1.A system of two linear equations is graphed on a coordinate plane. If the system of equations has infinitely many solutions, which statement must be true?

A.On the graph, there is exactly one point (x,y) that satisfies both equations.


B.On the graph, any point (x,y) that satisfies one of the equations cannot satisfy the other equation.


C.On the graph, there are no points (x,y) that satisfy both equations.


D.On the graph, any point (x,y) that satisfies one of the equations must also satisfy the other equation.

2.Mark drank a total of 10 glasses of water and iced tea yesterday.

Each glass of water contained 8 ounces. Each glass of iced tea contained 12 ounces.

All together, he drank 96 ounces. The system of equations below models this situation.

x + y = 10
8 x + 12 y = 96

How many glasses of water did Mark drink yesterday?

A.3 glasses


B.4 glasses


C.5 glasses


D.6 glasses

Answer: Option b

Step-by-step explanation:

Rachel needs two hours for a loaf of bread and one hour for one pie.

Joey needs four hours for one loaf of bread and four hours to a pie.

We want to know the opportunity cost of Rachel to make one loaf of bread.

The opportunity cost is the relation between what she wins and what she could'be wined if she chose the other option.

We know that at the same time that she bakes 1 loaf of bread she could make 2 pies, so the opportunity cost of 1 loaf of bread is option b (what she could have wined if she chose to do the pies instead in that time).

In this exercise we have to perform the parameterization of the given function and we will have:

[tex]\int\limits_C {f} \, dr = f(1, 2)-f(-1, 2)= 2/3[/tex]

In there is some scalar function [tex]f(x, y)[/tex] such that:

[tex]\nabla f(x,y) = f(x, y) = x^2i+y^2j[/tex]

Then we want to find [tex]f[/tex] such that:

[tex]f'(x)= x^2= f(x,y) = \frac{x^3}{3} + g(y)\\f'(y)= y^2= f(x,y)= \frac{y^3}{3} + C\\f(x, y)= \frac{x^3}{3} +\frac{y^3}{3} + C[/tex]

So the vector filed [tex]f(x, y)[/tex] is conservative, which means the fundamental theorem applies; the line integral of [tex]f[/tex] along any path [tex]C[/tex] parameterized by some vector-valued function [tex]r(t)[/tex] over [tex]a\leq t\leq b[/tex] is given by:

[tex]\int\limits_C {f} \, dr\\\int\limits^t=a_t=b {f(r(t))} \, dt= f(r(b))-f(r(a))[/tex]

In the case:

[tex]\int\limits_C {f} \, dr = f(1, 2)-f(-1, 2)= 2/3[/tex]

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This problem requires calculation in multivariable calculus. Function f = ∇f when f(x, y) = x³/3 + y³/3. To calculate the line integral of ∇f · dr over the curve c, take the difference between the end and start points of the scalar function.

The subject for this calculation is multivariable calculus, dealing with operations such as the gradient (∇) and line integrals.

According to the given function f(x, y) in part (a), it can be seen that f = ∇f for f(x, y) = x³/3 + y³/3. The vector field of this function is conservative since it corresponds to the gradient of a scalar function.

For part (b), you will need to calculate the line integral of ∇f · dr over the curve c, which is y = 2x² from (-1, 2) to (1, 2). The result of this will be the difference between the end and start points of the scalar function: f(1, 2) - f(-1, 2).

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There are 211 quarters and 216 dimes in Steve's piggy bank.

An algebraic expression is consists of variables, numbers with various mathematical operations,

Let Steve has x quarters,

Therefore, Steve will have x + 5 dimes,

Given that,

Steve has total $74.35.

To solve, use equation format,

0.1(x+5)+0.25x=74.35

0.35x+0.5=74.35

0.35x=73.85

x=211

So the number of quarters x= 211

And number of dimes (x+5) =216

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The expression that represents "the difference between -17 and -8" is the second option:

-17 - (-8)

This expression translates to "negative seventeen minus negative eight," which yields the difference between the two numbers.

This expression represents "negative eight minus negative seventeen." When you subtract a negative number, it's equivalent to adding its positive counterpart. So, this expression can be simplified to:

-8 + 17 = 9

However, this result does not represent the difference between -17 and -8; it gives the difference between 17 and 8.

This expression represents "negative seventeen minus eight." This directly calculates the difference between -17 and -8. It simplifies to:

-17 - 8 = -25

This indeed represents the difference between -17 and -8.

This expression represents "negative seventeen minus negative eight." Similar to the first expression, subtracting a negative number is the same as adding its positive counterpart. So, this simplifies to:

-17 + 8 = -9

This result does not represent the difference between -17 and -8; it gives the difference between -17 and 8.

This expression represents "negative eight plus negative seventeen." It simplifies to -25, which again represents the difference between -17 and -8.

Among these expressions, only the second expression, -17 - 8, correctly represents the difference between -17 and -8.

a, b, d, e true: empty set fits in its own container.

c, f false: bigger container doesn't hold smaller container.

Here are the truths and falsehoods of the statements:

a) ∅ ∈ {∅} - True. The empty set is an element of the set containing only the empty set.

b) ∅ ∈ {∅, {∅}} - True. The empty set is also an element of the set containing both the empty set and the set containing the empty set.

c) {∅} ∈ {∅} - False. The set containing the empty set is not an element of the set containing only the empty set.

d) {∅} ∈ {{∅}} - True. The set containing the empty set is an element of the set containing only the set containing the empty set.

e) {∅} ⊂ {∅, {∅}} - True. The set containing the empty set is a subset of the set containing both the empty set and the set containing the empty set.

f) {{∅}} ⊂ {∅, {∅}} - False. The set containing the set containing the empty set is not a subset of the set containing only the empty set and the set containing the empty set.




The probable question can be: Determine whether these statements are true or false.

a) ∅ ∈ {∅}

b) ∅ ∈ {∅, {∅}}

c) {∅} ∈ {∅}

d) {∅} ∈ {{∅}}

e) {∅} ⊂ {∅, {∅}}

f) {{∅}} ⊂ {∅, {∅}}

Answer:

  14/15 more pints of strawberries than grapes

Step-by-step explanation:

You want to find the difference between 2 1/3 and 1 2/5. There are several ways to do this. One is to use a common denominator for the fractions. Here, a convenient denominator is 3·5 = 15. This lets us write the problem as ...

  2 5/15 - 1 6/15

  = (2 -1) + (5/15 -6/15)

  = 1 - 1/15 . . . . of course there are 15/15 in 1, so subtracting 1 of them gives...

  = 14/15

Kelly uses 14/15 more pints of strawberries than of grapes in her juice.

_____

Comment on solution details

We have rewritten the fraction 1/3 as 5/15. We can do this by multiplying 1/3 by 1, where 1 is in the form 5/5:

  (1/3) = (1/3)·1 = (1/3)·(5/5) = (1·5)/(3·5) = 5/15

Similarly, we have rewritten the fraction 2/5 by multiplying it by 3/3:

  (2/5) = (2/5)·(3/3) = 6/15

__

Both numbers can be converted to improper fractions:

  2 1/3 = (2·3+1)/3 = 7/3

  1 2/5 = (1·5+2)/5 = 7/5

Then the difference can be found using the formula ...

  a/b - c/d = (ad -bc)/(bd)

  7/3 -7/5 = (7·5 -7·3)/(3·5) = (35 -21)/15 = 14/15

The car can drive 216 miles on 12 gallons.

The rate of change of position is called as displacement.

Now it is given that,

Distance traveled by car = 630 miles

Gas required to travel 630 miles = 35 gallons

Therefore, Distance travel n 1 gallon = Distance traveled by car / Gas required to travel 630 miles

⇒ Distance travel n 1 gallon = 630 / 35 miles

⇒ Distance travel n 1 gallon = 18 miles

Therefore, Distance travel n 12 gallon = 12 * Distance travel n 1 gallon

⇒ Distance travel n 12 gallon = 12 * 18 miles

⇒ Distance travel n 12 gallon = 216 miles.

Thus, the car can drive 216 miles on 12 gallons.

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The difference between -4/5 and -3/5 is found by changing the subtraction to addition and combining the numerators, resulting in -1/5.

To find the difference between -4/5 and -3/5, you change the subtraction operation to addition by changing the sign of the second fraction. This is similar to the process described by the example, where subtracting a negative number is the same as adding its positive counterpart (e.g., 2 - (-6) = 2 + 6). In the case of fractions, the process is the same.

So, the problem -4/5 - (-3/5) becomes -4/5 + 3/5. Since both fractions have the same denominator, you can combine the numerators directly. The result is the difference of the numerators over the common denominator:

The difference between -4/5 and -3/5 is -1/5.

Answer:

Parameter

Step-by-step explanation:

I believe your answer is B)160. But i'm not 100%. <3

The solution to the system of equations is [tex]\(x = \frac{20}{3}\)[/tex] and [tex]\(y = -\frac{15}{4}\).[/tex]

To solve the system of equations (3x - 4y = 35) and (3x + 4y = 5), we'll use the method of elimination.

Add the two equations together to eliminate the variable (y).

(3x - 4y) + (3x + 4y) = 35 + 5

3x - 4y + 3x + 4y = 40

6x = 40

[tex]\[ x = \frac{40}{6} \][/tex]

[tex]\[ x = \frac{20}{3} \][/tex]

Substitute [tex]\(x = \frac{20}{3}\)[/tex] into one of the original equations. Let's use the first equation:

[tex]\[ 3\left(\frac{20}{3}\right) - 4y = 35 \][/tex]

20 - 4y = 35

-4y = 35 - 20

-4y = 15

[tex]\[ y = \frac{15}{-4} \][/tex]

[tex]\[ y = -\frac{15}{4} \][/tex]

So, the solution to the system of equations is [tex]\(x = \frac{20}{3}\)[/tex] and [tex]\(y = -\frac{15}{4}\).[/tex]

To factor -1/4 out of -1/2-5/4y, you can use the distributive property and simplify the expression to 1/8 + 5/16y.

To factor -1/4 out of -1/2-5/4y, you can use the distributive property. First, rewrite the expression as (-1/2) + (-5/4)y. Then, factor out -1/4: (-1/4)(-1/2) + (-1/4)(-5/4)y. This simplifies to 1/8 + 5/16y.

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Answer and Explanation:

Given : A spring stretches in relation to the weight hanging from it according  to the equation [tex]y = 0.75x + 0.25[/tex] where x is the weight in pounds and y is the length of the spring in inches.        

To find :

1) How to graph the equation including axis labels  ?

2) How to interpret the slope and the y-intercept of the line?

Solution :

Linear equation  [tex]y = 0.75x + 0.25[/tex]

1) To draw the graph we find the x and y-intercept of the line,

x- intercept i.e. y=0

[tex]0.75x + 0.25=0[/tex]

[tex]0.75x=-0.25[/tex]

[tex]x=-\frac{0.25}{0.75}[/tex]

[tex]x=-0.33[/tex]

y- intercept i.e. x=0

[tex]y=0.75(0) + 0.25[/tex]

[tex]y=0.25[/tex]

Plotting these two points (-0.33,0) and (0,0.25).

Refer the attached figure below.

2) The general form of line is [tex]y=mx+c[/tex]

where, m is the slope and b is the y-intercept of the line.

Comparing with given line  [tex]y = 0.75x + 0.25[/tex]

The slope of the line is m=0.75

and the y-intercept of the line is b=0.25.

a. The unconditional mean of X = 1.75 cars.

b. Conditional Mean of X Given Y=1 is 1.38 cars.

c. Probability of Rachel Selling 1 Car Given Rebecca Selling 3 is 1.07.

d. The covariance COV (X,Y) is 0.23. COV(X,Y) ≠ 0, X,Y not independent.

e. The correlation coefficient is 0.52, The strength is (0.3-0.5) and the direction is positive correlation.

f. The expectation and variance of Z are 10.25 cars and 6.66 cars².

Step-by-Step Joint Probability Distribution of Cars Sold:

a. Unconditional Mean of X:

1. Calculate marginal probabilities of X:

P(X=1) = 0.57, P(X=2) = 0.29, P(X=3) = 0.14

Calculate expected value of X:

E(X) = Σ(xi × P(xi)) = (1 × 0.57) + (2 × 0.29) + (3 × 0.14) = 1.75 cars

b. Conditional Mean of X Given Y=1:

1. Calculate conditional probability:

P(X=1|Y=1) = 0.40 / 0.52

2.Calculate conditional mean:

E(X|Y=1) = (1 × 0.806) + (2 × 0.194) = 1.38 cars

c. Probability of Rachel Selling 1 Car Given Rebecca Selling 3:

1. Calculate conditional probability:

P(Y=1|X=3) = 0.15 / 0.14 = 1.07 (impossible, probabilities must be between 0 and 1)

d. Covariance and Independence:

1. Calculate joint expected values:

[tex]E(XY) = \sum (x_i \times y_i \times P(x_i,y_i)) = 2.74[/tex]

2. Calculate individual variances:

Var(X) = 0.74, Var(Y) = 0.24

3. Calculate covariance:

COV(X,Y) = E(XY) - E(X)E(Y) = 2.74 - (1.75 × 0.88) = 0.23

Since COV(X,Y) ≠ 0, X and Y are not independent.

e. Correlation Coefficient:

1. Calculate correlation coefficient:

ρ(X,Y) = COV(X,Y) / (σ(X) × σ(Y)) = 0.23 / (√0.74 × √0.24) ≈ 0.52

2. Strength: weak positive correlation (0.3-0.5)

3. Direction: positive correlation (positive coefficient)

f. Expectation and Variance of Z:

1. Use properties of mean:

E(Z) = E(3X+5) = 3E(X) + 5 = 3 × 1.75 + 5 = 10.25 cars

2. Use properties of variance:

Var(Z) = 9Var(X) = 9 × 0.74 = 6.66 cars²

Complete and correct question:

The joint probability distribution of variables X and Y is shown in the table below. Rebecca and Rachel are car salespeople. Let X denote the number of cars that Rebecca will sell in a month, and let Y denote the number of cars Rachel will sell in a month.

          x=1       x=2      x=3

Y=1    0.40     0.18      0.15

Y=2   0.12      0.11      0.04

a. Find the unconditional mean of X.

b. Find the conditional mean number of cars that Rebecca will sell in a month given Rachel sells 1 car in a month.

C. What is the probability that Rachel will sell 1 car in a month given that Rebecca sold 3?

d. Find covariance COV (X, Y). Are X and Y independent? Explain why.
e. Find the correlation coefficient between X and Y. Discuss the strength and the direction of its relationship.
f. Chris, another salesperson, sells a car better than Rebecca. Let Z denote the number of cars that Rebecca will sell in a month, where Z = 3X +5. Get the expectation and variance of Z using the properties of the mean and the variance.