 5\times1=4 - 5=-1$; for $\begin{vmatrix}5&5\-2&2\end{vmatrix}=5\times2-5\times(-2)=10 + 10 = 20$; for $\begin{vmatrix}5&2\-2&1\end{vmatrix}=5\times1-2\times(-2)=5 + 4 = 9$. [ \begin{align*} &10\times(-1)+1\times20 + 3\times9\ &=-10+20 + 27\ &=37 \end{align*} ]

Answer:

$37$

9. Fabian owns a small - business selling clothing. He knows that in the last week 56 customers paid cash, 9 customers used a debit card, and 14 customers used a credit card. Based on these results, express the probability that the next customer will pay with a credit card as a fraction in simplest form.

Explanation:

Step1: Find total number of customers

The total number of customers is $56 + 9+14=79$.

Step2: Calculate probability

The probability $P$ that the next customer will pay with a credit card is the number of credit - card users divided by the total number of customers. So $P=\frac{14}{79}$.

Answer:

$\frac{14}{79}$

10. Evaluate $\log_{9}\frac{1}{27}$

Explanation:

Step1: Let $\log_{9}\frac{1}{27}=x$, then rewrite in exponential form

$9^{x}=\frac{1}{27}$. Since $9 = 3^{2}$ and $27=3^{3}$, we have $(3^{2})^{x}=3^{- 3}$.

Step2: Simplify the left - hand side

Using the power - of - a - power rule $(a^{m})^{n}=a^{mn}$, we get $3^{2x}=3^{-3}$.

Step3: Solve for $x$

Since the bases are the same, $2x=-3$, so $x =-\frac{3}{2}$.

Answer:

$-\frac{3}{2}$

11. Evaluate $\log_{27}81$

Explanation:

Step1: Let $\log_{27}81 = x$, then rewrite in exponential form

$27^{x}=81$. Since $27 = 3^{3}$ and $81 = 3^{4}$, we have $(3^{3})^{x}=3^{4}$.

Step2: Simplify the left - hand side

Using the power - of - a - power rule $(a^{m})^{n}=a^{mn}$, we get $3^{3x}=3^{4}$.

Step3: Solve for $x$

Since the bases are the same, $3x = 4$, so $x=\frac{4}{3}$.

Answer:

$\frac{4}{3}$

12. Solve the following equation for $A$: $6=-G+\frac{1}{5}A$

Explanation:

Step1: Isolate the term with $A$

Add $G$ to both sides of the equation: $6 + G=\frac{1}{5}A$.

Step2: Solve for $A$

Multiply both sides by $5$: $A = 5(6 + G)=30+5G$.

Answer:

$A = 30 + 5G$

13. Solve the following equation for $M$: $RM=q + 4nM$

Explanation:

Step1: Move terms with $M$ to one side

$RM-4nM=q$.

Step2: Factor out $M$

$M(R - 4n)=q$.

Step3: Solve for $M$

$M=\frac{q}{R - 4n}$ (assuming $R\neq4n$).

Answer:

$M=\frac{q}{R - 4n}$

14. Write the exponential equation $5^{2}=25$ as a logarithmic equation.

Explanation:

Step1: Recall the conversion rule

If $a^{b}=c$, then $\log_{a}c=b$. Here $a = 5$, $b = 2$, $c = 25$. $\log_{5}25=2$

Answer:

$\log_{5}25=2$