Tim has 42 marbles. He divides them equally into 7 bags. How many marbles are in each bag?

Chris's time 3.058 sec faster than Michael's.

What is meaning of faster?

Fast and fast refer to moving quickly. Fast can be used as an adverb or an adjective. Quick is an adjective, while rapidly is its adverbial counterpart.

Given that,

Chris completed a 100-meter breaststroke swimming race in 92.542 seconds.

Michael completed the 100-meter breaststroke swimming race in 95.6 seconds.

The difference of time between Chris and Michael to complete 100-meter breaststroke swimming is = (95.6 - 92.542) = 3.058 sec.

Since both are completed 100 meter breaststroke, thus the difference of time denotes the faster time.

Thus Michael's timing was 3.058 seconds slower than Chris'.  

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Step-by-step explanation:

an arithmetic operation (like "-") on functions is the same as using the same operation on the functional excursions themselves.

(f - g)(x) = -x + 3 - (-10x - 9) = -x + 3 + 10x + 9 =

= 9x + 12

(f - g)(8.5) = 9×8.5 + 12 = 76.5 + 12 = 88.5

The Perimeter of the Quadrilateral will be P = 3s³ + s² + 5s + 1

What is Quadrilateral?

A quadrilateral is a four-sided polygon with four edges and four corners in geometry. The name derives from the Latin words quadri (a variation of four) and latus (a side).

Solution:

Given:

Length of Longest side is three times more than the cube of shortest side

Length of Another side is 6cm less than square of shortest side

Length of Third side is 7cm more than four times the length of shortest side

To Find:

Perimeter of the Quadrilateral in terms of Shortest Side

Let, Shortest Side be s

So, longest side will be 3 * s³

Another Side will be s² - 6

Third Side will be 4s + 7

Perimeter of the Quadrilateral will be s + 4s +7 + s² - 6 + 3s³

5s + 1 + s² + 3s³

P = 3s³ + s² + 5s + 1

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Answer:

The time when the rocket is at its highest point is 8.5 seconds.

Answer: x = 8 1/2 i believe

Step-by-step explanation:

The grand mean fir the observations of the age of three different types of vehicles currently on the road: trucks, cars, and vans is Eight . So, correct option is option B.

The overall or pooled average is the average of multiple subsample averages, as long as the subsamples have the same number of data points. In ANOVA, the overall mean for a set of multiple subsamples is the mean of all observations. that is, each data point divided by the common sample size.

∑x = represented by the sum of the averages of all sets. Overall grand mean formula is

Grand mean = sum of all sample mean / total number of sentences

We have given that,

Autotrader would like to test difference exists in the age of three different types of vehicles.

let set A , set B and set C represents the ages of three different vichles i.e trunks , cars and vans respectively.

From table we get,

Mean of ages of trunks , A = 10

Mean of ages of cars , B = 8

Mean of ages of vans , C = 6

Grand mean = (10+8+6)/3 = 8

Hence, the grand mean is 8..

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The length (in inches) of the second tablecloth is 43.75 inches

From the question, we have the following parameters that can be used in our computation:

Width = 48 inches. and Length = 72 inches ⇒ Small tablecloth

Width = 30 inches. and Length = ?? inches ⇒ Large tablecloth

Represent the unknown length with x

So, we have

Width = 48 inches. and Length = 72 inches ⇒ Small tablecloth

Width = 30 inches. and Length = x inches ⇒ Large tablecloth

This shows a direct variation

So, we cross multiply the equation

x * 48 = 70 * 30

So, we have

x = 70 * 30/48

Evaluate

x = 43.75

Hence, the length is 43.75 inches

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Answer:

x = 1

Step-by-step explanation:

x + 7 = 8 ( isolate x by subtracting 7 from both sides )

x = 8 - 7 = 1

The equivalent fraction for the given fraction is 1/5.

Equivalent fractions are two or more fractions that are all equal even though they different numerators and denominators.

The given fraction is 3/15.

The equivalent fraction is 3/15 =1/5

Therefore, the equivalent fraction for the given fraction is 1/5.

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There is insufficient evidence to conclude that there is a significant linear relationship between

x and y because the correlation coefficient is not significantly different from zero.

The correlation coefficient, r, tells us about the strength and direction of the linear relationship between x and y.

However, the reliability of the linear model also depends on how many observed data points are in the sample.

We need to look at both the value of the correlation coefficient r and the sample size n, together.

We perform a hypothesis test of the “significance of the correlation coefficient” to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute r, the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only have sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r, is our estimate of the unknown population correlation coefficient.

The symbol for the population correlation coefficient is ρ, the Greek letter “rho.”

The hypothesis test lets us decide whether the value of the population correlation coefficient

ρ is “close to zero” or “significantly different from zero”. We decide this based on the sample correlation coefficient r and the sample size n.

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is “significant.”

There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero. What the conclusion means: There is a significant linear relationship between x and y. We can use the regression line to model the linear relationship between x and y in the population.

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is “not significant.”

“There is insufficient evidence to conclude that there is a significant linear relationship between

x and y because the correlation coefficient is not significantly different from zero.” What the conclusion means: There is not a significant linear relationship between x and y. Therefore, we CANNOT use the regression line to model a linear relationship between x and y in the population.

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We must consider the sample size n together with the correlation coefficient r's value.

To determine whether the linear relationship in the sample data is strong enough to utilize to model the relationship in the population, we conduct a hypothesis test of the "significance of the correlation coefficient."

The correlation coefficient for the sample, r, is calculated using the sample data. The population correlation coefficient could be discovered if we had data for the entire population. However, we are unable to determine the population correlation coefficient because we only have sample data. Our best guess for the unknown population correlation coefficient is the sample correlation coefficient, or r.

The Greek letter "rho" serves as the sign for the population correlation coefficient.

is "near zero" or "quite different from zero." Based on the sample correlation coefficient r and the sample size n, we make this determination.

We refer to the correlation coefficient as "significant" if the test results show that it differs considerably from zero.

The fact that the correlation coefficient differs sufficiently from zero is sufficient evidence that x and y are meaningfully related linearly. The result indicates that there is a significant linear relationship between x and y. The regression line can be used to model the population's linear relationship between x and y.

Because the correlation coefficient does not deviate considerably from zero, x and y. What the conclusion says is that x and y do not have a meaningful linear connection.

The triangles are considered to be congruent if one right triangle's hypotenuse and another right triangle's hypotenuse are equal.

Given, assuming that the hypotenuse of one right triangle is equivalent to the hypotenuse of another right triangle, then the triangles are harmonious.

The given statement must be verified as true or false by us.

According to the RHS congruence theorem, two right-angled triangles are congruent if the hypotenuse and side of one triangle are equal to the hypotenuse and side of another.

The hypotenuse and the side of the triangle that corresponds to it must be equal according to the RHS rule.

The hypotenuses of the two triangles are equal, per the question.

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Guests who appear on talk shows and converse with one another for the benefit of the audience make up a (C) panel discussion.

Participants in a panel discussion are guests who appear on talk shows and engage in conversation with one another for the benefit of the audience.

A panel discussion, or simply a panel, is a gathering of people who talk about something in front of an audience, usually at fan conventions, conferences for business or academia, or on television.

Panels often have a moderator who leads the conversation and occasionally solicits questions from the audience with the intention of being both educational and entertaining.

Fan convention film panels have been attributed with increasing box office revenues by creating anticipation.

Therefore, guests who appear on talk shows and converse with one another for the benefit of the audience make up a (C) panel discussion.

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By using distance formula, it can be calculated that

Length of the rectangle is 4 units

Width of the rectangle is 2 units

Perimeter of the rectangle is 12 units

Area of the rectangle = 8 sq. units

Distance between (0, 0) and (4, 0) = [tex]\sqrt{(4-0)^2 + (0 - 0)^2}[/tex]

                                                         = [tex]\sqrt{16}[/tex]

                                                         = 4 units

Length of the rectangle is 4 units

Distance between (0, 0) and (0, 2) = [tex]\sqrt{(0-0)^2 + (0 - 2)^2}[/tex]

                                                         = [tex]\sqrt{4}[/tex]

                                                         = 2 units

Width of the rectangle is 2 units

Perimeter of the rectangle = 2 [tex]\times[/tex] (4  +2)

                                            = 12 units

Area of the rectangle = 4 [tex]\times[/tex] 2 = 8 sq. units

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A set of two or more linear equations involving the same set of variables make up a linear system of equations, which is the type of system of equations in question.

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In order to enclose the garden's perimeter, vintage fence will be used. The perimeter of a rectangle-shaped garden is equal to the sum of the garden's two lengths and two widths.

In order to enclose the garden's perimeter, vintage fence will be used. The perimeter of a rectangle-shaped garden is equal to the sum of the garden's two lengths and two widths.

According to their estimates, our perimeter fence measures 220 feet, hence 220 = 2L + 2W.

We can make this more manageable if we divide everything by two: 110 = L + W

The width is now specified as being "exactly two-thirds of the length."

In math, the letter "=" equals the word "IS". Consequently, "the width is" becomes "W =," and "the two-thirds of the length" is precisely what it says, "(2/3) L."

W = (2/3)L is the result of adding these.

and we will add this W to the formula above (110 = L + W) to make 110 = L + (2/3)L.

If we substitute L with (3/3) L (that is, 1*L with 1 written 3/3) now we can add (3/3) as adding fractions requires a "common denominator."

L + (2/3)

L = (5/3)L

At this point, our equation is 110 = (5/3)L.

The right side (3/5) is now multiplied by the reciprocal on both sides, yielding (3/5)(110) = (3/5)(5/3)

L66 = L and W=2/3

L = (2/3)(66) = 44 = W

66 feet by 44 feet are the measurements of the square flower garden.

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Problem 1

With simple interest, we take a percentage of the loan amount (or deposit depending how you frame things) to determine the interest paid or earned.

The simple interest formula is

i = P*r*t

Let's look at an example: You deposit $100 into an account earning 5% simple interest. You want to know how much interest you earn over 2 years.

i = P*r*t

i = 100*0.05*2

i = 10

You've earned $10 in simple interest. Notice how this is 10% of the original deposit ($100). You can think of the 5% interest rate doubling to 10% because we're working over a 2 year period.

---------------------------------

Compound interest is where we can't take a simple percentage of the deposit. This is because we must use this formula

A = P*(1+r/n)^(n*t)

variables are:

An example: You deposit $100 at an interest rate of 5%. The interest is compounded quarterly (4 times a year). How much interest have you made in 2 years?

The input values will be

So,

A = P*(1+r/n)^(n*t)

A = 100*(1+0.05/4)^(4*2)

A = 110.448610118141

A = 110.45

i = interest

i = A - P

i = 110.45 - 100

i = 10.45

You've made $10.45 in interest over the two year period.

As you can see, we used an exponential function to determine compound interest. Whereas with simple interest, the function was linear. Exponential functions grow much faster compared to linear functions.

======================================================

Problem 2

Savings accounts offer higher interest rates, which encourages people to save their money (hence the name). Checking accounts on the other hand are used to write checks and are there for immediate withdrawals of cash. Checking accounts are also used for debit cards, such as for online shopping.

Both types of accounts are useful in their own different ways. Linking them together is handy to quickly move money without having to deal with tons of paperwork and/or lots of time spent.

======================================================

Problem 3

APR = annual percentage rate

APY = annual percentage yield

The APR value is often the stated item on a loan, since it is the smaller of the two values. When it comes to savings accounts in banks, they tend to state the APY value since it is larger.

Let's look at an example:

The APR is 5% and the money is compounded monthly (12 times a year). Compute the APY.

----------

Solution:

The formula we use is

y = (1+r/n)^n

where y is the APY and r is the APR

In this case we have

Giving us the following

y = (1+r/n)^n - 1

y = (1+0.05/12)^12 - 1

y = 1.05116189788173 - 1

y = 0.0512

Then multiply that by 100, aka move the decimal point two spots to the right, to arrive at an APY of approximately 5.12%

This example demonstrates how compound interest slightly bumps up the amount of return. We've gone from 5.00% to 5.12%

This slight increase is because the money is compounded 12 times a year. Each time we compound, we're adding to the pile ever so slightly. There's a limit to this process (see the continuous compounding interest formula); i.e. we can't grow the APY forever.

If we were using simple interest, then the APY formula shown above would not apply. For simple interest situations, the APY and APR are the same value.

The the equation of line in slope - intercept form is [tex]y = -\frac{1}{3}x-3[/tex] and the slope for the given table is 0.

What is slope intercept form of the line?

When you know the slope of the line to be examined and the given point is also the y intercept, you can use the slope intercept formula, y = mx + b. (0, b). The y value of the y intercept point is denoted by the symbol b in the formula.

Given:

The line passes through the point (3, -4) and having the slope is -1/3.

Consider the point - slope form of the line,

[tex]y-y_1=m(x-x_1)[/tex]

where, [tex](x_1, x_2)[/tex] is the given point and m is the slope.

Plug [tex](x_1, y_1) = (3, -4)[/tex] and m = -1/3 in the above equation.

[tex]y-(-4)=-\frac{1}{3}(x-3)\\ y+4=-\frac{1}{3}x+1\\ y=-\frac{1}{3}x+1-4\\ y=-\frac{1}{3}x-3[/tex]

This is the equation of line in slope - intercept form.

Now to find the slope of the line that passes through the given points on the table.

Since,

Slope = [tex]\frac{Change In y}{Change In x}[/tex]

From the given table the change in y is 0, and the change in x is 5 for all the values.

So,

Slope = 0/5 = 0

Hence, the the equation of line in slope - intercept form is [tex]y = -\frac{1}{3}x-3[/tex] and the slope for the given table is 0.

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The value of ATC, AFC, TVC if TC = 6Q² + 5Q + 10, are 6Q + 5 + 10/Q, 5 + 6Q,  5Q + 6Q² respectively.

Average total cost (ATC) is calculated by dividing the entire cost by the total amount of output produced. The additional cost incurred by creating one more unit of output is known as marginal cost (MC).

Given:

TC = 6Q² + 5Q + 10,

Calculate the value of AFC as shown below,

AFC = 10 / Q

Calculate the value of TVC as shown below,

TVC = TC - TFC

TVC = 5Q + 6Q²

Calculate the value of AFC as shown below,

AFC = TVC / Q

AFC = 5Q + 6Q² / Q

AFC = 5 + 6Q

Calculate the value of ATC as shown below,

ATC = TC / Q

ATC = 6Q² + 5Q + 10 / Q

ATC = 6Q + 5 + 10/Q

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Answer:

--------------------------------

Find the following:

Answer:

[tex]h(-3)=\boxed{\dfrac{5}{4}}[/tex]

[tex]h(2)=\boxed{3}[/tex]

[tex]h(5)=\boxed{-\dfrac{19}{4}}[/tex]

Step-by-step explanation:

Given piecewise function:

[tex]h(x) \begin{cases} -\dfrac{3}{4}x-1\; &\text{if}\;x\neq 2\\\\3 \; &\text{if}\;x=2 \end{cases}[/tex]

To find h(-3), substitute x = -3 into the function:

[tex]\begin{aligned}\implies h(-3)&=-\dfrac{3}{4}(-3)-1\\\\&=\dfrac{9}{4}-1\\\\&=\dfrac{9}{4}-\dfrac{4}{4}\\\\&=\dfrac{5}{4}\end{aligned}[/tex]

To find h(2), substitute x = 2 into the function:

[tex]\implies h(2)=3[/tex]

To find h(5), substitute x = 5 into the function:

[tex]\begin{aligned}\implies h(5)&=-\dfrac{3}{4}(5)-1\\\\&=-\dfrac{15}{4}-1\\\\&=-\dfrac{15}{4}-\dfrac{4}{4}\\\\&=-\dfrac{19}{4}\end{aligned}[/tex]

The graph of the line segment after the translation (x, y) → (x-2, y-3) and (x, y) → (x-4, y + 5) has been drawn.

Transformation is rearranging a graph by a given rule it could be either increment of coordinate or decrement or reflection.

If we reflect any graph about y = x then the coordinate will interchange it that (x,y) → (y,x).

As per the given coordinate,T(1,2) and U(4,6)

Apply the translation (x, y) → (x-2, y-3)

T(1,2) → T'(1 - 2,2 - 3) = T'(-1,-1)

U(4,6) →  U'(4 - 2,6 - 3) = U'(2,3).

Apply the translation (x, y) → (x-4, y + 5)

T(1,2) → T'(1 - 4,2 + 5) = T'(-3,7)

U(4,6) →  U'(4 - 4,6 + 5) = U'(0,11)

All points have been plotted below.

Hence"After the translations (x, y) → (x-2, y-3)  and (x, y) → (x-4, y + 5), the graph of the line segment has been drawn.".

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Answer:

Translation (x, y) → (x - 2, y - 3):

Translation (x, y) → (x - 4, y + 5):

Step-by-step explanation:

Given endpoints of line segment TU:

Translation 1 (blue on attached diagram)

This means to subtract 2 from the x-values of the endpoints, and subtract 3 from the y-values of the endpoints:

Therefore:

Translation 2 (red on attached diagram)

This means to subtract 4 from the x-values of the endpoints, and add 5 to the y-values of the endpoints:

Therefore:

Answer:

Step-by-step explanation:

To solve this inequality graphically, we need to find the values of x that make the inequality true. We can start by looking at the absolute value expression on its own.

The absolute value of a number is always non-negative, so we can split the inequality into two cases: when the expression inside the absolute value is positive, and when it is negative.

For the first case, we have:

3|2x - 4| > 6

2x - 4 > 0

x > 2

For the second case, we have:

3|2x - 4| > 6

2x - 4 < 0

x < 2

So, the solution to the inequality is the union of these two cases: x > 2 or x < 2. In interval notation, this is written as (-infinity, 2) U (2, infinity).

Answer:

  (-∞, 1) ∪ (3, ∞)

Step-by-step explanation:

You want a graphical solution to 3|2x -4| > 6.

The attached graph is of the expression ...

  3|2x -4| -6

This is greater than 0 where x is a solution to the given inequality. It is greater than 0 for x < 1 or for 3 < x. In interval notation, the solution is ...

  (-∞, 1) ∪ (3, ∞)

Answer:

A. Reflection

The expression that represents the length of Ana Maria's ribbon is r/3

A word problem is a mathematical exercise where significant background information on the problem is presented in ordinary language rather than in mathematical notation. It should be noted that an expression shows the relationship between the variables.

Ana Maria has the length of 1/3 ribbon Jolie has.

Let r represent the length of ribbon Jolie has

Since Ana Maria's ribbon length is 1/3 that of Jolie ribbon. We can write:

Ana Maria's ribbon length = 1/3 × r = r/3

Thus, the expression for Ana Maria's ribbon length is r/3.

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The expression that represents the length of Ana Maria's ribbon is r/3

How to determine which expression can be used to represent the length?

A word problem is a mathematical exercise where significant background information on the problem is presented in ordinary language rather than in mathematical notation. It should be noted that an expression shows the relationship between the variables.

Ana Maria has the length of 1/3 ribbon Jolie has.

Let r represent the length of ribbon Jolie has

Since Ana Maria's ribbon length is 1/3 that of Jolie ribbon. We can write:

Ana Maria's ribbon length = 1/3 × r = r/3

Thus, the expression for Ana Maria's ribbon length is r/3. :)

The least total amount Lewis spends to buy 34 cartons of juice is £12.84 for Alternative B instead of £15.30 for Alternative A.

We can determine the total amount for each alternative and compare the two using two mathematical operations of multiplication and division.

We can also compare the alternatives by determining their unit cost per carton using division.

The number of cartons of juice required = 34

The unit cost of a carton = £0.45 or 45p

The total cost of 4 cartons = £1.80 (£0.45 x 4)

The total amount Lewis would spend for Alternative A = £15.30 (£0.45 x 34)

Cost of a pack of 4 cartons = £1.51

Unit cost per carton of the 4-carton pack = £0.3775 (£1.51/4)

The total amount Lewis spends for Alternative B = £12.835 (34/4 x £1.51)

Thus, mathematically Alternative B presents a lower cost than Alternative A, Lewis would choose to spend £12.84.

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Note that to achieve a volume of 64cm³, the height of the cuboidal packet will equal 0.33cm²; If three layers of such are stacked on each other the total volume gained is: 192cm³.

The volume of a cuboid is given as L x B x H.

Where L = Lenght

B = Breath (or Width); and

H = Height.

To solve this problem, we need to first find the height of the box.

Since the volume of a Cuboid is given as L x B X H,

and we have

L = 16 and

B = 12, and

V= 64cm³

Then the expression for getting H will evolve from:

64 = 16 x 12 x H

64 = 192 x H

H = 64/192

H = 0.33cm

If three layers are placed on top of the volume amounts to:

64 x 3 = 192cm³

As a result, it is reasonable to declare that the height of the cuboidal packet will equal 0.33cm² to derive a volume of 64cm³; if three layers of such are piled on top of each other, the total volume acquired is: 192cm³.

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To calculate the expected value of playing this game, we need to multiply the value of each possible outcome by its probability of occurring and then add these values together. In this case, there is 1 gold marble, 6 silver marbles, and 28 black marbles in the bag, for a total of 35 marbles. So, the probability of selecting a gold marble is 1/35, the probability of selecting a silver marble is 6/35, and the probability of selecting a black marble is 28/35. The value of winning $3 if a gold marble is selected is $3 * (1/35) = $0.09. The value of winning $2 if a silver marble is selected is $2 * (6/35) = $0.34. And the value of losing $1 if a black marble is selected is -$1 * (28/35) = -$0.80. Adding these values together, we get $0.09 + $0.34 - $0.80 = -$0.37. Therefore, the expected value of playing this game is approximately -$0.37.

The rule of dilation is (4x, 4y)

From the question, we have the following parameters that can be used in our computation:

Scale factor = 4

This means that

Image = point * scale factor

So, we have

Image = (x, y) * 4

Evaluate

Image = (4x, 4y)

Here, we have

Point = (-2, 8)

Scale factor = 1/2

So, we have

Image = (-2, 8) * 1/2

Image = (-1, 4)

Here, we have

Point = (2, 3)

Transformation = reflected across the y-axis.

So, we have

Image = (-x, y)

Image = (-2, 3)

Here, we have

A (4, 3) to point A' (12, 9)

The scale factor is

Scale factor = A'/A

So, we have

Scale factor = (12, 9)/(4, 3)

Evaluate

Scale factor = 3

Hence, the scale factor is 3

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Answer:

56 points

Step-by-step explanation:

Let's say that for the next 5 days, you earn x points each day. This means that you will earn 5x more points before the month ends.

Since you already earned 1300 points, you will have earned 1300 + 5x points, which through the question we know is equal to 1580 points. Using this knowledge, we can set up the equation below:

1300 + 5x = 1580

5x = 280

x = 56

Therefore, you will need 56 points each day to reach your goal.

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The amount Damian earns on Friday if he works for 2.75 hours is $43

From the question, we have the following parameters that can be used in our computation:

f(x) = 12x + 10

For working for 2.75 hours, the value of x is

x = 2.75

Substitute x = 2.75 in the equation f(x) = 12x + 10

So, we have the following representation

f(2.75) = 12 x (2.75) + 10

Evaluate the sum of products

f(2.75) = 43

hence, the earnings is $43

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Answer:

Step-by-step explanation: There are three possible solutions for a linear equation: no solution, one unique solution, and infinitely many solutions.

A linear equation is an algebraic equation in which the highest exponent of the variables is 1. For example, 2x + 3 = 5 is a linear equation because the highest exponent of the variable x is 1.

If a linear equation has no solutions, this means that the equation is contradictory and there is no possible value of the variable that can make the equation true. For example, the equation 2x + 3 = 5 has no solutions because it is already true for all possible values of x.

If a linear equation has a unique solution, this means that there is only one possible value of the variable that can make the equation true. For example, the equation 2x + 3 = 5 has a unique solution because there is only one value of x that can make the equation true (in this case, x = 1).

If a linear equation has infinitely many solutions, this means that there are an infinite number of possible values of the variable that can make the equation true. This typically happens when the equation has the form of 0 = 0, which is always true for any value of the variable. For example, the equation 0 = 0 has infinitely many solutions because it is true for all possible values of x.

In general, you can tell whether a linear equation has no solutions, a unique solution, or infinitely many solutions by solving the equation. Solving a linear equation involves isolating the variable on one side of the equation and then using algebraic techniques to find its value. If you are able to find a unique value for the variable, then the equation has a unique solution. If you are unable to find a unique value for the variable, then the equation either has no solutions or infinitely many solutions.