What is the justification for each step in the solution of the equation?

−2(x−1)+12=4x−6−2(x−1)+12=4x−6
Select from the drop-down menus to correctly justify each step.


−2(x−1)+12=4x−6−2(x−1)+12=4x−6 Given
−2x+2+12=4x−6−2x+2+12=4x−6
−2x+14=4x−6−2x+14=4x−6 Combine like terms.
−2x=4x−20−2x=4x−20
−6x=−20−6x=−20 Addition or Subtraction Property of Equality
x=206x=206
x=103x=103 Simplify.

The probability of an American roulette wheel landing on a green pocket is 1/19, which is calculated by dividing the number of green pockets by the total number of pockets on the roulette wheel.

The student is asking about the probability of a specific outcome in a random event, specifically, the likelihood of landing on a green pocket in an American roulette game. To find this probability, we need to look at the total number of pockets and how many of them are green. With 38 total pockets and 2 being green, the probability that the roulette wheel will land on a green pocket is calculated by the ratio of green pockets to total pockets.

To calculate it: Probability = Number of green pockets / Total number of pockets = 2 / 38 = 1/19

This simple division computes the likelihood of a single spin resulting in a green pocket landing. Importantly, this calculation assumes each pocket has an equal chance of being landed on and that there are no biases in the wheel.

Final answer:

To find the month when Pedro's exponentially growing ant farm and Bobby's linearly decreasing ant farm have the same number of ants, we set the exponential growth formula equal to the linear decrease formula and solve for the number of months, using graphical or numerical methods.

Explanation:

We need to determine after how many months Pedro's exponentially growing ant farm will have the same number of ants as Bobby's linearly decreasing ant farm.

Let's denote the number of months as m. Pedro's ant farm starts with 100 ants and grows at a rate of 15% per month, so the population at month m is PPedro = 100 × (1 + 0.15)m.

Bobby's ant farm starts with 350 ants and decreases by 5 ants per month, so the population at month m is PBobby = 350 - 5m.

We need to find the month m where PPedro is approximately equal to PBobby. This can be solved by equating and solving the two expressions:

100 × (1 + 0.15)m ≈ 350 - 5m

To solve this, we can use either graphical methods or numerical methods such as iterative approximation, since an exact solution may not be possible through algebraic methods. We are looking for the value of m that makes both sides approximately equal.

By setting up the proportion [tex]\(\frac{15}{x} = \frac{x}{135}\)[/tex] and solving for [tex]\( x \)[/tex], we find that [tex]\( x = 45 \).[/tex]

To find the value of [tex]\( x \)[/tex] when the terms 15, [tex]\( x \)[/tex], and 135 are in proportion, we set up the proportion:

[tex]\[\frac{15}{x} = \frac{x}{135}\][/tex]

This represents the relationship where the first ratio is equal to the second ratio. Cross-multiplying, we get:

[tex]\[15 \times 135 = x \times x\][/tex]

[tex]\[2025 = x^2\][/tex]

Taking the square root of both sides, we find:

[tex]\[x = \sqrt{2025}\][/tex]

[tex]\[x = \pm 45\][/tex]

However, since 15,  x , and 135 are in proportion, the value of  x  must be positive. Therefore,  x = 45 .

The question probable maybe:

If the three terms 15, x and135 are in proportion then what is the value of x

Final answer:

The rule that represents the given function with ordered pairs (1, 4), (2, 16), (3, 64), (4, 256), (5, 1,024) is f(x) = 4^(x-1), which is determined by observing the pattern of the ordered pairs.

Explanation:

Identifying the Rule of a Function

The ordered pairs (1, 4), (2, 16), (3, 64), (4, 256), and (5, 1,024) represent a function where each input from the domain is mapped to a unique output in the range. Observing the pattern of the ordered pairs, we can infer a rule that corresponds to this function. Each second element in the pair appears to be an exponentiation of 4 to the power of the first element minus one.

The general rule for this function can be expressed as f(x) = 4^(x-1). We can verify this by plugging in the values from the domain (1 through 5) into the function:

For x = 1, f(1) = 4^(1-1) = 4^0 = 1

For x = 2, f(2) = 4^(2-1) = 4^1 = 4

For x = 3, f(3) = 4^(3-1) = 4^2 = 16

For x = 4, f(4) = 4^(4-1) = 4^3 = 64

For x = 5, f(5) = 4^(5-1) = 4^4 = 256

Therefore, the rule representing this function is f(x) = 4^(x-1).

The average speed in miles per hour for the entire trip of Emily is 38 miles per hour.

The correct answer is Option D.

Given data:

To find Emily's average speed during her entire trip, calculate the total time she spent traveling on both the highway and local roads.

To determine the speed on the highway.

On the highway, she traveled 30 miles faster than on the local roads, and her speed on the local roads is 20 miles per hour.

Speed on the highway = Speed on the local roads + 30 miles per hour

Speed on the highway = 20 miles per hour + 30 miles per hour

Speed on the highway = 50 miles per hour

Now, calculate the time spent on each segment of the trip:

Time spent on the highway = Distance on the highway / Speed on the highway

Time spent on the highway = 60 miles / 50 miles per hour

Time spent on the highway = 1.2 hours

Time spent on local roads = Distance on local roads / Speed on local roads

Time spent on local roads = 16 miles / 20 miles per hour

Time spent on local roads = 0.8 hours

Now, add the times for both segments to get the total time:

Total time = Time spent on the highway + Time spent on local roads

Total time = 1.2 hours + 0.8 hours

Total time = 2 hours

So, the average speed is determined as:

Average speed = Total distance / Total time

Average speed = (60 miles + 16 miles) / 2 hours

Average speed = 76 miles / 2 hours

Average speed = 38 miles per hour

Hence, Emily's average speed during her entire trip was 38 miles per hour.

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Answer:

Settlers in the territory wanted the same approach to slavery as in the states they came from

Step-by-step explanation:

Two step algebraic equations are relatively quick and easy -- after all, they should only take two steps. To solve a two step algebraic equation, all you have to do is isolate the variable by using either addition, subtraction, multiplication, or division. If you want to know how to solve two step algebraic equations in a variety of ways, just follow these steps.

1. Write the problem. The first step to solving a two step algebraic equation is just to write the problem so you can start to visualize the solution. Let's say we're working with the following problem: -4x + 7 = 15.

2. Decide whether to use addition or subtraction to isolate the variable term. The next step is to find a way to keep "-4x" on one side and to keep the constants (whole numbers) on the other side. To do this, you'll have to do the "Additive Inverse," finding the opposite of +7, which is -7. Subtract 7 from both sides of the equation so that the "+7" on the same side as the variable term is canceled out. Just write "-7" below the 7 on one side and below the 15 on the other so the equation remains balanced. Remember the Golden Rule of Algebra. Whatever you do to one side of an equation must be done to the other side to maintain the balance. That is why 7 is subtracted from the 15 as well. We only need to subtract 7 once per side, which is why the 7 is not subtracted from the -4x as well.

To solve two-variable equations, identify unknowns, find related equations, substitute known values, and solve for one variable before finding the other. Use substitution or elimination methods in case of a system of equations.

To solve equations with 2 variables, follow these steps:

Regarding a 2x2 system (two equations with two unknowns), you can either use substitution—solving one equation for one variable and then substituting into the other—or elimination—adding or subtracting equations to cancel one of the variables. If an equation has more than one unknown, additional equations are necessary to solve the system.

For instance, consider a pair of equations:
1) x + y = 10
2) x - y = 6

By adding these two equations, y is eliminated, and we can solve for x quickly. Then, we can substitute the value of x into either equation to solve for y.

-6.5(n-5) = 18.2

-6.5n + 32.5 = 18.2

-6.5n = -14.3

n = -14.3 / -6.5 = 2.2

n = 2.2

Answer:

  x = -3/2 ± i(√11)/2

Step-by-step explanation:

The quadratic formula can be used directly, but I find it convenient to divide by 14 first:

  x^2 +3x +5 = 0

The solution to ...

  ax^2 +bx +c = 0

is given by the formula ...

  x = (-b ±√(b^2 -4ac))/(2a)

Here, we have a=1, b=3, c=5, so the formula gives the solutions ...

  x = (-3 ±√(3^2 -4·1·5))/(2·1) = -3/2 ± i(√11)/2 . . . . . solutions are complex

The angle which is 24° more than the measure of its complementary angle is 66°.

Two angles whose sum is equivalent to 90 degrees are called complementary angles. Mathematically -

∠x + ∠y = 90°

Assume that the angle which is 24° more than the measure of its complementary angle be [x].

Then, we can write -

x + 24 = 90

x = 90 - 24

x = 66°

Therefore, the angle which is 24° more than the measure of its complementary angle is 66°.

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