Find the slope of the line segment that joins the points A(−2,−4) and B(3, 1)

Question

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13) To find the slope of the line segment joining points $A(-2, -4)$ and $B(3, 1)$ , use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ .

Identify the coordinates.
$(x_1, y_1) = (-2, -4)$ $(x_2, y_2) = (3, 1)$
Substitute the values into the formula.
$m = \frac{1 - (-4)}{3 - (-2)}$ $m = \frac{1 + 4}{3 + 2}$ $m = \frac{5}{5}$ $m = 1$
The correct option is B.

14) Two sets are equal if they contain exactly the same elements.
List the elements of each set.
$X = \{a, b, 1, 2, 3\}$ $Y = \{2, d, b, c\}$ $Z = \{1, b, c, d\}$ $W = \{b, 2, 1, 3, a\}$
Compare the sets.
- Option a) $Y \neq Z$ because $2 \in Y$ but $2 \notin Z$ (and $1 \in Z$ but $1 \notin Y$ ).
- Option b) $X \neq Z$ because $a \in X$ but $a \notin Z$ (and $c, d \in Z$ but $c, d \notin X$ ).
- Option c) $Y \neq W$ because $d, c \in Y$ but $d, c \notin W$ (and $1, 3, a \in W$ but $1, 3, a \notin Y$ ).
- Option d) $X = W$ because both sets contain the elements $\{a, b, 1, 2, 3\}$ . The order of elements does not matter for set equality. The correct option is D.
15) Given $\tan A = \frac{3}{4}$ . In a right-angled triangle, $\tan A = \frac{\text{opposite}}{\text{adjacent}}$ .
Identify the opposite and adjacent sides.
$\text{Opposite} = 3$ $\text{Adjacent} = 4$
Use the Pythagorean theorem to find the hypotenuse.
$\text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2$ $\text{Hypotenuse}^2 = 3^2 + 4^2$ $\text{Hypotenuse}^2 = 9 + 16$ $\text{Hypotenuse}^2 = 25$ $\text{Hypotenuse} = \sqrt{25} = 5$
Find $\sin A$ .
In a right-angled triangle, $\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$ .
$\sin A = \frac{3}{5}$
The correct option is D.

16) To find the mean (average) score, sum all the scores and divide by the number of scores.
Sum the scores.
$\text{Sum} = 78 + 81 + 69 + 82 + 75 = 385$
Divide by the number of scores, which is 5.
$\text{Mean} = \frac{385}{5} = 77$
The correct option is B.

17) To solve the proportion $(r+1):7 = 20:28$ :
Write the proportion as an equation of fractions.
$\frac{r+1}{7} = \frac{20}{28}$
Simplify the fraction on the right side.
$\frac{20}{28} = \frac{5 \times 4}{7 \times 4} = \frac{5}{7}$
Set the simplified fractions equal and solve for $r$ .
$\frac{r+1}{7} = \frac{5}{7}$
Since the denominators are equal, the numerators must be equal.
$r+1 = 5$ $r = 5 - 1$ $r = 4$
The correct option is A.

18) To find the volume of a right cylinder, use the formula $V = \pi r^2 h$ .
Identify the given values.
Base diameter $= 28$ cm, so radius $r = \frac{28}{2} = 14$ cm.
Height $h = 40$ cm.
Use $\pi = \frac{22}{7}$ .
Substitute the values into the formula.
$V = \frac{22}{7} \times (14 \text{ cm})^2 \times 40 \text{ cm}$ $V = \frac{22}{7} \times 196 \text{ cm}^2 \times 40 \text{ cm}$
Perform the calculation.
$V = 22 \times \frac{196}{7} \times 40 \text{ cm}^3$ $V = 22 \times 28 \times 40 \text{ cm}^3$ $V = 616 \times 40 \text{ cm}^3$ $V = 24640 \text{ cm}^3$
The correct option is C.

19) This question requires data from a table or context not provided in the image. Therefore, it cannot be answered.

20) This question requires data from a table or context not provided in the image. Therefore, it cannot be answered.

21) To find the sum of $8144_9$ and $1542_9$ :
Add the numbers column by column from right to left, remembering that this is base 9 (digits 0-8).
$\begin{array}{r} 8144_9 \\ + 1542_9 \\ \hline \end{array}$
Add the rightmost column ( $4+2$ ).
$4 + 2 = 6_9$
Add the next column ( $4+4$ ).
$4 + 4 = 8_9$
Add the next column ( $1+5$ ).
$1 + 5 = 6_9$
Add the leftmost column ( $8+1$ ).
$8 + 1 = 9$
Since 9 is equal to the base, write 0 and carry 1.
$9 = 1 \times 9 + 0 \implies 10_9$
So, the sum is $10686_9$ .
The correct option is A.

22) To find the prime factorization of 588:
Divide by the smallest prime number, 2.
$588 \div 2 = 294$
Divide by 2 again.
$294 \div 2 = 147$
Divide by the next smallest prime number, 3.
$147 \div 3 = 49$
Divide by the next smallest prime number, 7.
$49 \div 7 = 7$
The prime factors are $2, 2, 3, 7, 7$.
$588 = 2^2 \times 3^1 \times 7^2$
The correct option is D.

22) (This is a duplicate question number, but a new problem) To find Mr. Grabrah's commission:
Identify the selling price and commission rate.
Selling price = \43,770.00 $Commission rate$ = 3%$
Calculate the commission.
$\text{Commission} = 3\% \text{ of } \$43,770.00$ $\text{Commission} = 0.03 \times \$43,770.00$ $\text{Commission} = \$1313.10$ The correct option is C.

23) To calculate the simple interest, use the formula $I = PRT$ .
Identify the principal ( $P$ ), rate ( $R$ ), and time ( $T$ ).
$P = \$37.00$
$R = 2\% = 0.02$ $T = 9 \text{ years}$
Substitute the values into the formula.
$I = \$37.00 \times 0.02 \times 9$
Perform the calculation.
$I = \$0.74 \times 9$ $I = \$6.66$ The correct option is C.

24) The logarithm property for the product of two numbers is $\log_b (XY) = \log_b X + \log_b Y$ .
Apply the product rule for logarithms.
$\log_b MN = \log_b M + \log_b N$
The correct option is A.

25) To evaluate the expression $P(1+rt)$ with the given values:
Substitute $P=2$ , $r=-5$ , and $t=3$ into the expression.
$P(1+rt) = 2(1 + (-5)(3))$
Perform the multiplication inside the parenthesis first.
$2(1 + (-15))$
Perform the addition inside the parenthesis.
$2(1 - 15)$ $2(-14)$
Perform the final multiplication.
$2 \times (-14) = -28$
The final answer is $\boxed{-28}$ .

Find the slope of the line segment that joins the points A(-2,-4) and B(3, 1)

Answer

Step-by-step solution

Identify the coordinates.

Substitute the values into the formula.

List the elements of each set.

Compare the sets.

Identify the opposite and adjacent sides.

Use the Pythagorean theorem to find the hypotenuse.

Find sin⁡A\sin AsinA.

Sum the scores.

Divide by the number of scores, which is 5.

Write the proportion as an equation of fractions.

Simplify the fraction on the right side.

Set the simplified fractions equal and solve for rrr.

Identify the given values.

Substitute the values into the formula.

Perform the calculation.

Add the numbers column by column from right to left, remembering that this is base 9 (digits 0-8).

Add the rightmost column (4+24+24+2).

Add the next column (4+44+44+4).

Add the next column (1+51+51+5).

Add the leftmost column (8+18+18+1).

Divide by the smallest prime number, 2.

Divide by 2 again.

Divide by the next smallest prime number, 3.

Divide by the next smallest prime number, 7.

The prime factors are $2, 2, 3, 7, 7$.

Identify the selling price and commission rate.

Calculate the commission.

Identify the principal (PPP), rate (RRR), and time (TTT).

Substitute the values into the formula.

Perform the calculation.

Apply the product rule for logarithms.

Substitute P=2P=2P=2, r=−5r=-5r=−5, and t=3t=3t=3 into the expression.

Perform the multiplication inside the parenthesis first.

Perform the addition inside the parenthesis.

Perform the final multiplication.

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More Mathematics Questions

Find $\sin A$ .

Set the simplified fractions equal and solve for $r$ .

Add the rightmost column ( $4+2$ ).

Add the next column ( $4+4$ ).

Add the next column ( $1+5$ ).

Add the leftmost column ( $8+1$ ).

Identify the principal ( $P$ ), rate ( $R$ ), and time ( $T$ ).

Substitute $P=2$ , $r=-5$ , and $t=3$ into the expression.