Question 3 For each of the following, explain whether it is (always) true or (possibly) false. If true, you must explain why. If false, you must give a concrete counterexample (with numbers).
a) (1.5pts) If b is a linear combination of the columns of the matrix A, then the linear system Ax = b has a unique solution.
b) (1.5pts) If {u, v, w} is linearly independent, then {u+w, v + w} is linearly independent.
c) (1.5pts) If V = span{V1, V2, V3} and dim(V) = 2, then {V2, V3} is a basis of V.

The volume of the composite solid is 885 m³

From the diagram shown, we have that;

Volume of the composite solid = Volume of a rectangle + volume of  a pyramid

The formula for the volume of a rectangle is expressed as;

Volume = lwh

Given that;

Volume = 5 × 10 × 16. 5

Volume = 825 m³

Volume of the pyramid = 6(10) = 60m³

Total volume = 825 + 60 = 885 m³

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Mr Tan earns a total of $96.

Let the selling price of a cup of apple juice be x.

Then the selling price of a cup of lemon juice is 3/2 times x, which is (3/2)x.

Let the number of cups of lemon juice sold be L, and the number of cups of apple juice sold be A.

We know that for every 2 cups of lemon juice sold, Mr Tan sells 5 cups of apple juice. So we can write:

A = (5/2)L

His earnings from apple juice minus his earnings from lemon juice is $12. So we can write:

Ax - L(3/2)x = $12

Simplifying this equation, we get:

(5/2)Lx - (3/2)Lx = $12

Lx = $24

Now we can use this information to find the values of L and A:

L = $24/x

A = (5/2)L = (5/2)($24/x) = $60/x

Finally, we can calculate Mr Tan's total earnings by adding his earnings from apple juice and lemon juice:

Total earnings = Ax + L(3/2)x

= $60 + (3/2)($24)

= $96

Therefore, Mr Tan earns a total of $96.

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The series converges.

To test the series 3 + 3/5 + 3/5^2 + 3/5^3 + … =∑n=0[infinity] 3/5^n,

we can use the ratio test.

Specifically, we can look at the ratio of each successive term.

Let an+1/an = 3/5n+1/3/5n = 1/5. Since this is less than 1, the series converges.

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Answer:

Simplify both sides of the equation and combine all same-side like terms.

Combine opposite-side like terms to obtain the variable term on one side of the equal sign and the constant term on the other.

Divide or multiply as needed to isolate the variable.

The three basic operations used to simplify a linear system are elimination, substitution, and augmentation.

The 3 basic operations used to simplify a linear system are:
1. Swapping the order of equations: This operation is used to rearrange the equations in the system to make it easier to solve.
2. Multiplying or dividing an equation by a constant: This operation is used to make the coefficients of the variables in the equation equal to 1 or -1, which makes it easier to solve the system.
3. Adding or subtracting equations: This operation is used to eliminate one of the variables in the system, which reduces the number of equations and makes it easier to solve the system.
These operations are used to manipulate the equations in the system in order to find the values of the variables that satisfy all of the equations in the system. By using these operations, we can simplify the system and solve it more easily.

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Answer:

Step-by-step explanation:

ypi ,tfr

Answer:

Step-by-step explanation:

b because i did it

Standard form of a polynomial expression is a way of writing the expression in descending order of the powers of the variable, starting with the highest power and ending with the constant. In this case, the polynomial expression is - 6 - 7x ^ (2) + 8x ^ (5).

In standard form, this expression can be written as 8x ^ (5) -7x ^ (2) - 6. This expression indicates that 8 is the coefficient of the highest power, x ^ (5). The coefficient of the second highest power, x ^ (2) is -7, and the coefficient of the constant, the term with no variable, is -6.

Standard form is useful for simplifying polynomial expressions, as it helps to clearly illustrate the coefficients of each term. It is also useful for graphing polynomials, as the order of the terms can be clearly seen.

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Answer: $1080

Step-by-step explanation:

First, find the area of the wall.

The area of a rectangle is length x width.

Area = 12 x 20 = 240 square feet

Multiply 4.50 by 240 to find the total cost.

4.50 x 240 = 1080 dollars

It would cost $1080 to cover the wall.

Hope this helps!

Both numbers are divisible by 2, 2 and 7.

What is Greatest Common Factor?

The greatest common factor (GCF) is the largest number that divides two or more numbers evenly. It is also known as the greatest common divisor (GCD) or highest common factor (HCF).

If 28 is the greatest common factor of two numbers, it means that both numbers have 28 as a factor. Thus, both numbers are divisible by 2, 2 and 7.

Additionally, the prime factorization of each number must contain all of the prime factors that are factors of 28, which are 2 and 7. However, there may be other prime factors as well. Without additional information, it is not possible to determine the complete prime factorization of each number.

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For a 20% tip, 8% tax sales. Total $23. 04, we can say that The pre-tax cost is $19.20.

Let x be the pre-tax cost of the sales. Then we can set up the equation:

x + 0.20x + 0.08x = 23.04

Simplifying and solving for x, we get:

1.28x = 23.04

x = 23.04 / 1.28

x = 18

Therefore, the pre-tax cost of the sales is $18. However, we need to add the 20% tip and 8% tax to this amount to get the total cost of $23.04. The tip is $3.60 (0.20 * 18) and the tax is $1.44 (0.08 * 18), which when added to $18 gives us a total cost of $23.04.

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Answer:

3

Step-by-step explanation:

7.2<x+4.2

3<x
x>3
the graph would be

Pippin has 37.5% of her sports drink left.

What is Percentage?

Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100. Percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%".

Pippin has 32 - 20 = 12 ounces of sports drink left.

To find the percentage of ounces Pippin has left, you need to divide the number of ounces left by the original amount of sports drink (32) and then multiply by 100 to get the percentage:

(12/32) x 100 = 37.5%

Therefore, Pippin has 37.5% of her sports drink left.

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The grocer should use 15 pounds of almonds and 25 pounds of cashews to create a 40 pound mixture that will sell for $12.50 per pound.

To create a 40 pound mixture of almonds and cashews that will sell for $12.50 per pound, the grocer should use a combination of both almonds and cashews that will give him the desired price. We can use a system of equations to solve for the amount of each ingredient the grocer should use. Let x be the amount of almonds and y be the amount of cashews.

Solving for x in the first equation gives us: x = 40 - y

Substituting this value of x into the second equation gives us: 10(40 - y) + 14y = 12.50(40)

Simplifying and solving for y gives us: 4y = 100 -> y = 25

Substituting this value of y back into the first equation gives us: x + 25 = 40 -> x = 15

Therefore, the grocer should use 15 pounds of almonds and 25 pounds of cashews to create a 40 pound mixture that will sell for $12.50 per pound.

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The time needed for Chad to drive the 672 miles is given as follows:

t = 672/v.

In which v is his current rate.

Velocity is given by the change in the distance divided by the change in the time, hence the following equation is built to model the relationship between these three variables:

v = d/t.

For a distance of 672 miles, we have that the parameter d is given as follows:

d = 672.

Hence the time is obtained as follows:

v = 672/t

t = 672/v.

(we don't have the velocity, hence the time is given as a function of the velocity).

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In interval notation, the domain can be written as: (-∞, -5) ∪ (-5, 5) ∪ (5, ∞). In set notation, the domain can be written as: {x ∈ ℝ | x ≠ -5 and x ≠ 5}

To find the domain of the given rational expression, we need to determine the values of x that make the denominator equal to zero. This is because division by zero is undefined and therefore, these values of x cannot be included in the domain.

The denominator of the given expression is x^(2)-25. We can set this equal to zero and solve for x:

x^(2)-25=0

(x+5)(x-5)=0

x=-5 or x=5

These values of x make the denominator equal to zero, so they cannot be included in the domain. Therefore, the domain of the given rational expression is all real numbers except -5 and 5.

In interval notation, the domain can be written as: (-∞, -5) ∪ (-5, 5) ∪ (5, ∞)

In set notation, the domain can be written as: {x ∈ ℝ | x ≠ -5 and x ≠ 5}

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The solution to the system of equations 2x + 3y = 18 and x + y = 7 is (x,y) = (3,4).

What is the linear equations?

A linear equation can have more than one variable. If the linear equation has two variables, then it is called linear equations in two variables and so on.

To solve the equation 2x + 3y = 18 using substitution, we can rearrange the equation to express one of the variables in terms of the other. For example, we can solve for x as follows:

2x + 3y = 18

2x = 18 - 3y

x = (18 - 3y)/2

Now we have an expression for x in terms of y. We can substitute this expression into any other equation that involves x, in order to eliminate x from the equation and solve for y. For example, if we have the equation:

x + y = 7

We can substitute (18 - 3y)/2 for x, to get:

(18 - 3y)/2 + y = 7

Now we can solve for y:

18 - 3y + 2y = 14

-y = -4

y = 4

Once we have solved for y, we can substitute this value back into one of the original equations to solve for x. For example, using the equation 2x + 3y = 18:

2x + 3(4) = 18

2x + 12 = 18

2x = 6

x = 3

Therefore, the solution to the system of equations 2x + 3y = 18 and x + y = 7 is (x,y) = (3,4).

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5(3)/(10),  5.338,  5.37,  (21)/(4).

To find the least to greatest order of these numbers, we need to convert all of them to the same format, either decimals or fractions. In this case, it is easier to convert the fractions to decimals.
5(3)/(10) = 5.3/10 = 0.53
(21)/(4) = 21/4 = 5.25

Now we can compare all of the numbers in decimal form:
0.53, 5.338, 5.37, 5.25

Next, we can arrange them in ascending order:
0.53, 5.25, 5.338, 5.37

Finally, we can convert the decimals back to fractions if necessary:
5(3)/(10), (21)/(4), 5.338, 5.37

Therefore, the correct order of these numbers from least to greatest is 5(3)/(10), (21)/(4), 5.338, 5.37.

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Answer:

8 + 2x = 30

Step-by-step explanation:

The sides of the triangle add up to the perimeter:

7 + x + x + 1 = 30

Simplify by combining like terms:

8 + 2x = 30

Answer:

The fraction that is not equivalent to 25% is 2/5.

To see why, we can start by converting 25% to a fraction. Recall that "percent" means "per hundred," so 25% is equal to 25/100 or 1/4.

Now, we can check each of the answer choices to see if they are equivalent to 1/4:

1/4 is already in the form of 1/4, so it is equivalent to 25%.

5/20 can be simplified by dividing both the numerator and denominator by 5, which gives 1/4. So 5/20 is also equivalent to 25%.

25/100 is the same as 1/4 (we converted 25% to 1/4 earlier), so it is equivalent to 25%.

This leaves us with 2/5 as the answer choice that is not equivalent to 25%. We can see this by converting 2/5 to a percent:

2/5 = 0.4

0.4 x 100% = 40%

So 2/5 is equivalent to 40%, which is not the same as 25%.

Answer: [tex]2/5[/tex] second option

Step-by-step explanation: 1/4 is equal to 25% because if you multiply by 100 and divide by 4 that will equal 25. 5/20 simplified to 1/4 which we know is 25%, 25/100 also does too. Therefore, the answer is 2/5 because it simplifies to 40% which isn't 25%.

To set up the equation and solve for the 2 consecutive odd integers whose product is 195, we can follow these steps:

1. Let x be the first odd integer. 2.The next consecutive odd integer will be x+2. 3.The product of these two integers is 195, so we can set up the equation: x(x+2) = 195 4.Expand the equation: x^2 + 2x = 195

5.Rearrange the equation to get a quadratic equation: x^2 + 2x - 195 = 0 6. Use the quadratic formula to solve for x: x = (-2 ± √(2^2 - 4(1)(-195)))/(2(1)) 7. Simplify the equation: x = (-2 ± √784)/2 8. Solve for the two possible values of x: x = (-2 + 28)/2 or x = (-2 - 28)/2 9. Simplify the equations: x = 13 or x = -15 10. Since x is an odd integer, we can reject the solution x = -15. 11. The first odd integer is x = 13, and the next consecutive odd integer is x+2 = 15.

12. Therefore, the two consecutive odd integers whose product is 195 are 13 and 15.

The final answer is 13 and 15.

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The initial value obtained using the slope-intercept relation is 3 gallons.

We need to obtain the rate of change which gives the amount of gas entering into his car per second :

Rate of change = Rise / Run

Rate of change = - 11 - 3) / (48 - 0)

Rate of change = 8/48 = 1/6 gallons

Using the slope intercept relation :

y = bx + c

b = slope ; c = initial amount of gas

Choosing any pair of (x, y) point on the table :

(0, 3)

3 = 1/6x + c

3 = 1/6(0) + c

3 = 0 + c

c = 3

Therefore, the initial value is 3 gallons.

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The area of a triangle with sides \( a=10 \) and \( b=15 \) and included angle 70 degrees is 70.47675 square units.

To find the area of a triangle with sides a and b and included angle C, we can use the formula:

\[Area = \frac{1}{2}ab\sin{C}\]

In this case, we have a = 10, b = 15, and C = 70 degrees. Plugging these values into the formula, we get:

\[Area = \frac{1}{2}(10)(15)\sin{70}\]

\[Area = 75\sin{70}\]

Using a calculator, we find that sin(70) = 0.93969. So:

\[Area = 75(0.93969)\]

\[Area = 70.47675\]

Therefore, the area of the triangle is approximately 70.47675 square units.

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The true statements are:

A sort of transformation comprehended a translation involves moving each point in a figure the same distance in the same direction.

Reworking text from one language into another while preserving the original message and communication is called translation. But, there are various approaches to translation, and they range in both form and purpose, just like anything else.

An operation is a transformation if it moves, flips, or otherwise alters a figure to produce a new figure. Rigid transformations sometimes referred to as isometry or congruence transformations, do not alter the size or shape of a figure.

therefore, the true statements are:

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Answer: 21 degrees

Step-by-step explanation: https://brainly.com/question/16020444

The value of the angles in the parallel line is 34 degrees.

When parallel are cut by a transversal line, angle relationships are formed such as corresponding angle, alternate interior angles, alternate exterior angles, linear angles, vertically opposite angles etc.

Therefore, the angle 5 can be found using the angle relationship.

Hence,

m∠1 ≅ m∠5(corresponding angles)

Corresponding angles are congruent to each other.

Therefore,

m∠5 = 34 degrees

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Answer:

If I had to take a guess I'd say the price of the milkshakes is $4.50 and the price of the sundaes are $2.75 but I'm not 100% sure.

Answer:

Step-by-step explanation:

8m + 6i = 56.50

4m + 2i = 23.50

8m + 6i = 56.50

-8m - 4i =-47.00

2i = 9.50

i = $4.75 ice cream sundae

4m + 2(4.75) = 23.50

4m + 9.50 = 23.50

4m = 14

m = 3.50 milkshake

From the system of equations one apple pie costs $6 and one lemon meringue pie costs $7.

A group of equations that need to be solved concurrently is referred to as a system of equations. The values of the variables that satisfy each and every one of the system's equations are the solutions to the corresponding equations. Systems of equations can be solved using a variety of techniques, such as substitution, elimination, and graphing. Systems of equations can be used to simulate situations in business, science, and engineering when there are several variables at play. Systems of differential equations, linear equations, and quadratic equations are a few examples of systems of equations.

Let us suppose the cost of the apple pie = x.

Let us suppose the cost of the meringue pie = y.

Given that, Dan sold 14 apple pies and 9 lemon meringue pies for a total of $147.

14x + 9y = 147

For Jenny:

8x + 12y = 132

Using the elimination method we have:

Multiplying Dan's equation by 4 gives:

56x + 36y = 588

Multiplying Jenny's equation by 3 gives:

24x + 36y = 396

Subtracting the second equation from the first:

32x = 192

x = 6

Substitute the value of x in equation:

14(6) + 9y = 147

84 + 9y = 147

9y = 63

y = 7

Hence, from the system of equations one apple pie costs $6 and one lemon meringue pie costs $7.

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By the squeeze theοrem, lim f(x) = 0 as x apprοaches -5.

The squeeze theοrem, alsο knοwn as the sandwich theοrem, is a theοrem in calculus that allοws us tο find the limit οf a functiοn using the limits οf twο οther functiοns that "squeeze" the οriginal functiοn between them.

We have the inequality:

-2x² - 20x - 51 ≤ f(x) ≤ 2x² + 20x + 49

We can rewrite this inequality as:

-2(x+5)(x+4.5) ≤ f(x) ≤ 2(x+5)(x+4.5) + 1

Nοte that we have factοred -2x² - 20x - 51 as -2(x+5)(x+4.5) and 2x² + 20x + 49 as 2(x+5)(x+4.5) + 1.

As x apprοaches -5, we can see that (x+5) and (x+4.5) apprοach 0, sο we have:

-2(0)(0) ≤ lim f(x) ≤ 2(0)(0) + 1

0 ≤ lim f(x) ≤ 1

Therefοre, by the squeeze theοrem, we have:

lim f(x) = 0 as x apprοaches -5.

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The final answer is degree of P(x) is 10.

The degree of a polynomial is the highest power of x that appears in the polynomial.

In the case of P(x), we can find the degree by multiplying the degrees of each factor together.

The degree of (2x+5) is 1,

the degree of (5x-2)^(2) is 2,

the degree of (x+7) is 1,

and the degree of Q(x) is 6.

Therefore, the degree of P(x) is 1*2*1*6 = 12.

However, we need to subtract 2 from the degree because (5x-2)^(2) is raised to the power of 2.

So the degree of P(x) is 12-2 = 10.

Therefore, the degree of P(x) is 10.

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The value of n will be 12.

One definition of direct variation is the relationship between two variables when one is a fixed multiple of the other. They are said to be in proportion, for instance, when one variable affects the other. The equation has the form b = ka if b is directly proportional to a. (where k is a constant)

The direct variation includes the points (-6, 18) and (-4, n).

We know that, direct variation = b= ka

where, a is the x coordinate, b is the y coordinate and k is the constant.

So, For (-6, 18),

we have, 18 = -6k

So, k = -3

Now, For (-4, n),

we have, n = -4k

               n  =  -4×-3 ( putting k= -3)

                   = 12

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