Estimate the solution of the linear system graphically and check the solution algebraically: A. A line graph showing 2 intersecting lines, the first line passes through the points (0, 5), (-1, 1), and (-3, -6) and the second line passes through the points (-3, 5), (-2, 3), (-1, 1), (0, -1), (1, -3), and (2, -5). B. A line graph showing 2 intersecting lines, the first line passes through the points (3, 4), (2, 1), (1, -2), and (0, -5) and the second line passes through the points (-5, 4), (-4, 3), (-3, 2), (-2, 1), (-1, 0), (0, -1), (1, -2), (2, -3), and (3, -4). C. A line graph showing 2 intersecting lines, the first line passes through the points (3, 4), (2, 1), (1, -2), (0, -5) and (-1, -8) and the second line passes through the points (5, 4), (4, 3), (3, 2), (2, 1), (1, 0), (0, -1), (-1, -2), (-2, -3), and (-3, -4). D. A line graph showing 2 intersecting lines, the first line passes through the points (2, 5), (1, 2), (0, -1), (-1, -4), and (-2, -7) and the second line passes through the points (-6, 5), (-5, 4), (-4, 3), (-3, 2), (-2, 1), (-1, 0), (0, -1), (1, -2), and (2, -3).

Answer:

1

Step-by-step explanation:

1 / 1 = 1

The required, graph of the function f(x) = √[x+3]-1 is a curve that starts at (-3,-1) and extends to the right, increasing in value but at a decreasing rate due to the square root function.

Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.

Here,
The graph of the function f(x) = √[x+3]-1 is a curve that starts at the point (-3,-1) and extends to the right indefinitely. The square root function √x has a domain of x ≥ 0, so in this case, the domain of the function is x ≥ -3.

The graph is always above the x-axis, as the square root function can only output non-negative values. The graph also approaches but never touches the horizontal line y = 0 as x increases without bound, since the -1 term in the function only shifts the graph downward by one unit.

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Synthetic Division of the polynomials (x^(4)-5x^(3)+4x-17) and (x-5) gives answer "x^(3)+4+3/(x-5)."

Step 1: Write the coefficients of the polynomial in descending order. In this case, the coefficients are 1, -5, 0, 4, -17.

Step 2: Write the value of x from the divisor (x-5) in the left column. In this case, the value of x is 5.

Step 3: Bring down the first coefficient to the bottom row.

Step 4: Multiply the value of x by the number in the bottom row and write the result in the next column.

Step 5: Add the numbers in the second column and write the result in the bottom row.

Step 6: Repeat steps 4 and 5 until you have completed all the columns.

Step 7: The numbers in the bottom row are the coefficients of the quotient, and the last number is the remainder.

The synthetic division will look like this:

5 | 1 -5 0 4 -17
 |   5 0 0 20
 ----------------
   1  0 0 4   3

So, the quotient is x^(3)+0x^(2)+0x+4, or simply x^(3)+4, and the remainder is 3.

Therefore, the answer is (x^(4)-5x^(3)+4x-17)÷(x-5) = x^(3)+4+3/(x-5).

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Answer:

Rise and run

Step-by-step explanation:
Lets have it in slope intercept form
y=3/5x

Therefore our slope is 3/5 and y-intercept is (0,0)

Now we can start to graph. Start on our Y-intercept which is zero. Let's count 3 spaces up, which is our rise, then lets count 5 spaces to our right, which is our positive run, and plot a dot on our final point. You can do the same negative to graph our opposing points. Start on our 0,0 y-intercept, and count down 3 spaces. Then count 5 spaces to the left, and plot a dot on your final point. Now, let's draw a line through our points (-5, -3), (0,0), and (5,3)!

As per the concept of integral, the series ∑ne⁻ⁿ converges.

Let's consider the series ∑ne⁻ⁿ. Since each term of the series is positive, the first condition of the Integral Test is satisfied. To check the second condition, we need to determine whether the function f(x) = xe⁻ˣ is decreasing for x ≥ 1.

To do this, we can take the derivative of f(x) with respect to x:

f'(x) = e⁻ˣ - xe⁻ˣ

Setting f'(x) = 0, we get:

e⁻ˣ - xe⁻ˣ = 0

x = 1

So f(x) has a maximum value at x = 1. Since f'(x) < 0 for x ≥ 1, f(x) is decreasing for x ≥ 1.

Now, we can set up the corresponding integral:

∫₁^∞ xe⁻ˣ dx

To evaluate this integral, we can use integration by parts:

u = x, dv = e⁻ˣ dx

du = dx, v = -e⁻ˣ

∫₁^∞ xe⁻ˣ dx = -xe⁻ˣ │₁^∞ + ∫₁^∞ e⁻ˣ dx

= 0 + e⁻ˣ │₁^∞

= 1

Since the integral converges, the series ∑ne⁻ⁿ also converges by the Integral Test.

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Complete Question:

How do you use the Integral Test to determine convergence or divergence of the series: ∑ne⁻ⁿ from n=1 to infinity?

Ship B is 22.8 km away from ship A on a bearing of N50°W.

The topmost point of the town hall is 28 metres above ground level.
The distance travelled by a point on the roundabout in one second is 113 cm for point A and 63 cm for point B.
Ship B is 22.8 km away from ship A on a bearing of N50°W.

Explanation:
1) For the first question, we can use trigonometry to find the height of the town hall. Let h be the height of the topmost point above ground level, and d be the distance between point A and the base of the town hall. Then we have:
tan(35°) = h/d and tan(60°) = h/(d-30)
Solving for h, we get:
h = d*tan(35°) and h = (d-30)*tan(60°)
Equating the two expressions for h, we get:
d*tan(35°) = (d-30)*tan(60°)
d = 30*tan(60°)/(tan(60°)-tan(35°)) ≈ 48.5 metres
Substituting back into the first equation, we get:
h = 48.5*tan(35°) ≈ 28 metres
Therefore, the topmost point of the town hall is 28 metres above ground level.

2) For the second question, we can use the formula for the circumference of a circle to find the distance travelled by a point on the roundabout in one second. The circumference of a circle is given by C = 2πr, where r is the radius of the circle. The distance travelled by a point in one second is then given by C/5, since the roundabout makes one revolution every five seconds. For point A, which is 1.8 metres from the centre of rotation, we have:
C = 2π*1.8 = 11.31 metres
Distance travelled in one second = 11.31/5 = 2.26 metres ≈ 226 cm ≈ 113 cm to the nearest centimetre
For point B, which is 1 metre from the centre of rotation, we have:
C = 2π*1 = 6.28 metres
Distance travelled in one second = 6.28/5 = 1.26 metres ≈ 126 cm ≈ 63 cm to the nearest centimetre

3) For the third question, we can use the law of cosines to find the distance between ship A and ship B. The law of cosines states that c^2 = a^2 + b^2 - 2ab*cos(C), where a, b, and c are the lengths of the sides of a triangle, and C is the angle opposite side c. Let x be the distance between ship A and ship B, and let θ be the angle between the two ships. Then we have:
x^2 = 17.2^2 + 14.1^2 - 2*17.2*14.1*cos(θ)
To find the angle θ, we can use the fact that the sum of the angles in a triangle is 180°. We have:
θ = 180° - 60° - 80° = 40°
Substituting back into the equation, we get:
x^2 = 17.2^2 + 14.1^2 - 2*17.2*14.1*cos(40°) ≈ 520.3
x ≈ 22.8 km
To find the bearing of ship B from ship A, we can use the law of sines. The law of sines states that a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the angles opposite those sides. Let α be the bearing of ship B from ship A. Then we have:
14.1/sin(α) = 22.8/sin(40°)
Solving for α, we get:
α = sin^-1(14.1*sin(40°)/22.8) ≈ 30°
Since ship B is to the west of ship A, the bearing of ship B from ship A is N50°W.
Therefore, ship B is 22.8 km away from ship A on a bearing of N50°W.

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cos 59° = ±([tex]\sqrt{1 - sin^2 59}[/tex]) ≈ ±0.515 in terms of the sine function.

To express cos 59° in terms of the sine function, we can use the trigonometric identity:

[tex]sin^2[/tex]θ + [tex]cos^2[/tex]θ = 1

Rearranging this identity, we get:

[tex]cos^2[/tex]θ = 1 - [tex]sin^2[/tex]    θ

Taking the square root of both sides, we get:

cosθ = ±[tex]\sqrt{(1 - sin^2θ)}[/tex]

In the case of cos 59°, we know that sin 59° can be calculated using the sine function since sin 59° is the opposite side of a right triangle divided by its hypotenuse, where the angle opposite to the opposite side measures 59 degrees. Therefore:

sin 59° = 0.85717 (rounded to 5 decimal places)

Substituting sin 59° into the equation for cosθ, we get:

cos 59° = ±([tex]\sqrt{1 - sin^2 59}[/tex]) = ±[tex]\sqrt{(1 - 0.85717^2) }[/tex]≈ ±0.515

Note that since the angle 59° is in the first quadrant, cos 59° is positive. Therefore, we can write:

cos 59° ≈ 0.515 in terms of the sine function.

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1 and 1/7 multiplied by 3/5 is equal to 24/35 or 0.6857 (rounded to four decimal places).

What do you mean by decimal?

In mathematics, a decimal is a number that represents a fraction or a part of a whole using a base-ten positional numeral system.

To multiply 1 and 1/7 by 3/5, we can first convert the mixed number to an improper fraction.

1 and 1/7 can be written as:

(7/7 * 1) + 1/7 = 7/7 + 1/7 = 8/7

So, we have:

1 and 1/7 = 8/7

Now, we can multiply 8/7 by 3/5 as follows:

(8/7) * (3/5) = (8 * 3) / (7 * 5) = 24/35

Therefore, 1 and 1/7 multiplied by 3/5 is equal to 24/35 or 0.6857 (rounded to four decimal places).

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The required value of x for the triangle is 9 units.

The theorem you are referring to is known as the "Parallel Line Theorem" or "Thales' Theorem".

If a line is drawn parallel to one side of a triangle, then the other two sides of the triangle are divided proportionally.

Consider a triangle ABC with a line parallel to one of its sides, say side AB. Let the parallel line intersect sides AC and BC at points D and E, respectively.

Then,

AB/BD = CB/BE

In triangle;

[tex]$\frac{4x}{x-1} =\frac{27}{6}[/tex]

24x = 27x - 27

= 3x = 27

= x = 9

Thus, required value of x is 9.

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The minimum value of the objective function is -15, which occurs at the corner point (0, 5).

The corner points of a linear programming problem are the points where the constraints intersect. These points can be found by solving the system of inequalities for each pair of constraints.

For this problem, we can find the corner points by solving the system of inequalities for each pair of constraints:

3x – y >= 2 and x + y <= 5:
- Add y to both sides of the first inequality: 3x >= 2 + y
- Subtract 2 from both sides of the first inequality: 3x - 2 >= y
- Substitute 3x - 2 for y in the second inequality: x + (3x - 2) <= 5
- Simplify: 4x <= 7
- Divide by 4: x <= 7/4
- Substitute 7/4 for x in the first inequality: 3(7/4) - 2 >= y
- Simplify: 5/4 >= y

The first corner point is (7/4, 5/4).

3x – y >= 2 and x >= 0:
- Set x = 0 and solve for y: 3(0) - y >= 2, y <= -2
- Set y = 0 and solve for x: 3x - 0 >= 2, x >= 2/3

The second corner point is (2/3, 0).

x + y <= 5 and x >= 0:
- Set x = 0 and solve for y: 0 + y <= 5, y <= 5
- Set y = 0 and solve for x: x + 0 <= 5, x <= 5

The third corner point is (0, 5).

x + y <= 5 and y >= 0:
- Set x = 0 and solve for y: 0 + y <= 5, y <= 5
- Set y = 0 and solve for x: x + 0 <= 5, x <= 5

The fourth corner point is (5, 0).

Now we can plug each corner point into the objective function to find the minimum value:

C = 3x – 3y

C = 3(7/4) - 3(5/4) = 3
C = 3(2/3) - 3(0) = 2
C = 3(0) - 3(5) = -15
C = 3(5) - 3(0) = 15

Therefore, the solution to the linear programming problem is (0, 5).

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The compound interest on N2,000 for 2 years at the rate of 10% per annum is N441.00.

Compound interest is the interest earned not only on the principal amount but also on the accumulated interest. To calculate the compound interest, we use the formula:

where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.

In this case, P = N2,000, r = 0.1 (10%), n = 1 (compounded annually), and t = 2 years.

So, A = 2000[tex](1 + 0.1/1)^{(1*2)[/tex] = N2,441.00

Therefore, the compound interest is N2,441.00 - N2,000.00 = N441.00.

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Answer:$72 Total

Step-by-step explanation:

20%=1/5

1/5 + 5/5(total) = 6/5

60 * 6/5 = $72

Therefore, to have enough for one application per person, the ensemble of 15 needs to buy about 8.55 ounces of self-tanning lotion.

To answer mathematical issues involving proportional relationships between quantities, one uses the unitary method. This approach involves calculating the value of a single unit of a quantity and using that value to calculate the value of any number of additional units. For instance, if we know that five items cost $20 each, we can calculate the price of any other quantity of items by first determining the price of one item, then increasing that figure by the desired quantity of items.

given

The quantity of lotion needed for each application, assuming a 4-oz container of self-tanning lotion yields seven applications, is:

4 oz / 7 = 0.57 oz (rounded to two decimal places) (rounded to two decimal places)

The quantity needed for each application must be multiplied by the number of applications needed by each individual, which equals 1, to determine how many ounces of self-tanning lotion the ensemble of 15 needs to buy:

15. People at 0.57 ounce each equals 8.55 oz.

Therefore, to have enough for one application per person, the ensemble of 15 needs to buy about 8.55 ounces of self-tanning lotion.

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The formula for calculating the effective annual rate of interest with quarterly compounding is:

(1 + r/4)^4 - 1 = A/P

where r is the quarterly interest rate, A is the final amount, and P is the principal.

In this case, P = $2,000, A = $3,000, and the time period is 7 years or 28 quarters.

So we have:

(1 + r/4)^4 - 1 = 3000/2000

(1 + r/4)^4 = 1.5

1 + r/4 = (1.5)^(1/4)

r/4 = (1.5)^(1/4) - 1

r = 4[(1.5)^(1/4) - 1]

To get the effective annual rate, we need to convert the quarterly rate to an annual rate by multiplying by 4:

effective annual rate = 4[(1.5)^(1/4) - 1] ≈ 8.84%

Therefore, the effective annual rate of interest at which $2,000 will grow to $3,000 in seven years if compounded quarterly is approximately 8.84%, rounded to 2 decimal places.

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The probability of escape for the guest is approximately 9 - √27/4.

This can be calculated using the principle of inclusion-exclusion. To calculate the probability of escape, we can calculate the probability of the guest being unable to move from Room 0 to any of the other rooms.

This is the probability of all doors from Room 0 being locked, which is equal to 1/3 x 1/3 = 1/9. Therefore, the probability of escape is 1 - 1/9 = 8/9.

Using the inclusion-exclusion principle, we can calculate the probability of the guest being able to move from Room 0 to any of the other rooms. The probability of the guest being able to move from Room 0 to any of the other rooms is equal to 1 - (1/3 + 1/3 - 1/9) = 9 - √27/4.

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Answer:

Step-by-step explanation:

0.16x + 0.20y = 11

we can write 0.16 = 16/100 = 4/25 and 0.20 = 20/100 = 1/5

[tex]\frac{4}{25}x + \frac{1}{5}y = 11[/tex] --------(1)

[tex]x + y = 60 \\[/tex] --------(2)

multiply both side by 5 in equation (1) we get

[tex]\frac{4}{5}x + y =55[/tex]   --------(3)

Subtract equation 3 from 2

[tex]x - \frac{4}{5}x + y - y = 60 - 55\\\frac{x}{5} = 5\\ x = 25[/tex]

put the value of x in equation 2

25 + y = 60

y = 35

The equation 4(5r-15)=13+11+8r has a unique i.e., one solution.

The given equation is: 4(5r-15) = 13+11+8r.

Simplifying the left-hand side, we get 20r-60 = 32+8r.

Bringing all the r terms to the left-hand side and the constants to the right-hand side, we get:

20r - 8r = 32 + 60

12r = 92

r = 7.67

Therefore, we get a unique solution for the given equation, which is r = 7.67.

There are a few possible reasons why an equation may not have any solution or may have infinitely many solutions. In this case, since we obtained a unique solution, neither of these situations applies.

One common reason why an equation may not have any solution is that the equation may be inconsistent, meaning that the left-hand side and right-hand side of the equation cannot be made equal for any value of the variable. For example, the equation 2x + 3 = 2x + 4 has no solution since the two sides of the equation can never be equal for any value of x.

On the other hand, an equation may have infinitely many solutions if it is an identity, meaning that the left-hand side and right-hand side of the equation are always equal, regardless of the value of the variable. For example, the equation x + 1 = x + 1 is an identity since the left-hand side and right-hand side are always equal for any value of x.

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Answer:

3 Marbles were left.

Step-by-step explanation:

To find the number of marbles Ronen gave to each teammate, we can divide the total number of marbles by the number of teammates:

1003 ÷ 8 = 125 with a remainder of 3

Therefore, each teammate received 125 marbles, and there were 3 marbles left over.

So, 125 marbles were shared with each teammate, and 3 marbles were left.

The sample space includes the following outcomes: HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT.

a. A simple event is an event that has only one possible outcome. An example of a simple event is getting a head on the third coin. This is represented by option C: "Getting a head on the third coin and a head on the fourth coin."
b. A joint event is an event that involves two or more simple events happening at the same time. An example of a joint event is getting a head on the first coin and a tail on the third coin. This is represented by option C: "Getting a head on the first coin and a tail on the third coin."
c. The complement of getting a tail on the fourth coin is getting a head on the fourth coin. This is represented by option A: "Getting a head on the fourth coin."
d. The sample space consists of all possible outcomes of the four flipped coins. This is represented by option D: "Getting a head or a tail on any of the four coins." The sample space includes the following outcomes: HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT.

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Rearranging this equation gives a quadratic equation[tex]length^2 - 3 length - 75 = 0[/tex]

Let's start by using the formula for the volume of a rectangular prism:

Volume = length x width x height

We know that the volume is 900 cubic meters and the width is 12 meters. Let's substitute these values into the formula:

[tex]900 = length \times 12 \times height[/tex]

Now we need to use the information about the height. We know that the height is 3 meters shorter than the length, so we can write:

height = length - 3

Substituting this into the formula gives:

[tex]900 = length \times 12 \times (length - 3)[/tex]

Simplifying this equation gives:

[tex]900 = 12 length^2 - 36 length[/tex]

Dividing both sides by 12 gives:

[tex]75 = length^2 - 3 length[/tex]

Therefore, Rearranging this equation gives a quadratic equation:

[tex]length^2 - 3 length - 75 = 0[/tex]

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The product of the polynomials (4x-7)^(2) is 16x^(2)-56x+49. This is the final answer expressed as a single polynomial in standard form.

To multiply the polynomials using a special product formula, we can use the formula (a-b)^(2)=a^(2)-2ab+b^(2). In this case, a=4x and b=7. Plugging these values into the formula gives us:
(4x-7)^(2)=(4x)^(2)-2(4x)(7)+(7)^(2)

Simplifying the terms on the right side of the equation gives us:
(4x-7)^(2)=16x^(2)-56x+49

Simplifying, we have (4x-7)2 = 16x2 - 56x + 49, which is our answer expressed as a single polynomial in standard form.

Multiplying polynomials require only three steps.

First, multiply each term in one polynomial by each term in the other polynomial using the distributive law.

Add the powers of the same variables using the exponent rule.

Then, simplify the resulting polynomial by adding or subtracting the like terms.

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The possible number of distinct complex roots for a third-degree polynomial function are 0, 1, 2, or 3, making option (B) the correct answer.

A polynomial function of degree n can have at most n distinct roots. For a third-degree polynomial, n=3, so it can have at most 3 roots.

Complex roots of a polynomial come in conjugate pairs. So if a third-degree polynomial has one real root, then the other two roots must be a conjugate pair of complex roots. If the polynomial has two real roots, then the third root must be a complex root that is not real.

Therefore, the possible number of distinct complex roots for a third-degree polynomial function is 0, 1, 2, or 3, making option (B) the correct answer.

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There are a total of 26 letters in the English alphabet and 10 digits (0-9). Therefore, the total number of car number plates that can be made if each plate contains 3 different letters followed by 3 different digits can be calculated as follows:

- For the first letter, there are 26 options.

- For the second letter, there are 25 options (since the first letter has already been used).

- For the third letter, there are 24 options (since the first and second letters have already been used).

- For the first digit, there are 10 options.

- For the second digit, there are 9 options (since the first digit has already been used).

- For the third digit, there are 8 options (since the first and second digits have already been used).

Therefore, the total number of car number plates that can be made is:

26 × 25 × 24 × 10 × 9 × 8 = 11,232,000

So, the answer is 11,232,000 car number plates can be made if each plate contains 3 different letters followed by 3 different digits.

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Answer:88 miles

Step-by-step explanation:

1.1x80+88

Step-by-step explanation:

How many Oz make 1 Pound

Answer is 16

Divide 87 by 16

Answer is 5Lb 7oz

The value of k is ∛√199.

Consider the expansion of x^2(3x^2 + k/x)^8. The constant term is 16128. We need to find the value of k.

The expansion of (3x^2 + k/x)^8 will have terms of the form (3x^2)^a(k/x)^b, where a + b = 8. The constant term will be the term where the powers of x cancel out, so we need to find a and b such that 2a - b = 0.

Solving for a and b, we get a = 4 and b = 8. So the constant term will be (3x^2)^4(k/x)^8 = 81x^8(k^8/x^8) = 81k^8.

Setting this equal to 16128 and solving for k, we get:

81k^8 = 16128
k^8 = 16128/81
k^8 = 199
k = ∛√199

Therefore, the value of k is ∛√199.

k = ∛√199.

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Answer:

the greatest number of groups you can make is 4

Step-by-step explanation:

First we have to find the greatest number which can divide 8 and 12 exactly.

there is none but G.C.F is 4

That is, 8 U.S stamps can be displayed in 4 groups at 2 groups.

And 12 international stamps can be displayed in 4 groups at 3 groups

In this way, each of the 4 groups would have 2 U.S stamps and 3 international stamps. And all the 4 groups would be identical.

Answer: 7

Step-by-step explanation:

The slope is the number being multiplied by x

Answer: The angles are classified as interior angles, where x=9

Step-by-step explanation:

The angle measure of a straight line is 180°, meaning that 7x-10 and 12x+19 must add up to 180.

It is clear from the visual that the former is an acute angle, and the latter is an obtuse angle. This means that when solving each, 7x-10 must be less than 90, and 12x+19 must be greater than 90, but less than 180.

If you set up the equation to solve for x; (7x-10) + (12x+19) = 180, you must combine like terms.

19x+9=180 is the new equation. Since x needs to be isolated, subtract 9 from each side, to get 19x=171. Dividing each side by 19 results in x=9.

Answer:

1369.1

Step-by-step explanation:

Answer:

1369.1 feet

Step-by-step explanation: