F(x) =x^2-3x+9 for x=2 and f(x) =kx +1 for x>2 for what value of k , if any. Is f continuous at x=2?

Banks and financial institutions offer card loans to people who do not apply for them for a variety of reasons, including to extend credit to those with a high risk of default, to attract potential customers, and to build customer loyalty.

A loan is an agreement between a borrower and a lender in which the borrower receives an amount of money (principal) from the lender, and agrees to repay the lender with interest. The terms of the loan usually specify the repayment period, interest rates, and fees associated with the loan. Examples of loans include mortgages, student loans, car loans, and personal loans.

Card loans are a type of revolving credit that allow people to borrow money from a bank or financial institution and then pay it back, with interest, over time. Banks and financial institutions offer card loans to people who do not apply for them for a variety of reasons.

First, card loans are offered to those who are considered high credit risks, such as those with no prior credit history or a low credit score. By offering card loans, banks and financial institutions can extend credit to these individuals and help them establish a credit history. The loan may come with a higher interest rate than those offered to individuals with good credit, but it can still be beneficial for those who need access to credit.

Second, card loans are often offered as a way to entice potential customers to open accounts with the bank or financial institution. Banks may offer a low introductory interest rate and other incentives, such as rewards or cash back, to encourage new customers to open an account with them.

Finally, banks and financial institutions may offer card loans to help build customer loyalty. When customers use a bank's card for their purchases and pay off their balance each month, they may become more likely to use the bank's other services in the future.

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Complete questions as follows-
Why, do you think, do banks and financial institutions offer card loans to people that do not apply for it?

Answer: 1. 3(17-y)

Step-by-step explanation: multiply 3 to 17 and y, which you get (51) and (3). Subtract it to get 48, which is equivalent to 60 - 3y - 9 = 48

if it's wrong, i'm sorry

hope this helps!!

Answer:

To find the intersection points between the two functions, we need to solve the equation:

cos(2x) + 2 = sin(x) + 3

Rearranging, we get:

cos(2x) = sin(x) + 1

Using the identity cos(2x) = 1 - 2sin^2(x), we can rewrite the equation as:

1 - 2sin^2(x) = sin(x) + 1

Simplifying, we get:

2sin^2(x) - sin(x) = 0

Factoring out sin(x), we get:

sin(x)(2sin(x) - 1) = 0

This gives us two solutions: sin(x) = 0 and sin(x) = 1/2.

For sin(x) = 0, we get the values x = 0, π, and 2π.

For sin(x) = 1/2, we get the values x = π/6 and 5π/6.

Out of these solutions, only π/2 and 3π/2 are in the interval [0, 2π), so the intersection points are:

f(π/2) = cos(2(π/2)) + 2 = 1 + 2 = 3

g(π/2) = sin(π/2) + 3 = 1 + 3 = 4

f(3π/2) = cos(2(3π/2)) + 2 = 1 + 2 = 3

g(3π/2) = sin(3π/2) + 3 = -1 + 3 = 2

Therefore, the answer is: x equals pi over 2 and 3 times pi over 2.

Step-by-step explanation:

The functions f (x) = cos 2x + 2 and g(x) = sin x + 3 intersect at x = π/2, x = 7π/6, x = 3π/2, and x = 11π/6 in the interval [0, 2π).

In order for the functions f (x) = cos 2x + 2 and g(x) = sin x + 3 to intersect, the two equations must equal each other. Hence:

cos 2x + 2 = sin x + 3

By subtracting 2 from both sides of the equation, we are left with:

cos 2x = sin x + 1

This equation can be solved by applying the trigonometric function properties. However, in order to avoid any potential confusion, it can be much easier to use an online graphing tool to find the points of intersection. In the interval [0, 2π), the solutions are x = π/2, x = 7π/6, x = 3π/2, and x = 11π/6.

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Answer:

Rs 183.75

Step-by-step explanation:

1: 24.5

7.5/1 = 7.5

7.5*24.5 = 183.75

115500 divided by 7

115.500:7 =16.500

Answer:

8 students are in the Environmental Club

Step-by-step explanation:

Answer: 5% of 160 is 8, so there are 8 students in the environmental club.

Step-by-step explanation:

Accοrding tο the given infοrmatiοn, a linear relatiοnship between time and pοpulatiοn οccurs when the change in pοpulatiοn οver a given periοd is cοnstant οr prοpοrtiοnal tο the pοpulatiοn size at the beginning οf the periοd.

In a graph, a linear relatiοnship is a pattern in which the pοints οn a scatter plοt tend tο fall alοng a straight line. A linear relatiοnship means that there is a cοnstant rate οf change between twο variables, such that when οne variable increases by a certain amοunt, the οther variable increases οr decreases by a cοnstant amοunt as well.

A situatiοn in which a city's pοpulatiοn grοws at a cοnstant rate οver time wοuld exhibit a linear relatiοnship between time and pοpulatiοn. As time passes, the pοpulatiοn wοuld increase by a fixed amοunt. This is an example οf linear grοwth, and it cοuld be mοdeled by a linear equatiοn such as P = mt + b, where P is the pοpulatiοn, m is the slοpe (the rate οf grοwth), t is time, and b is the initial pοpulatiοn at time t=0.

Similarly, if a pοpulatiοn is decreasing by a fixed percentage each year, then the relatiοnship between time and pοpulatiοn will alsο be linear. As time passes, the pοpulatiοn will decline by a cοnstant percentage οf the previοus year's pοpulatiοn.

In general, a linear relatiοnship between time and pοpulatiοn οccurs when the change in pοpulatiοn οver a given periοd is cοnstant οr prοpοrtiοnal tο the pοpulatiοn size at the beginning οf the periοd.

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[tex]y = 1.2x + 6[/tex]

When the equation intersects with the x-axis, y = 0

when y = 0

[tex]1.2x + 6 = 0[/tex]

[tex]1.2x = -6[/tex]

[tex]x = -6 \div 1.2 = -5[/tex]

[tex]\implies \text{The coordinate is} \ (-5, 0)[/tex]

When the equation intersects with the y-axis, x = 0

When x = 0

[tex]y = 1.2(0) + 6[/tex]

[tex]y = 6[/tex]

[tex]\implies \text{The coordinate is} \ (0, 6)[/tex]

Answer: The intersection with the x − axis and y − axis is the point with coordinates (-5 , 0) and (0 , 6)

Ordering the probabilities from most likely to occur to least likely to occur:

P(football, then volleyball) = 1/49

P(volleyball, then volleyball) = 1/25

P(baseball, then football, then volleyball) = 3/1000

P(baseball, then football, then football) = 5/2944

P(event) = (number of ways the event can occur) / (total number of possible outcomes)

P(volleyball, then volleyball) if the ball is replaced after the first selection:

The probability of selecting a volleyball on the first draw is:

P(volleyball on first draw) = 10/50 = 1/5

Since the ball is replaced after the first selection, the probability of selecting a volleyball on the second draw is also 1/5. Therefore:

P(volleyball, then volleyball) = P(volleyball on first draw) * P(volleyball on second draw) = 1/5 * 1/5 = 1/25

P(baseball, then football, then volleyball) if the balls are replaced after each selection:

The probability of selecting a baseball on the first draw is:

P(baseball on first draw) = 15/50 = 3/10

Since the ball is replaced after each selection, the probability of selecting a football and a volleyball on the second and third draw respectively is also 1/10. Therefore:

P(baseball, then football, then volleyball) = P(baseball on first draw) * P(football on second draw) * P(volleyball on third draw) = 3/10 * 1/10 * 1/10 = 3/1000

P(baseball, then football, then football) if the balls are not replaced after each selection:

The probability of selecting a baseball on the first draw is:

P(baseball on first draw) = 15/50 = 3/10

Since the ball is not replaced after the first selection, the probability of selecting a football on the second draw is:

P(football on second draw) = 5/49

Since the ball is not replaced after the second selection, the probability of selecting a football on the third draw is:

P(football on third draw) = 4/48

Therefore: P(baseball, then football, then football) = P(baseball on first draw) * P(football on second draw) * P(football on third draw) = 3/10 * 5/49 * 4/48 = 5/2944

P(football, then volleyball) if the ball is not replaced after the first selection:

The probability of selecting a football on the first draw is:

P(football on first draw) = 5/50 = 1/10

Since the ball is not replaced after the first selection, the probability of selecting a volleyball on the second draw is:

P(volleyball on second draw) = 10/49

Therefore: P(football, then volleyball) = P(football on first draw) * P(volleyball on second draw) = 1/10 * 10/49 = 1/49

Ordering the probabilities from most likely to occur to least likely to occur:

P(football, then volleyball) = 1/49

P(volleyball, then volleyball) = 1/25

P(baseball, then football, then volleyball) = 3/1000

P(baseball, then football, then football) = 5/2944

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The effect of changing the sample sizes on the interval would depend on the relative size of the populations. Generally, a larger sample size will lead to a more precise interval, but if population X is much larger than population Y, then the size of the interval will not be affected much by the change.

If a random sample of 52 is selected from population x, and a random sample of 63 is selected from population y and when all other things remain constant, such a change in sample sizes will cause the margin of error to decrease.

When the sample size increases, the standard error of the mean decreases as well. Since the margin of error is directly linked to the standard error of the mean, this is the case. As a result, when the sample size is raised, the margin of error decreases.

The margin of error is a statistical concept that reflects the degree of uncertainty in a sample estimate. As a result, when the margin of error decreases, the precision of the results increases. As a result, when the sample size is increased, the data becomes more accurate.

So, when all other factors remain constant, an increase in sample size produces a more accurate result.

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The expected number of full days hereafter that the machine will remain operational before the next breakdown is still 10 days

We calculated that the expected number of full days until the next breakdown was 10 days. Now, suppose that the machine has already gone 15 full days without a breakdown since the last repair was completed. This means that the machine has already exceeded the expected number of full days until the next breakdown.

However, the fact that the machine has already gone 15 full days without a breakdown does not change the underlying probability distribution of the time until the next breakdown. The distribution is still exponential with a mean of 10 days. Therefore, even though the machine has already exceeded the expected number of full days until the next breakdown, the expected number of full days hereafter that the machine will remain operational before the next breakdown is still the same as before: 10 days.

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Answer:xplanation:

the additive inverse of a number is the opposite of the number. It is the amount you add to equal zero. 5 + -5 = 0 so -5 is the additive inverse of 5.

-2/3 + 2/3 =0 so 2/3 is the additive inverse of -2/3

Answer:

(i) To find the actual area of the field represented on the map by a rectangle 6 cm long and 4 cm wide, we need to first convert the dimensions of the rectangle to actual units. Since the scale of the map is 1:20,000, this means that 1 cm on the map represents 20,000 cm in actuality.

The length of the rectangle on the map is 6 cm, so its actual length is:

6 cm x 20,000 cm/cm = 120,000 cm

Similarly, the width of the rectangle on the map is 4 cm, so its actual width is:

4 cm x 20,000 cm/cm = 80,000 cm

The actual area of the rectangle is then:

Actual area = actual length x actual width

Actual area = 120,000 cm x 80,000 cm

Actual area = 9,600,000,000 cm^2

Therefore, the actual area of the field represented on the map by a rectangle 6 cm long and 4 cm wide is 9,600,000,000 cm^2.

(ii) To convert the actual area to hectares, we need to divide by 10,000 m^2/ha, since 10,000 m^2 is equal to 1 hectare.

Actual area in hectares = (9,600,000,000 cm^2) / (10,000 m^2/ha)

Actual area in hectares = 960,000 m^2 / 10,000 m^2/ha

Actual area in hectares = 96 hectares

Therefore, the area of the field in hectares is 96 hectares.

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Answer:

is this a quize theres no insructions

Answer:

The sample space of randomly selecting 2 lollipops with replacement is

(GG,  GC, GL, CG, CC, CL, LG, LC, LL}

The sample space of randomly selecting 2 lollipops without replacement is:

{GC, GL, CG, CL, LG, LC}

The experiment of randomly selecting 2 lollipops without replacement shows dependent events because the probability of selecting the second lollipop depends on what was selected in the first draw.

hope this helps

The line of reflection for the transformation between polygon ABCDE and A' B' C' D' E' is the x-axis and the y-axis.

we need to look at the position of their corresponding vertices.

If we reflect polygon ABCDE across the x-axis, the y-coordinates of all its vertices will be negated.

For example, the first vertex A(-1, -4) will become A'(-1, 4).

We can see that the y-coordinates of the corresponding vertices of the two polygons are negated with respect to each

other. Therefore, the line of reflection is the x-axis.

If we reflect polygon ABCDE across x = 6, the x-coordinate of vertex A(-1, -4) will become 13, which is not the x

coordinate of any vertex in A' B' C' D' E'. Therefore, this cannot be the line of reflection.

If we reflect polygon ABCDE across y = -3, the y-coordinate of vertex A(-1, -4) will become 1, which is not the y-

coordinate of any vertex in A' B' C' D' E'.

Therefore, this cannot be the line of reflection.

If we reflect polygon ABCDE across the y-axis, the x-coordinates of all its vertices will be negated. For example, the first

vertex A(-1, -4) will become A'(1, -4). We can see that the x-coordinates of the corresponding vertices of the two

polygons are negated with respect to each other. Therefore, the line of reflection is the y-axis.

Therefore, the line of reflection for the transformation between polygon ABCDE and A' B' C' D' E' is the x-axis and the

y-axis.

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The level of the liquid is dropping at a rate of 9/(4π) inches per second when the liquid is 1 inch deep.

We are given that the filter is in the shape of a vertex-down cone, with a height of 6 inches and a diameter of 4 inches at its open (upper) end. We are also given that the liquid drips out the bottom of the filter at a constant rate of 3 cubic inches per second.

Let's let h be the height of the liquid in the cone, and let V be the volume of the liquid in the cone.

The volume of a cone is given by the formula:

V = (1/3)πr²h,

where r is the radius of the cone. Since the diameter of the cone at its open end is 4 inches, the radius is 2 inches. Thus, we have:

V = (1/3)π(2²)h = (4/3)πh.

Now, we need to find dh/dt, the rate at which the height of the liquid is dropping, when the liquid is 1 inch deep. Since the liquid is dripping out of the filter at a constant rate of change 3 cubic inches per second, we have:

dV/dt = -3.

Taking the derivative of the equation for V with respect to t, we get:

dV/dt = (4/3)π dh/dt.

Substituting -3 for dV/dt, and substituting 1 for h, we get:

-3 = (4/3)π dh/dt.

Solving for dh/dt, we get:

dh/dt = (-3)/(4/3)π = -9/(4π) inches per second.

Therefore, the level of the liquid is dropping at a rate of 9/(4π) inches per second when the liquid is 1 inch deep.

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The answer for the first triangle is 1 triangle(s) and the answer for the second triangle is 0 triangle(s).

According to the given measures, for the triangle with m∠E = 64°, g = 9, and e = 10, only 1 distinct triangle can be formed. In this case, the given angle is opposite the longest side, and hence one distinct triangle can be formed.

On the other hand, for the triangle with m∠J = 129°, k = 8, and j = 3, no distinct triangle can be formed as the given angle is greater than 90 degrees, which violates the triangle inequality theorem.

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In hypothesis testing, if the test value is in the critical region, the researcher should fail to reject the null hypothesis. The given statement is true.

Hypothesis testing is a statistical process used to determine the validity of a statement that is made about a population. It's a statistical procedure that aids in making inferences about population parameters from sample data. It's used to test the null hypothesis against the alternative hypothesis, which determines whether or not a sample statistic and the claim made by a population parameter are in agreement.

The null hypothesis is that the population parameter is equal to a particular value, while the alternative hypothesis is that it is different from that value.

In a hypothesis test, the calculated value of a test statistic is compared to a critical value. The null hypothesis is rejected if the calculated value exceeds the critical value, implying that the alternative hypothesis is correct. However, if the calculated value of the test statistic falls within the critical region, the null hypothesis cannot be rejected, implying that there is insufficient evidence to support the alternative hypothesis.

Thus, the given statement is true.

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As a result, Class B's average math test score is roughly 57.67 as rounded to 2 decimal places.

The word "mean" is typically used in mathematics to describe one of the various averages that are used to represent a set of numerical data. The arithmetic mean, usually referred to as the average, is the most widely used kind of mean. For calculating the arithmetic mean, all the values in a set of data are added up, and the sum is then divided by the total number of values. It provides a general understanding of the "core" value in a collection of data. For instance, adding up each person's height and dividing it by the total number of individuals would yield the average height of the group.

given

Use this equation to determine the mean (average) of a group of numbers:

mean = (all numbers added together) / (number of numbers)

Since we are aware that the mean score for Class A is 21, we can use the following calculation to determine the total of all Class A scores:

Sum of Class A Grades = Mean * Number of Class A Students = 21 * 28 = 588

Also, we are aware that the average score for both classes is 39, thus we can use the following formula to determine the total score for both classes:

Average of the students' combined test results in both classes is 39 * (28 + 27) = 39 * 55, or 2145.

Lastly, we can apply the mean calculation to determine the mean grade in Class B:

Sum of Class B students' scores divided by the total number of Class B students equals 1557 divided by 27.

= 57.67 (rounded to 2 decimal places) (rounded to 2 decimal places)

As a result, Class B's average math test score is roughly 57.67 as rounded to 2 decimal places.

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So, Isaac should pour 81 cubic inches of mixture into the pan.

Volume is a physical quantity that refers to the amount of space occupied by an object or a substance. It is typically measured in cubic units, such as cubic meters, cubic centimeters, or cubic feet.

The volume of an object or substance can be determined by measuring its length, width, and height (in the case of a solid object), or by measuring the amount of space it occupies (in the case of a liquid or gas).

In mathematical terms, the volume of an object is given by the formula V = l x w x h, where l is the length, w is the width, and h is the height. For irregularly shaped objects, the volume may be determined by displacement - that is, by measuring the amount of water or other fluid displaced by the object when it is immersed in the fluid.

To find the volume of the mixture that Isaac should pour into the pan, we need to find the volume of the pan itself and subtract the volume of the space that needs to be left empty.

The pan is in the shape of a rectangular prism, so its volume is:

V_pan [tex]= l *w * h = 9 in. * 9 in. * 2 in. = 162 in^{3}[/tex]

To leave room for the casserole to rise, Isaac should only fill the pan to a height of 1 inch. So, the volume of the space that needs to be left empty is:

V_empty [tex]= l *w * h = 9 in. * 9 in. * 2 in. = 81 in^{3}[/tex]

Therefore, the volume of the mixture that Isaac should pour into the pan is:

V = V_pan - V_empty[tex]= 162 in^{3} - 81 in^{3} = 81 in^{3}[/tex]

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Answer:

5 Bikes, 3 Scooters

Step-by-step explanation:

(5*15.20) + (8.50*3) = 101.50

Answer:

Step-by-step explanation:

Para encontrar el precio del videojuego después de que primero aumentó un 25 % y luego se redujo un 20 %, debemos seguir estos pasos:

Paso 1: Calcular el nuevo precio después del aumento del 25%.

Para ello, podemos multiplicar el precio original por 1,25 (que representa el 25% de incremento):

Nuevo precio tras la subida del 25% = 55,32€ x 1,25 = 69,15€

Paso 2: Calcula el precio después de la reducción del 20%.

Para ello, podemos multiplicar el nuevo precio (después de la subida del 25 %) por 0,80 (que representa la rebaja del 20 %):

Precio después de la reducción del 20% = 69,15 € x 0,80 = 55,32 €

Por lo tanto, el precio final del videojuego tras subir un 25% y luego reducirlo un 20% es de 55,32€. Este es el mismo que el precio original del videojuego.

R'(-6, 0), S'(-1, 3), T'(2, -2), and U' are the vertices of the converted square R'S'T'U' (-3, -5).

A transformation in arithmetic is the process of altering the positioning, scale, form, or alignment of a polyhedron. You can think of transformations as functions that translate coordinates from one system to another, which might or might not be the same.

In mathematics, there are various different kinds of transformations, including: Translation: A figure is modified when it is moved a particular distance in one direction without affecting its symmetry or size. Reflection:

A transformation called a reflection flips an image across a line known as the line of reflective thinking. The image appears as just a mirror image of the original figure because the figure is projected across this line.

given

Applying the given transformation to each vertex of the original square RSTU will allow us to graph the transformed square R'S'T'U.

Vertex R'(-6, 0) changes from R(-2, 1):

-4 = -2 - 4, and -1 = 1 - 1

S(3, 4) converts to S'(-1, 3) at the vertex.

-1 = 3 - 4, and 2 = 4 - 1

Transformation of vertex T(6, -1) to T'(2, -2):

2 = 6 - 4, and -1 = -1 - 1

Vertex U(1, -4) changes to U'(-3, -5) because -3 = 1 - 4 and -4 = -4 - 1

R'(-6, 0), S'(-1, 3), T'(2, -2), and U' are the vertices of the converted square R'S'T'U' (-3, -5).

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question is -

Square RSTU is translated to form R'S'T'U', which has vertices R'(–8, 1), S'(–4, 1), T'(–4, –3), and U'(–8, –3). If point S has coordinates of (3, –5), which point lies on a side of the pre-image, square RSTU?

(–5, –3)

(3, –3)

(–1, –6)

(4, –9)

The direction and magnitude of the resultant force is 7,870.1 lbs at 5.3 ° south of west.

We can start by breaking down the forces into their x and y components:

The force of 3700 lbs pulling west has an x-component of -3700 lbs and a y-component of 0 lbs.

The force of 4200 lbs pulling 10° north of west has an x-component of -4200 cos(10°) lbs and a y-component of 4200 sin(10°) lbs.

The x-component of the resultant force is;

Fx = -3700 lbs - 4200 cos(10°) lbs

Fx = -7,836.19 lbs

The y-component of the resultant force is;

Fy =  4200 sin(10°) lbs

Fy = 729.32 lbs

To find the magnitude and direction of the resultant force, we can use the Pythagorean theorem and inverse tangent function:

The magnitude of the resultant force is;

F = √(Fx² + Fy²)

F = √ (-7836.19² + 729.32²)

F = 7,870.1 lbs

The direction of the resultant force is;

θ = arc tan (Fy/Fx)

θ = arc tan (729.32 / 7836.19)

θ = 5.3⁰

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Answer:

To write the equation y - 7 = 8(x + 2) in slope-intercept form, we need to solve for y:

y - 7 = 8(x + 2)

Distribute the 8:

y - 7 = 8x + 16

Add 7 to both sides:

y = 8x + 23

Now, we have the equation in the form y = mx + b, where m is the slope and b is the y-intercept. Therefore, the slope of the line is 8, and the y-intercept is 23.

The slope-intercept form of the equation is: y = 8x + 23.

Step-by-step explanation:

Answer:

Step-by-step explanation:

[tex]y-7=8(x+2)[/tex]

[tex]y-7=8x+16[/tex]       (expanded brackets)  

[tex]y=8x+23[/tex]              (+7 both sides)

Solution: [tex]y=8x+23[/tex]

Therefore, the best prediction for the number of times the spinner will land on blue in 6 spins is 1. However, it's important to note that this is just a prediction based on probability, and the actual number of times the spinner lands on blue may vary from this prediction.

Probability refers to the measure or quantification of the likelihood of an event or outcome occurring. It is a numerical value between 0 and 1, where 0 represents an impossible event (never occurring) and 1 represents a certain event (always occurring).

Probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if we flip a fair coin, the probability of getting heads is 0.5, because there is one favorable outcome (heads) out of two possible outcomes (heads or tails).

by the question.

Assuming that the spinner is fair and unbiased, and that the probability of landing on blue is equal to the probability of landing on any other color, the best prediction for the number of times it will land on blue is simply the expected value of the number of times it will land on blue in 6 spins.

Since there are six equally likely outcomes each time the spinner is spun, the probability of landing on blue is 1/6. Thus, the expected number of times the spinner will land on blue in 6 spins is:

Expected number of times landing on blue = probability of landing on blue × number of spins

Expected number of times landing on blue = 1/6 × 6

Expected number of times landing on blue = 1

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To find where the line crosses the x-axis, we need to find the value of x the line crosses the x-axis at the point   [tex](-5/3, 0).[/tex]

a) The gradient of a line is the coefficient of the x term in its equation in the form y = mx + c, where m is the gradient. Therefore, the gradient of the line with equation [tex]y = 3x + i is 3[/tex]  .

b) Since the line passes through the point (2, 11), we can substitute these values into the equation  [tex]y = 3x + i[/tex] to find the value of i:

[tex]11 = 3(2) + i[/tex]

[tex]11 = 6 + i[/tex]

[tex]i = 11 - 6[/tex]

[tex]i = 5[/tex]

Therefore, the equation of the line can be written as   [tex]y = 3x + 5.[/tex]

c) The value of k in the equation   [tex]11 = 6 + k[/tex] is simply the y-intercept of the line. From the equation [tex]y = 3x + 5[/tex]  , we can see that the y-intercept is 5. Therefore, k = 5, and the equation of the line is [tex]y = 3x + 5.[/tex]

d) To find where the line crosses the x-axis, we need to find the value of x when y = 0. Substituting y = 0 into the equation [tex]y = 3x + 5[/tex]  , we get:

[tex]0 = 3x + 5[/tex]

[tex]x = -5/3[/tex]

Therefore, the line crosses the x-axis at the point  [tex](-5/3, 0).[/tex]

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a) The gradient of the line is 3.

c) The Value of k is 6.

d) The line crosses the y-axis at (0, 5).

The line's equation is y = 3x + k,

a) Compare this with the standard equation of line y = mx + c

where 3 is the coefficient of x.

The slope or gradient of a line is represented by the coefficient of x in its equation.

As a result, the gradient of the line y = 3x + k is 3.

b) We now know that the line goes through the point. (2, 11). Using the equation y = 3x + k, we can find the value of the y-intercept (k) at this moment. When we substitute x = 2 and y = 11 into the equation, we get:

11 = 3(2) + k

When we simplify the right side, we get:

11 = 6 + k

c) When we subtract 6 from both sides, we get:

k = 11 - 6 = 5

As a result, the equation of the line is as follows:

y = 3x + 5.

d) So, the y-intercept value is 5, indicating that the line crosses the y-axis at the location. (0, 5).

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We need to find the measure of angle m∠2.

Given that AB∥DE, C is the point of intersection of line AE and line DB, m∠1 = 80°, and m∠2 : m∠3 = 2 : 3.

Step 1: Identify the alternate interior angles. Since AB∥DE, we know that ∠1 and ∠2 are alternate interior angles.

Step 2: Use the property of alternate interior angles. The property of alternate interior angles states that if two parallel lines are cut by a transversal, the alternate interior angles are congruent. Therefore, m∠1 = m∠2.

Step 3: Substitute the given values. We know that m∠1 = 80°, so m∠2 = 80°. Hence, the measure of angle 2 (m∠2) is 80°.

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Final cost of the item with an original price of $23 and a discount of 15% is $19.5 for given expression.

To calculate the final cost of an item that has a discounted price of 15%, we can use the expression p - 0.15p, where p is the original price of the item. Simplifying:

p - 0.15p = 0.85p

This means that the final cost of the item will be 85% of the original price.

If original price of the item is $23, substitute value into expression to find final cost:

0.85 x $23 = $19.55

Final cost of item with an original price of $23 and a discount of 15% is $19.55.

It's important to note that this calculation assumes that the discount is applied before any taxes or other fees. If there are additional costs associated with the item, such as sales tax, shipping, or handling fees, these should be added to the final cost separately.

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