help me figure out the first part of this problem!!

There are 4 people in Molly's group.

Step-by-step explanation:

Let,

x be the number of people.

Individual ticket price = $42

Price after discount = $30

Discount per ticket = $3

Discount for x people = 3x

According to given statement;

42-3x=30  

Subtracting 42 from both sides

[tex]42-42-3x=30-42\\-3x=-12\\[/tex]

Dividing both sides by -3

[tex]\frac{-3x}{-3}=\frac{-12}{-3}\\x=4[/tex]

There are 4 people in Molly's group.

Keywords: Subtraction, division

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Answer:

0.7 is the probability that a randomly selected person either has high blood pressure or is a runner or both.

Step-by-step explanation:

We are given the following information in the question:

Probability that a randomly selected person has high blood pressure = 0.4

[tex]P(H) = 0.4[/tex]

Probability that a randomly selected person is a runner = 0.4

[tex]P(R) = 0.4[/tex]

Probability that a randomly selected person has high blood pressure and is a runner = 0.1

[tex]P(H \cap R) = 0.1[/tex]

If the events of selecting a person with high blood pressure and person who is a runner are independent then we can write:

[tex]P(H \cup R) = P(H) + P(R)-P(H\cap R)[/tex]

Probability that a randomly selected person either has high blood pressure or is a runner or both =

[tex]P(H \cup R) = P(H) + P(R)-P(H\cap R)\\P(H \cup R) = 0.4 + 0.4 -0.1 = 0.7[/tex]

0.7 is the probability that a randomly selected person either has high blood pressure or is a runner or both.

Final answer:

The probability that a randomly selected person either has high blood pressure or is a runner or both is 0.7.

Explanation:

To find the probability that a randomly selected person either has high blood pressure or is a runner or both, we use the formula for the probability of either event A or event B occurring, which is:

P(A or B) = P(A) + P(B) - P(A and B).

Given:

P(H) = probability of high blood pressure = 0.4

P(R) = probability of being a runner = 0.4

P(H and R) = probability of both high blood pressure and being a runner = 0.1

Using the formula, we plug in the given probabilities:

P(H or R) = 0.4 + 0.4 - 0.1 = 0.7.

So, the probability that a randomly selected person either has high blood pressure, is a runner, or both events occur is 0.7.

Answer:

a) 0.3277

b) 0.0128

Step-by-step explanation:

We are given the following information in the question:

N(2750, 560).

Mean, μ = 2750

Standard Deviation, σ = 560

We are given that the distribution of distribution of birth weights is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

a) P (less than 2500 grams)

P(x < 2500)

[tex]P( x < 2500) = P( z < \displaystyle\frac{2500 - 2750}{560}) = P(z < -0.4464)[/tex]

Calculation the value from standard normal z table, we have,  

[tex]P(x < 2500) = P(z < -0.4464) = 0.3277 = 32.77\%[/tex]

b) P ((less than 1500 grams)

P(x < 1500)

[tex]P( x < 1500) = P( z < \displaystyle\frac{1500 - 2750}{560}) = P(z < -2.2321)[/tex]

Calculation the value from standard normal z table, we have,  

[tex]P(x < 1500) = P(z < -2.2321) = 0.0128 = 1.28\%[/tex]

Final answer:

The probability of a late-preterm baby having a low birth weight (< 2500 grams) is approximately 32.74%, and the probability of a very low birth weight (< 1500 grams) is about 1.29%.

Explanation:

To answer the given questions, we will use the properties of the normal distribution, where the mean birth weight (μ) for late-preterm babies is 2750 grams, and the standard deviation (σ) is 560 grams. The distribution of birth weights is assumed to be normal (N(2750, 560)).

1. Probability of a Low Birth Weight (< 2500 grams)

To find the probability of a baby having a low birth weight (less than 2500 grams), we use the Z-score formula: Z = (X - μ) / σ, where X is the value of interest (2500 grams). Plugging in the values gives us Z = (2500 - 2750) / 560 = -0.4464. Using a Z-table or a normal distribution calculator, we find the probability corresponding to Z = -0.4464, which is approximately 0.3274. Therefore, the probability of a late-preterm baby having a low birth weight is about 0.3274 or 32.74%.

2. Probability of a Very Low Birth Weight (< 1500 grams)

To calculate the probability of a very low birth weight (less than 1500 grams), again we calculate the Z-score: Z = (1500 - 2750) / 560 = -2.2321. The probability of Z = -2.2321, referring to the Z-table or calculator, is extremely low, approximately 0.0129 or 1.29%. This indicates that the chances of a late-preterm baby having a very low birth weight is about 1.29%.

Answer:

Their intercepts are unique.

Explanation:

[tex]\displaystyle x^3 - 24x^2 + 192x - 508 = (x - 8)^3 + 4[/tex]

This graph's x-intercept is located at approximately [6,41259894, 0], and the y-intercept located at [0, −508].

[tex]\displaystyle g(x) = x^3[/tex]

The parent graph here, has both an x-intercept and y-intercept located at the origin.

I am joyous to assist you anytime.

Answer:

There were 333 seniors in 1990

153 seniors had jobs in 2010.

Step-by-step explanation:

Let the number of high school seniors in 1990 = x

75% of x had jobs in 1990

In 2010 there were 50 more seniors. That is, the number of seniors is x+50.

40% of (x+50) had jobs in 2010.

Total number of jobs in both years = 403.

\[0.75 * x + 0.4 * (x + 50) = 403\]

\[0.75 * x + 0.4 * x  + 20 = 403\]

=> \[1.15 * x  = 403 - 20\]

=> \[1.15 * x  = 383\]

=> \[x  = 383/1.15\]

=> \[x  = 333\]

Number of seniors having job in 2010 = 0.4 * (333 + 50) = 0.4 * 383 = 153.2 = 153(approx)

Answer:

x = 1

y = -4

Step-by-step explanation:

We will multiply the 2nd equation by 4 first:

4 * [-x + y = -5]

-4x + 4y = -20

Now we will add this equation with 1st equation given. Shown below:

4x + 3y = -8

-4x + 4y = -20

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

7y = -28

Now, we can solve for y easily:

7y = -28

y = -28/7

y = -4

Now, we take this value of y and put it in 1st original equation and solve for x:

4x + 3y = -8

4x + 3(-4) = -8

4x - 12 = -8

4x = -8 + 12

4x = 4

x = 4/4

x = 1

So, this is the only solution to this problem ( 1 intersection point at x = 1 and y = -4)

Answer:

Step-by-step explanation:

The population P of a certain city can be modeled by

P=19000e0.0215t

where T represent the number of years since 2000.

When t=1 the year is 2001

when t =2 the year is 2002. This means that

When t=0 the year is 2000

To determine the year when the population will be 40,000 , we will substitute P = 40,000 and solve for t. It becomes

40000 = 19000e^0.0215t

40000 / 19000 = e^0.0215t

2.105 = e^0.0215t

Take ln of both sides

0.7443 = 0.0215t

t = 0.7443/0.0215

t = 34.62

Approximately 35 years

So the year will be 2035

Answer:each pound of jelly beans cost $2.5

Each pound of trailmix cost $1.25

Step-by-step explanation:

Let x represent the cost of one pound of jelly bean.

Let y represent the cost of one pound of trail mix.

A stores having a sale on jelly beans and trail mix for 8 pounds of jelly beans and 4 pounds of trail mix the total cost is $25. This means that

8x + 4y = 25 - - - - - - -1

For 3 pounds of jelly beans and 2 pounds of trailmix the total cost is $10. This means that

3x + 2y = 10 - - - - - - - - 2

Multiplying equation 1 by 3 and equation 2 by 8, it becomes

24x + 12y = 75

24x + 16y = 80

Subtracting,

- 4y = -5

y = - 5/ -4 = 1.25

Substituting y = 1.25 into equation 2, it becomes

3x + 2×1.25 = 10

3x + 2.5 = 10

3x = 10 - 2.5 = 7.5

x = 7.5/3 = 2.5

Answer:

C

Step-by-step explanation:

2x²-4x=0

2x(x-2)=0

either x=0

or x-2=0

x=2

so x=0,2

The real zeroes of given function is [tex]-\frac{3}{2},-1, \text { and } 2[/tex]

Given that, [tex]f(x)=2 x^{3}+x^{2}-7 x-6[/tex]

We have to find the real and imaginary zeroes

This can be found out by equating the function to zero and finding the roots "x"

Now, let us use trail and error method.

So put x = 1 in f(x)

f(1) = 2 + 1 – 7 – 6 = - 10  

1 is not a root.  Since f(1) is not equal to 0

Now put x = -1

f(-1) = -2 + 1 + 7 – 6 = 0  

-1 is a root.  Since f(-1) is equal to 0

So, one of the roots is -1. Let the other roots be a, b.

[tex]\text { Sum of roots }=\frac{-x^{2} \text { coefficient }}{x^{3} \text { coefficient }}[/tex]

[tex]\begin{array}{l}{a+b+(-1)=\frac{-1}{2}} \\\\ {a+b=1-\frac{1}{2}} \\\\ {a+b=\frac{1}{2} \rightarrow(1)}\end{array}[/tex]

[tex]\begin{array}{l}{\text {Product of roots }=\frac{-\text {constant}}{x^{3} \text {coefficient}}} \\\\ {a b(-1)=\frac{-(-6)}{2}} \\\\ {a b(-1)=3} \\\\ {a b=-3 \rightarrow(2)}\end{array}[/tex]

Now, we know that, algebraic identity,

[tex]\begin{array}{l}{(a-b)^{2}=(a+b)^{2}-4 a b} \\\\ {(a-b)^2=\left(\frac{1}{2}\right)^{2}-4(-3)} \\\\ {(a-b)^2=\frac{1}{4}+12} \\\\ {(a-b)^2=\frac{49}{4}} \\\\ {a-b=\frac{7}{2} \rightarrow(3)}\end{array}[/tex]

Add (1) and (3)

[tex]\begin{array}{l}{2 a=\frac{7+1}{2} \rightarrow 2 a=4 \rightarrow a=2} \\\\ {\text { Then, from }(2) \rightarrow b=-\frac{3}{2}}\end{array}[/tex]

Hence, the roots of the given equation are [tex]-\frac{3}{2},-1, \text { and } 2[/tex]

Answer:

b. Exactly two outcomes are possible on each trial.

Step-by-step explanation:

The correct answer is option B: Exactly two outcomes are possible on each trial.

We can define the binomial probability in any binomial experiment, as the probability of getting exactly x successes for n repeated trials, that can have 2 possible outcomes.

The binomial probability distribution is applicable when an experiment has exactly two outcomes. These trials are independent and the probabilities of the outcomes remain the same for each trial, referred to as Bernoulli trials. Using this distribution, we can calculate mean and standard deviation for the function.

The characteristic of an experiment where the binomial probability distribution is applicable is when exactly two outcomes are possible on each trial, referred to as 'success' and 'failure.' These are termed as Bernoulli trials for which binomial distribution is observed. In this situation, 'p' denotes the probability of a success on one trial, while 'q' represents the failure likelihood. The trials are independent, which means the outcome of one trial does not affect the results of subsequent trials. Moreover, the probabilities of the outcomes remain constant for every trial. The random variable X signifies the number of successes in these 'n' independent trials. The mean and standard deviation can be calculated using the formulas µ = np and √npq, respectively.

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The plot is missing. I have written the stem-leaf plot below.

Answer:

(a) 35 participants

(b) 22 participants traveled more than 14 km.

Step-by-step explanation:

Given:

(a)

The stem-leaf plot is given as:

12| 3  3  6 7 9 9

13| 1 1 4 5 5

14| 0 0 2 3 3 8 8 9

15| 2 2 2 2 2 3 5 5 7

16| 4 5 5 9 9

17| 3 5

The total number of numbers on the leaf side gives the total number of participants.

So, the number of participants is equal to the total number of elements on the leaf part. The total number of elements on the leaf part is 35.

Therefore, the total number of participants is 35.

(b)

Given:

12| 3  3   6 7 9 9

13| 1 1 4 5 5

14| 0 0 2 3 3 8 8 9

15| 2 2 2 2 2 3 5 5 7

16| 4 5 5 9 9

17| 3 5

One element of the stem part and one part of leaf part is represented as:

12 | 3 means 12.3 km

So, the number of walkers traveling more than 14.0 km is the list of all numbers greater than 14.0 km. Let us write all the numbers which are greater than 14.0 km. The numbers are:

14 | 2 3 3 8 8 9 = 6 participants

15 | 2 2 2 2 2 3 5 5 7 = 9 participants

16 | 4 5 5 9 9 = 5 participants

17 | 3 5 = 2 participants

Therefore, the total number is the sum of all the above which is equal to:

= 6 + 9 + 5 + 2 = 22 participants

Answer:

f(x) = ln(x^3 + e^4x) answer is explain in attachment .

Step-by-step explanation:

f(x) = ln(x^3 + e^4x) =(2x+3e³ˣ) 1/x² + e³ˣ

Answer:

Ans 1. [tex]x= 7[/tex]

Ans 2.a.

[tex]\overline {AC} \cong \overline {JL} \\\textrm{is the additional information required to prove the triangles are congruent by SAS postulate}[/tex]

Ans.2.b.

[tex]\overline {BC} \cong \overline {KL} \\\textrm{is the additional information required to prove the triangles are congruent by the HL theorem}[/tex]

Step-by-step explanation:

Solution:

1.

Vertically opposite angles are equal.

[tex]\therefore (x+16) = (4x-5)\\\therefore (4x-x) = (16+5)\\\therefore (3x) = (21)\\\therefore x = 7[/tex]

2.a.

proof for Δ BAC ≅ ΔKJL by SAS postulate.

InΔ BAC and Δ KJL

BA ≅  KJ               Given

∠ BAC ≅ ∠ KJL   {measure each angle is 90}

[tex]\overline{AC} \cong \overline{JL}\ \textrm{additional information require to prove the tangles are congruent by SAS postulate}\\\therefore \triangle BAC \cong \triangle KJL\ \textrm{By Side-Angle-Side postulate...PROVED}[/tex]

2.b.

proof for Δ BAC ≅ ΔKJL by HL theorem.

InΔ BAC and Δ KJL

BA ≅  KJ               Given

∠ BAC ≅ ∠ KJL   {measure each angle is 90}

[tex]\overline{BC} \cong \overline{KL}\ \textrm{additional information require to prove the tangles are congruent by HL theorem}\\\therefore \triangle BAC \cong \triangle KJL\ \textrm{By Hypotenuse Leg Theorem......PROVED}[/tex]

Answer:

  1

Step-by-step explanation:

(f+g)(1) = f(1) +g(1) = -1 + 2 = 1

__

The pair (1, -1) in the definition of f tells you f(1) = -1.

The pair (1, 2) in the definition of g tells you g(1) = 2.

He has a 26% of getting a strike, which means he has a 74% chance of not getting a strike ( 100% - 26% = 74%).

Multiply the chance of not getting a strike by the number of attempts:

0.74 x 0.74 x 0.74 x 0.74 x 0.74 = 0.22

The answer is B) 0.22

The probability of making a free throw is 77%, the probability of not making one would be 23% ( 100% - 77% = 23%).

Add the probability of making the first one ( 0.77) by the probability of making the second one multiplied by the probability of missing the second one ( 0.77x 0.23)

0.77 + (0.77 x 0.23)

0.77 + 0.18 = 0.95

The answer is D) 95%

Final answer:

To find the probability that Keyshawn doesn't get a strike until after his first five attempts, we need to find the probability of him not getting a strike on each individual attempt. The probability that Keyshawn doesn't get a strike until after his first five attempts is 0.19. To find the probability that Coram makes his first free throw within his first two shots, we need to find the probability of him making a free throw on the first or the second shot. The probability that Coram makes his first free throw within his first two shots is 95%.

Explanation:

To find the probability that Keyshawn doesn't get a strike until after his first five attempts, we need to find the probability of him not getting a strike on each individual attempt. Since each attempt is independent, we can multiply the probabilities together.

The probability of him not getting a strike on each attempt is 1 - 0.26 (the probability of getting a strike). So the probability of not getting a strike on the first five attempts is (1 - 0.26) * (1 - 0.26) * (1 - 0.26) * (1 - 0.26) * (1 - 0.26).

Calculating this expression, we get:

(0.74) * (0.74) * (0.74) * (0.74) * (0.74) = 0.192.

So the probability that Keyshawn doesn't get a strike until after his first five attempts is approximately 0.192.

Therefore, the correct answer is A) 0.19.

To find the probability that Coram makes his first free throw within his first two shots, we need to find the probability of him making a free throw on the first or the second shot. These events are mutually exclusive, so we can add the probabilities together.

The probability of him making a free throw on the first shot is 0.77, and the probability of him missing the first shot and making a free throw on the second shot is (1 - 0.77) * 0.77. So the total probability is 0.77 + (1 - 0.77) * 0.77.

Calculating this expression, we get:

0.77 + 0.23 * 0.77 = 0.77 + 0.1771 = 0.9471.

So the probability that Coram makes his first free throw within his first two shots is approximately 0.9471.

Therefore, the correct answer is D) 95%.

Final answer:

There are 371 students who have taken a course in either calculus or discrete mathematics, 157 students have taken calculus but not discrete mathematics, and 24 students have taken discrete mathematics but not calculus.

Explanation:

To determine how many students have taken a course in either calculus or discrete mathematics, we use the principle of inclusion and exclusion. The formula for two sets A (calculus students) and B (discrete mathematics students) is: |A∪B| = |A| + |B| - |A∩B|.

A. Using the provided numbers: |A∪B| = 347 (calculus students) + 214 (discrete mathematics students) - 190 (students in both) = 371 students have taken a course in either calculus or discrete mathematics.

B. The number of students who have taken calculus but not discrete mathematics is found by subtracting the number of students who have taken both from the total number of calculus students: 347 - 190 = 157 students.

C. Similarly, the number of students who have taken discrete mathematics but not calculus is found by subtracting the number of students who have taken both from the total number of discrete mathematics students: 214 - 190 = 24 students.

Answer:

x − 1

x + 2

Step-by-step explanation:

f(x) = 2x³ + 5x² − x − 6

Plug in the zero of each binomial.  If that zero is also a zero of f(x), then the binomial is a factor of f(x).

f(1) = 0

f(-1) = -2

f(2) = 28

f(-2) = 0

The correct options are option A and option D that is the factors of given binomial expressions are x  - 1 and x + 2.

What is binomial factor ?

Binomial factors are polynomial factors that have exactly two terms. Factoring a polynomial is the first step to finding its roots.

The given binomial expression is equal to :

2x³ + 5x² - x - 6

We know that factor theorem states that if [tex]x_{1}[/tex] , [tex]x_{2}[/tex], --- --- --- --- , [tex]x_{n}[/tex] are roots of a function , them (x - [tex]x_{1}[/tex]) , (x - [tex]x_{2}[/tex]) , --- --- --- --- , ( x - [tex]x_{n}[/tex]) are the factors of given binomial expression.

In the given question :

f(x) = 2x³ + 5x² - x - 6

If we put x = 1 , then we get f(x) = 0 . This implies the binomial factor will be (x - 1).

So ,

(ax² + bx + c ) ( x - 1) = 2x³ + 5x² - x - 6

ax³ + (b-a) x² + (c - b) x - c = 2x³ + 5x² - x - 6

If we compare both sides we get :

a  = 2 , b - a  = 5 , c - b  = 1 and c = 6

So , b = 7

We know that general form is given by :

ax² + bx + c

or

2x² + 7x + 6

Then :

Δ = 7² - 4 × 2 × 6 = 1

and other roots of solutions are :

x1 = [tex]\frac{-7 + \sqrt{1} }{4}[/tex] = [tex]\frac{-3}{2}[/tex]

x2 = [tex]\frac{-7 - \sqrt{1} }{4}[/tex] = -2

So , the other factors will be x + [tex]\frac{3}{2}[/tex] and x + 2.

Therefore , the correct options are option A and option D that is the factors of given binomial expressions are x  - 1 and x + 2.

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Answer:

x=4

Step-by-step explanation:

EF is half of the top and bottom lines comined so you would make an equation:

5x+8/2=2x+6

5x+8=4x+12

x=4

Answer:

Step-by-step explanation:

Phil is going to store to buy a hat and a coat.

Let $x represent the cost of the cos of the hat. The coat cost 3 times as much as the hat. It means that the cost of the coat will be 3×x = $3x

His mom tells him that he cannot spend more than $120. Assuming he spends exactly $120. It means that

3x + x = 120

4x = 120

x = 120/ 4 = 30

The hat costs $30

The coat costs 30×3 = $90

The most he can spend on a coat is $90 since he cannot spend more than $120

Answer:

The number of ways the license plates contain no repeated letters and no repeated digits = 1,404,000

Step-by-step explanation:

The total number of alphabet's = 26

The total number of digits  = 10 (0 - 9)

The number plate contains 3 letters followed by 2 digits.

The number of ways the license plates contain no repeated letters and no repeated digits.

The number of ways first letter can be filled in 26 ways.

The number of ways second letter can be filled in 25 ways.

The number of ways third letter can be filled in 24 ways.

The number of ways the first digit can be in 10 ways

The number of ways the second digit can be in 9 ways.

The number of ways the license plates contain no repeated letters and no repeated digits = 26 × 25 × 24 × 1 0 × 9

The number of ways the license plates contain no repeated letters and no repeated digits = 1,404,000

Answer:

Step-by-step explanation:

She uses all the food to feed the dogs in 6 days. She feeds the with an equal amount of food in 6 days. This means that whatever amount of food she buys, the least amount she would feed the dogs with is 1/6 of that amount.

There are 31 cups of dog food. This means that the amount she would feed them with each day is

1/6 × 31 = 5.1 cups

So the least amount is 5 cups

Therefore,

The number of cups she feeds her dogs with would be between 5 and 6 cups.

Ground: y = 0

- 16t² + 128t + 50 = 0

Apply quadratic equation.

t = 0, 8 (rounded to the nearest second)

8 seconds

The time it will take for the object to hit the ground is approximately 4 seconds (rounded to the nearest second).

The given function a(t) = -16t^2 + 128t + 50 represents the approximate altitude, in feet, of an object released from the top of a building t seconds after it is released. To find the time it will take for the object to hit the ground, we need to find the value of t when the altitude is 0.

Setting a(t) = 0, we get:

-16t^2 + 128t + 50 = 0

Using the quadratic formula, we can solve for t.

t = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values a = -16, b = 128, and c = 50, we get t = 0.54 s or t = 3.79 s. Since the object is already at a height of 0 at t = 0 (the time of release), the time it will take for the object to hit the ground is approximately 4 seconds (rounded to the nearest second).

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Answer:

$25 each ticket!

Step-by-step explanation:

6*4=24

124-24=100

100/4=25

Answer:

d. 29ft

Step-by-step explanation:

Using pythagoras theorem which states that

a^2 + b^2 = c^2

where a and b are the opposite and adjacent sides of a right angled triangle and c is the hypotenuse side

From the attached image, Let the length of the rope be x

[tex]x^{2} = 8^{2}+ 28^{2} \\[/tex]

[tex]x^{2} = 64 + 784[/tex]

[tex]x^{2} = 848[/tex]

[tex]x =\sqrt{848}[/tex]

[tex]x = 29.12[/tex]

x≈29

Answer:

The answer to your question is: Burger = $4.45, Pizza = $2.2

Step-by-step explanation:

Jason = $15.5 for 3 slices of pizza + 2 burgers

Susan = $20 for 1 slice of pizza + 4 burgers

Pizza = p

burger = b

System of equations

Jason                                   3p + 2b = 15.5        (I)

Susan                                     p + 4b = 20           (II)

Solve system by elimination

Multiply (II) by -3

                                          3p + 2b = 15.5

                                        -3p - 12b = -60

                                              -10b = -44.5

                                                   b = -44.5/-10

                                                  b = $4.45

                                             p + 4(4.45) = 20

                                             p + 17.8 = 20

                                             p = 20 - 17.8

                                                 p = 2.2

One slice of pizza costs $2.2 and one burger cost $4.45

The speed is 66.33 miles per hour

Step-by-step explanation:

Given

Total distance = 325 miles

They still have 126 miles to go. In order to find the distance they have covered, we will subtract the remaining miles from the total

So,

[tex]Distance\ covered=325-126=199\ miles[/tex]

the car has been driving for 3 hours

So,

t= 3 hours

d = 199 miles

[tex]Speed=\frac{d}{t}\\=\frac{199}{3}\\=66.33\ miles\ per\ hour[/tex]

The speed is 66.33 miles per hour

Keywords: Speed, Distance

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Answer:

Remember, a homogeneous system always is consistent. Then we can reason with the rank of the matrix.

If the system Ax=0 has only the trivial solution that's mean that the echelon form of A hasn't free variables, therefore each column of the matrix has a pivot.

Since each column has a pivot then we can form the reduced echelon form of the A, and leave each pivot as 1 and the others components of the column will be zero. This means that the reduced echelon form of A is the identity matrix and so on A is row equivalent to identity matrix.