2. Construct a line that is perpendicular to line m and that passes through point P.

Answer:

Step-by-step explanation:

We can find the numbers of male students and female students by making the system of equations using the given data:

Let:

Then:

Now we have the system of equations:

[tex]\begin{cases} m=7a+1 &\text{... [1]}\\f=\frac{1}{2} m &\text{... [2]}\\a+m+f=82 &\text{... [3]}\end{cases}[/tex]

[1] & [2]

By substituting [1] into [2], the [2] equation becomes:

[tex]f=\frac{1}{2} m[/tex]

[tex]f=\frac{1}{2} (7a+1)\ ...\ [4][/tex]

[1] & [3] & [4]

By substituting [1] & [4] into [3], the [3] equation becomes:

[tex]a+m+f=82[/tex]

[tex]a+(7a+1)+\frac{1}{2} (7a+1)=82[/tex]

[tex]\frac{23}{2} a+\frac{3}{2} =82[/tex]

[tex]23a+3=164[/tex]

[tex]23a=164-3[/tex]

[tex]a=161\div23[/tex]

[tex]\bf a=7[/tex]

Now, we can find the numbers of male and female students by substituting the value of a into equation [1] & [2]

[1] [tex]m=7a+1[/tex]

   [tex]m=7(7)+1[/tex]

   [tex]\bf m=50[/tex]

[2] [tex]f=\frac{1}{2} m[/tex]

    [tex]f=\frac{1}{2}(50)[/tex]

    [tex]\bf f=25[/tex]

The greatest common factor of the monomials 25x^5 and 20x^6 is 5x^5.

To find the greatest common factor (GCF) of the monomials 25x^5 and 20x^6, we first break down each monomial into its prime factors and include the variables.

25x^5 = 5 × 5 × x × x × x × x × x

20x^6 = 2 × 2 × 5 × x × x × x × x × x × x

Next, we find the common factors between the two:

The common prime number is 5.

The common variable factor is x raised to the minimum power between the two terms, which is x^5 (since x^5 is the lower power compared to x^6).

Therefore, the greatest common factor is 5x^5.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

The final expression for the expression

3(2x + 1) + 2(x + 4) is 8x + 7.

What is an expression?

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example:

2 + 3x + 4y = 7 is an expression.

3X + 4X + 4 is an expression.

We have,

3 (2x + 1) + 2 (x + 4)

Remove the parenthesis.

3 x (2x) + 3 + 2x + 4

6x + 3 + 2x + 4

8x + 7

Thus,

The final expression for the expression

3(2x + 1) + 2(x + 4) is 8x + 7.

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The graphs of the two equations differ from each other because they have different y intercepts.

A linear equation is in the form:

y = mx + b;

where m is the slope, b is the y intercept and y, x are variables.

The first equation is given by:

2x + 3y = 12

3y = -2x + 12

y = (-2/3)x + 12       (1)

Hence the first equation has a slope of -2/3 and y intercept of 12.

The second equation is given by:

2x + 3y = 6

3y = -2x + 6

y = (-2/3)x + 6       (2)

Hence the second equation has a slope of -2/3 and y intercept of 6.

The graphs of the two equations differ from each other because they have different y intercepts.

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The division rule for exponents demonstrates that when a number is raised to the zero power, it equates to 1 since the subtraction of equal exponents (as in division) yields an exponent of zero, and any number with zero instances in a multiplication chain results in the multiplicative identity, which is 1.

The division rule for exponents is integral to understanding why anything raised to the zero power is 1. The division rule for exponents states that if you divide two expressions with the same base, you can subtract the exponents.

For example, consider [tex]3^4[/tex] divided by [tex]3^4[/tex], which simplifies to [tex]3^{4-4} = 3^0[/tex]. According to the rule, when we have the same base with exponents subtracted during division, we subtract the exponents. This results in an exponent of zero for the base. Since [tex]3^4[/tex] divided by [tex]3^4[/tex] is equal to 1, and we expressed that as [tex]3^0[/tex], it logically follows that any non-zero number raised to zero power equals 1.

To further clarify, anything to the zero power signifies zero instances of the number in a multiplication chain and the implicit factor of 1 is all that remains, meaning that you're left with 1 as the result. This understanding is consistent with the properties of exponents across different types of numbers, be they integers, fractions, or irrational numbers.

The absolute value of the complex number -4 - square root of -2i is 2sqrt(6).

The absolute value of a complex number is the distance between the complex number and the origin on the complex plane. In this case, the complex number is -4 - square root of -2i. To find the absolute value, we calculate the magnitude of the complex number using the Pythagorean theorem.

The magnitude is given by the formula |z| = sqrt(a^2 + b^2), where a and b are the real and imaginary parts of the complex number, respectively.

By substituting the values a = -4 and b = -sqrt(-2i) into the formula, we have |z| = sqrt((-4)^2 + (-sqrt(-2i))^2)).

Since the square root of -2i can be simplified as 2sqrt(2), we have |z| = sqrt((-4)^2 + (-2sqrt(2))^2) = sqrt(16 + 8) = sqrt(24) = 2sqrt(6).

Answer:

There are a few ways of saying happy in Spanish such as:

-Feliz

-Alegre

-Contento

Any of the three opcións are help

Answer: D. w = 4xz - 2z/y - 2.

Step-by-step explanation: 2x - 1/y = w + 2/2z                                                             2z (2x - 1) = y (w+2)                                                                                     4xz - 2z/y = w +2                                                                                w = 4xz - 2z/y - 2

Answer:

(a)

Step-by-step explanation:

Two banks: First state bank and common bank are compared in the table based on the prices of both the banks.

After observing, we come to the conclusion that holding a first state bank will charge a monthly fee of $4 whereas in common bank there is no monthly fee.

Secondly, non sufficient bonds are cheaper in common bank as compared to first state bank.

Also, 2nd back statement is charged as $7 and $ 3 for each transaction under first state bank and common bank respectively.

Hence common bank is cheaper. Therefore,(a) common bank will save Sharon more money.

Answer:

Sharon is comparing two banks - First state bank and Common bank

The table below is comparing both the banks.

We can see that the first state bank has a monthly fee of $4 while there is no such charge in common bank. The second thing is non sufficient funds  which is less in common bank. Thirdly, the 2nd copy charges of the common bank is cheaper(though this is not needed by Sharon).The ATM usage charges of common bank is also cheaper.

Comparing everything we can say that option A is the answer.

Let

[tex] A(1,61) B(2,125) C(3,177) D(4,242) [/tex]

Using a graph tool

see the attached figure N 1

therefore

the answer is the option A

see the attached figure N 2

The correct answer is:

Graph A.

Explanation:

We can eliminate the last two graphs as they decrease; the numbers in this table continually increase.

To determine if it is graph A or graph B, we look at the spacing between the points. In the first graph, there is a steep jump between the first two points. The next two points increase more slowly, and there is another jump between the last two.

In the second graph, there is a slight increase between each pair of points. This does not match the data in the table; the first two points increase from 61 to 125, whereas the next two points go from 125 to 177, which is a smaller increase. This means the correct graph is graph A.

Answer:        

3)[tex]y = x^2[/tex], [tex]y = 2x^2[/tex], [tex]y = 3x^2[/tex]

4) [tex]y = -x^2[/tex],[tex]y = -5x^2[/tex], [tex]y = -4x^2[/tex]

5) It is shifted 1 unit up.

Step-by-step explanation:

3) Since, the general equation of parabola is,

[tex]f(x) = a(x-h)^2+k[/tex]

If 0<a<1 then f(x) is wider than [tex]f(x)=x^2[/tex]

If a>1 then f(x) is narrow that [tex]f(x)=x^2[/tex]

Thus, the required sequence of equations from widest to narrow graph,

[tex]y = x^2[/tex], [tex]y = 2x^2[/tex], [tex]y = 3x^2[/tex]

4) Again if a<0 then [tex]f(x)=-a(x-h)^2+k[/tex] is wider than [tex]f(x)= -x^2[/tex]

Thus, the required sequence from widest to narrowest graph is,

[tex]y = -x^2[/tex],[tex]y = -5x^2[/tex], [tex]y = -4x^2[/tex]

5) Since, In equation of parabola,

[tex]f(x) = a(x-h)^2+k[/tex]

k shows the y-coordinate of the vertex of the parabola,

⇒ k shows the shifting along y-axis.

Thus, If the graph [tex]y=4x^2[/tex] is transformed to [tex]y=4x^2+1[/tex]

Then we will say it is shifted 1 unit up or shifted vertically by the factor 1.

Therefore, First Option is correct.

Question (3): The correct option is [tex]\boxed{\bf option (d)}[/tex].

Question (4): The correct option is [tex]\boxed{\bf option (c)}[/tex].

Question (5): The correct option is [tex]\boxed{\bf option (a)}[/tex].

Further explanation:

Question (3):

Solution:

The best method to find which graph is widest and narrowest is graphing a quadratic function.

The vertically stretching of graph of [tex]f(x)[/tex] by a factor of [tex]c[/tex] can be obtained by [tex]g(x)=cf(x)[/tex] if [tex]c[/tex] is greater than [tex]1[/tex].

The graph is stretched as we increase the value of [tex]c[/tex] in first case the value of [tex]c[/tex] is [tex]1[/tex] and in the second case the value of [tex]c[/tex] is [tex]2[/tex] and the third case is value of [tex]c[/tex] is [tex]3[/tex].

It means that the third case has maximum stretching that leads to a narrow graph, and similarly in second case the value of [tex]c[/tex] is [tex]2[/tex] it is also stretches but less than the third case.

Therefore, the order of quadratic functions from widest to narrowest graph is as follows:

[tex]\boxed{(y=x^{2})>(y=2x^{2})>(y=3x^{2})}[/tex]

Figure 1 (attached in the end) represents the graph of the functions [tex]y=x^{2},y=2x^{2},y=3x^{2}[/tex].

Thus, the correct option is [tex]\boxed{\bf option (d)}[/tex].

Question (4):

Solution:

The reflection of graph of [tex]f(x)[/tex] about [tex]x[/tex]-axis can be obtained by [tex]g(x)=-f(x)[/tex].

The graph of the function [tex]y=-x^{2}[/tex] is obtained when each point on the curve of the function [tex]y=x^{2}[/tex] is reflected across the [tex]x[/tex]-axis.

Similarly, the graph of the function [tex]y=-4x^{2}[/tex] [tex]y=-5x^{2}[/tex] and is obtained when each point on the curve of the function [tex]y=4x^{2}[/tex] and [tex]y=5x^{2}[/tex] is reflected across the [tex]x[/tex]-axis respectively.

From figure 2 (attached in the end) it is observed that the graph of the function [tex]y=-x^{2}[/tex] is the widest and the graph of the function [tex]y=-5x^{2}[/tex] is the narrowest.

Therefore, the correct option is [tex]\boxed{\bf option (c)}[/tex].

Question (5):

Solution:

If a constant is added to a function, the graph of the function shifts vertically upwards if the constant is positive and it shifts vertically downwards if the constant is negative.

For example: The graph of the function of the form [tex]y=f(x)+a[/tex] is obtained when each point on the curve of [tex]y=f(x)[/tex] is shifted along the [tex]y[/tex]-axis. If [tex]a[/tex] is positive then the curve shifts vertically upwards and if [tex]a[/tex] is negative then the curve shifts vertically downwards.

Similarly, the graph of the function [tex]y=4x^{2}+1[/tex] is obtained when each point on the curve of [tex]y=4x^{2}[/tex] shifts [tex]1\text{ unit}[/tex] vertically upwards.

Figure 3 (attached in the end) represents the graph of the function [tex]y=4x^{2}[/tex] and [tex]y=4x^{2}+1[/tex].

Therefore, the correct option is [tex]\boxed{\bf option (a)}[/tex].

Learn more:

1. Representation of graph https://brainly.com/question/2491745

2. Quadratic equation: https://brainly.com/question/1332667

Answer details:

Grade: High school

Subject: Mathematics

Topic: Shifting of graph

Keywords: Graph,inequality ,y=x^2 ,y=-4x^2 ,y=-5x^2 ,y=x^2 ,y=2x^2 , y=3x^2,  shifted, stretches, widest, narrowest, quadratic function shifting, translation, curve.

The mathematical statement 4⊂A means 4 is a subset of A, option A is correct.

In set theory, the symbol "⊂" represents the subset relationship.

So, when we write "4⊂A," it means that the element 4 is a member of the set A and is included as part of the set.

It does not imply that 4 is an intersection or an element of the set, but rather that 4 is a subset of A.

Therefore, option A is the correct interpretation.

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Answer:

(C)

Step-by-step explanation:

It is given that JK is perpendicular to LM, since JK and LM both are lines which lie in the same plane, therefore JK and LM are coplanar.

Since JK is perpendicular to LM, then JK and LM will  intersect.

Thus, option A is not correct.

And, as JK is perpendicular to LM, then they will meet each other at a right angle as perpendicular lines make a right angle with each other.

Thus, Option C is correct.