During first semester, the ratio of the number of students in our class to the number of students in gym class was 2 to 7. However, the art classes were really small, and the gym classes with large, so the principal change the students classes for second semester. In second semester, the ratio of the number of students in our class to the number of students in gym class was 5 to 4. If 75 students were in our class second semester, how many were in our class in gym class first semester.

Answer:

Any proposed solution of a rational equation that causes a denominator to equal​ __ZERO__ is rejected.

Step-by-step explanation:

We will show this statement is true by an example:

Consider the expression :  [tex]\frac{x}{x-4}=\frac{x}{x-4}+4[/tex]

Now, solved the rational expression and check its proposed solution

[tex]\frac{x}{x-4}=\frac{x}{x-4}+4[/tex]

x cannot equal to 4, as it makes  both denominators equal to zero.

Multiply both the sides by (x-4),

[tex]\left ( x-4 \right )\cdot \frac{x}{x-4}=\left (x-4 \right )\cdot\left ( \frac{x}{x-4}+4  \right )\\[/tex]

Now, use the distributive property on Right hand side,

[tex]\left ( x-4 \right )\cdot \frac{x}{x-4}=\left (x-4 \right )\cdot \left ( \frac{x}{x-4} \right )+\left (x-4 \right )\cdot 4\\[/tex]

Simplify the above expression,

[tex]x=x+4x-16[/tex]

Combine like terms,

[tex]x-x-4x=-16[/tex]

[tex]-4x=-16[/tex]

Divide both sides by -4, we get

[tex]\frac{-4x}{-4}=\frac{-16}{-4}[/tex]

[tex]x=4[/tex].

As we know that x cannot equal to 4, replacing x=4 in the original expression causes the denominator equal to 0.

Check the solution: [tex]\frac{x}{x-4}=\frac{x}{x-4}+4[/tex]

Substitute the value of x=4 in the original expression,

[tex]\frac{4}{4-4}=\frac{4}{4-4}+4[/tex]

[tex]\frac{4}{0}=\frac{4}{0}+4[/tex]

Thus, 4 must be rejected as the solution, and the solution set is only 0.

A proposed solution of a rational equation that causes a denominator to equal zero is rejected because division by zero is not possible.

Any proposed solution of a rational equation that causes a denominator to equal zero is rejected. When the denominator of a rational equation is zero, the fraction becomes undefined because division by zero is not possible. In mathematics, division by zero is an undefined operation.

Answer: 120 different combinations  .

Step-by-step explanation:

The combination of n things taking m things at a time is given by :-

[tex]C(n,m)=\dfrac{n!}{m!(n-m)!}[/tex]

Given : The total numbers of errands = 6

Now, the combination of 6 errands taking 3 at a time is given by :-

[tex]C(6,3)=\dfrac{6!}{3!(6-3)!}\\\\=\dfrac{6\times5\times4\times3!}{3!}=120[/tex]

Hence, he could choose to run 120 different combinations of three errands .

The quadratic formula can be used to find the solutions of a quadratic equation. In this case, the equation is 2x² - 3x - 5 = 0. Using the quadratic formula, the two solutions are x = 2 and x = -1.

To find the solutions to the equation 2x² - 3x - 5 = 0, we can use the quadratic formula. The formula states that the solutions of any quadratic equation ax² + bx + c = 0 can be calculated using the formula:

x = (-b ± √(b² - 4ac)) / 2a

In this case, a = 2, b = -3, and c = -5. Substituting these values into the formula, we get:

x = (-(-3) ± √((-3)² - 4(2)(-5))) / (2(2))

Simplifying further, we have:

x = (3 ± √(9 + 40)) / 4

x = (3 ± √49) / 4

x = (3 ± 7) / 4

Therefore, the two solutions to the equation 2x² - 3x - 5 = 0 are x = (3 + 7) / 4 = 2 and x = (3 - 7) / 4 = -1.

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Hypothesis 1 supported (potentially adapted wings), 2 needs more data, 3 and 4 not supported (scavengers and traffic constant).

Here's how to proceed:

1. Match Data to Hypotheses:

Hypothesis 1:

Relevant Data: "The wing shapes of swallows killed on roads differ from those of the general population."

Hypothesis 2:

Relevant Data: Not directly provided in the given options. More information is needed about the actual population size of cliff swallows living near roads over time.

Hypothesis 3:

Relevant Data: "Avian scavengers did not increase during this time, and terrestrial scavengers probably did not increase."

Hypothesis 4:

Relevant Data: "Car traffic stayed the same or increased during this time."

2. Evaluate Hypothesis Support:

Hypothesis 1:

Supported (additional data may be needed). The wing shape difference suggests potential adaptation, but more evidence would strengthen the conclusion.

Hypothesis 2:

Cannot be determined without direct data on population size changes.

Hypothesis 3:

Not supported (probably not a factor). The data indicates scavenger populations haven't increased, suggesting they aren't significantly affecting the number of road-killed swallows found.

Hypothesis 4:

Not supported (probably not a factor). The data shows car traffic has remained steady or increased, making it less likely to be the cause of a decrease in observed road-killed swallows.

Complete Question:

The answer can be written in statement form as,"Time measured in minutes is an example of a(n) interval scale."

A numerical scale known as an interval scale is one in which both the order of the variables and their differences are known. Using the Interval scale, variables with recognizable, consistent, and calculable differences are categorized.

It is easy to remember the primary role of this scale too, ‘Interval’ indicates ‘distance between two entities’, which is what Interval scale helps in achieving.

The value of time is given in minutes.

The given quantity can be measured on interval scale.

Hence, the required scale is known as interval scale.

To know more about interval scale click on,

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what is the answer? this is hard