Emir, Katherine, Dylan and Anna Who earned enough money to attend the festival?

Solving equations and inequalities are essential skills in various real-world applications, from engineering and physics to business and finance.

These skills enable us to find solutions to problems and make informed decisions. For instance, engineers use equations to design buildings, bridges, and machines, while scientists use equations to model and predict the behavior of physical systems. In business and finance, equations and inequalities are used to solve problems related to budgets, investments, and financial planning.

They help individuals and organizations make informed decisions and manage resources efficiently. For example, a company can use equations to determine the optimal production level that maximizes profits or to evaluate the feasibility of a new project.

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Answer:

12

Step-by-step explanation:

Given that,

f(x) = 2x² - 3x + 7

To find the value of f ( 2.5 ), replace x with 2.5 and solve the equation.

Let us solve it now.

f(2.5) = 2(2.5)² - 3×2.5 + 7

f(2.5) = 12.5 - 7.5 + 7

f(2.5) = 12

Answer:

[tex]f(x) = 2 {x}^{2} - 3x + 7 \\ f(2.5) = 2 {(2.5)}^{2} - 3(2.5) + 7 \\ f(2.5) = 12.5 - 7.5 + 7 \\ \boxed{f(2.5) = 12}[/tex]

Answer:

y = 0.5 - x/2

Step-by-step explanation:

The -1 power means the inverse function of f(x). This means that you will write the function in terms of x and y, replaces x's with y's and vice versa, and then write the new function in terms of x.

f(x) = 1 - 2x

y = 1 - 2x

Switch variable places.

x = 1 - 2y

Solve for y.

2y = 1 - x

y = 0.5 - x/2

Answer:

C.

Step-by-step explanation:

Thus, the number of buses needed to carry the 2177 tourists is found as 61.

A number can be divided into an equal amount of component parts.

People may display a variety of signs to denote division. The backslash / is additionally employed, though the character is more typical. Numerous numbers may occasionally be written with a line separating them. It is also known as a fraction.

A division calculation has a name for each component. The dividend, the divisor, and indeed the quotient are the three key terms.

Given data:

Number of tourists carried by 1 bus = 36.

Total number of tourists = 2177.

Number of buses = (Total number of tourists)/(Number of tourists carried by 1 bus)

Number of buses = 2177/36

Number of buses = 60.47

Number of buses =  61

Thus, the number of buses needed to carry the 2177 tourists is found as 61.

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The expected value of the car representing the delay of the car at one of the light as per distribution of time is given by option d. 2/3.

For the probability density function of X.

Divide the interval from 0 to 3 minutes into three sub-intervals,

The first minute, during which the car will pass the intersection if the light is green.

The second minute, during which the car will be delayed by one minute if the light is red.

The third minute, during which the car will be delayed by two minutes if the light is red.

The probability of the car arriving during the first sub-interval is 1/3.

If the light is green, the car will pass immediately with no delay.

If the light is red, the car will be delayed by one minute.

So the probability density function of the delay during this sub-interval is,

f(x) ={  1/3, if 0 ≤ x ≤ 1

         0,     otherwise  }

Probability of the car arriving during the second sub-interval is also 1/3.

If the light is red, the car will be delayed by one minute.

If the light is green, the car will be delayed by two minutes.

So the probability density function of the delay during this sub-interval is,

f(x) = { 1/6,      if 1 ≤ x ≤ 2

         1/3,       if 2 ≤ x ≤ 3

         0,          otherwise }

Probability of the car arriving during the third sub-interval is also 1/3.

If the light is red, the car will be delayed by two minutes.

If the light is green, the car will be delayed by three minutes.

But since the car cannot wait for more than three minutes, the delay during this sub-interval is bounded by 2 minutes.

So the probability density function of the delay during this sub-interval is,

f(x) = {  1/6,     if 2 ≤ x ≤ 3

             0,      otherwise  }

Now,

calculate the expected value of X by integrating the product of the delay and the probability density function over the entire range of possible delays

E(X) = [tex]\int_{0}^{3}[/tex] x f(x) dx

= [tex]\int_{0}^{1}[/tex]0 dx + [tex]\int_{1}^{2}[/tex] x (1/6) dx + [tex]\int_{2}^{3}[/tex] x(1/6) dx

= (1/6)(1/2)  [(2^2 - 1^2) + (3^2 - 2^2)]

= (1/12) [ 3 + 5 ]

= 8/12

= 2/3

Therefore, the expected value of the delay of the car at the light is equal to option d. 2/3.

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Therefore, the approximate sample standard deviation that the organization must have used to compute the margin of error is approximately 4.174 complaints per week. Answer: a. 4.174 complaints

Simple deviation is a statistical term used to describe the difference between a data point and a reference value or average. It is calculated by subtracting the reference value or average from the data point and taking the absolute value of the result.

by the question.

The formula for margin of error is:

[tex]Margin of error = Z * (standard deviation / \sqrt(sample size))[/tex]

Where Z is the critical value from the standard normal distribution for the desired confidence level (99% in this case), standard deviation is the population standard deviation (which we do not know), and sample size is the number of observations in the sample (30 weeks in this case).

We can rearrange the formula to solve for the standard deviation:

[tex]Standard deviation = Margin of error * sqrt(sample size) / Z[/tex]

Plugging in the given values, we get:

[tex]Standard deviation = 2.1 * \sqrt(30) / 2.58 =4.174[/tex]

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If you choose a card at random from 1 through 8, there is an 87.5% probability that you will choose a card that is higher than 1.

There are 8 cards in total, and we want to find the probability of picking a card that is greater than 1.

Out of the 8 cards, 7 of them are greater than 1 (2, 3, 4, 5, 6, 7, and 8).

Therefore, the probability of picking a card that is greater than 1 is:

P(greater than 1) = 7/8

To write this as a percentage rounded to the nearest tenth, we can multiply by 100 and round to one decimal place:

P(greater than 1) = 7/8 * 100% = 87.5% (rounded to the nearest tenth)

Therefore, the probability of picking a card that is greater than 1 is 87.5%.

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Substitute a=7, b=5, and c=3 into the expression ab+c:

ab+c = (7)(5) + 3

ab+c = 35 + 3

ab+c = 38

Therefore, ab+c is equal to 38 when a=7, b=5, and c=3.

The average rate of change from year 1 to year 10 is 8.3 pounds per acre per year based on given function.

We must compute the change in crop yield (in pounds per acre) during that time period and divide by the number of years to determine the average rate of change from year 1 to year 10:

Average change rate is equal to (f(10) - f(1)) / (10 - 1)

where f(x): function that represents the crop yield over an x-year period in pounds per acre.

When we enter the numbers 10 and 1 as substitutes in the function, we obtain:

f(10) = [tex]-10^2[/tex] + 20(10) + 50 = 150

f(1) =[tex]-1^2[/tex] + 20(1) + 50 = 69

Thus, from year one to year ten, the average rate of change is:

(150 - 69) / (10 - 1), or 8.3 pounds per acre per year, is the average rate of change.

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The expression that corresponds to the provided sentence is 2sinx cos²x.

As an illustration, the expression x + y is one where both x and y have words with an addition function in between. There are two kinds of expressions in mathematics: numerical expressions, which only comprise integers, and algebraic expressions, which also include variables.

In mathematics, an expression is a combination of one or more numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, division, exponents, and roots) that can be evaluated or simplified to produce a single value.

For example, 2 + 3x is an expression that contains the variables x and the numbers 2 and 3, as well as the operations of addition and multiplication.

Another example is [tex](4a^2 - 3b)/c[/tex], which is an expression that contains the variables a, b, and c, as well as the numbers 4 and 3, and the operations of addition, subtraction, multiplication, and division.

sin2x cos x is equivalent to (2sinx cosx) cosx,

which simplifies to 2sinx cos²x.

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question-

Which of the following expressions is equal to sin(2x) 2(1 - cos^2(x))

Answer:

Step-by-step explanation:

34,000,000,000=4,005,000(1+0.041)^x

(1.041)^x=(34,000,000,000)/4,005,000

(1.041)^x=34,000,000/4,005

xlog(1.041)=log34,000,000-log 4005

=7.15348-3.60260

=3.55088

x=3.55088/log(1.041)=3.55088/0.01745

\approx 203.5~years

The area of ΔQRS is 26.10 square units.

What is triangle?

A triangle is a closed, two-dimensional geometric shape with three straight sides and three angles.

To find the area of [tex]$\triangle QRS$[/tex], we can use the formula:

[tex]$Area = \frac{1}{2} \times base \times height$[/tex]

where the base and height are the length of two sides of the triangle that are perpendicular to each other. We can find these sides using trigonometry.

First, we need to find the length of side [tex]$QR$[/tex]. We can use the Law of Cosines:

[tex]$QR^2 = QS^2 + RS^2 - 2(QS)(RS)\cos(R)$[/tex]

where [tex]$R$[/tex] is the angle at vertex [tex]$R$[/tex]. Substituting the given values, we get:

[tex]$QR^2 = 9^2 + 5^2 - 2(9)(5)\cos(57^\circ)$[/tex]

[tex]$QR \approx 8.02$[/tex]

Next, we need to find the height of the triangle, which is the perpendicular distance from vertex [tex]$R$[/tex] to side [tex]$QS$[/tex]. We can use the sine function:

[tex]$\sin(R) = \frac{opposite}{hypotenuse}$[/tex]

[tex]$\sin(57^\circ) = \frac{height}{8.02}$[/tex]

[tex]$height \approx 6.51$[/tex]

Now we can find the area of the triangle:

[tex]$Area = \frac{1}{2} \times QR \times height$[/tex]

[tex]$Area = \frac{1}{2} \times 8.02 \times 6.51$[/tex]

[tex]$Area \approx 26.10$[/tex] square units

Therefore, the area of [tex]$\triangle QRS$[/tex] is approximately [tex]$26.10$[/tex] square units.

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Answer: It would be 3:13

Step-by-step explanation:

Pls brainliest if this helped you! :D

Answer:

3:13

Step-by-step explanation:

First we have to find the GCF (greatest common factor of both numbers) which is 20

So 60 divided by 20 is 3 and 260 divided by 20 is 13

Your final answer should by 3:13 because you can't divide any further using those numbers or any other number

Hope this helps!

Also for your information to make things easier --> keep in mind that whenever it says "ratio" means "to divide" !!!

Therefore , the solution of the given problem of equation comes out to be the equation of the perpendicular bisector of that line segment (7, 3) and (-5, 9) is y = 2x + 4.

Complex algorithms frequently employ variable words to demonstrate coherence between two opposing assertions. Equations are academic expressions that are used to demonstrate the equality of different academic figures. In this case, leveling generates b + 7 rather than another algorithm that could analyze data provided by y + 7, split 12 into two parts, and create y + 7.

Here,

The midpoint formula can be used to determine the midpoint of the line segment whose ends are (7, 3) and (-5, 9):

=> ((7 + (-5))/2, (3 + 9)/2) = (1, 6)

the middle is thus (1, 6). The slope of the line going through (7, 3) and must now be determined. (-5, 9). The slope method is used to:

=>slope = (9 - 3)/(-5 - 7) = -6/12 = -1/2

The equivalent of -1/2 in the negative is 2.

=> y - y1 = m(x - x1)

where (x1, y1) is a location on the line and m denotes the slope. When we replace m = 2 with (1, x1, y1) = (1, 6), we obtain:

=> y - 6 = 2(x - 1)

By enlarging and condensing, we obtain:

=> y - 6 = 2x - 2

=> y = 2x + 4

the line segment whose ends are and the equation of the perpendicular bisector of that line segment (7, 3) and (-5, 9) is y = 2x + 4.

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The counting process was an essential component in evaluating the results and drawing conclusions based on the data gathered.

Hi! Bob and Aniyah's apples were counted by a designated individual, likely a teacher, supervisor, or event organizer, who was responsible for ensuring the accuracy and fairness of the counting process.

This person utilized their organizational skills and attention to detail to effectively tally the number of apples collected by both Bob and Aniyah.

By doing so, they were able to determine the final count, which was then used to either compare the performance of the two individuals or contribute to a larger total for a specific event or purpose.

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Therefore, the coordinates of the vertices of the image after the transformation Is,-1 ° r y-axis are J'(-1,-2), R'(3,6), and B'(-4,5).

In mathematics, a coordinate refers to a set of numbers that identifies the position or location of a point in a space. Coordinates can be used to specify the position of objects in one, two, or three-dimensional spaces. The most common types of coordinates used in mathematics are the Cartesian coordinates, which consist of an ordered pair of numbers representing the x and y coordinates of a point in a two-dimensional space, and the three-dimensional Cartesian coordinates, which consist of an ordered triple of numbers representing the x, y, and z coordinates of a point in a three-dimensional space. Coordinates are widely used in geometry, algebra, and other branches of mathematics.

Here,

To apply the transformation Is,-1 ° r y-axis, we need to reflect each point of the original figure across the y-axis. This is done by changing the sign of the x-coordinate and leaving the y-coordinate unchanged. The coordinates of the vertices of AJRB are J(1,-2), R(-3,6), and B(4,5). Applying the transformation, we get:

J'(-1,-2) (x-coordinate changes sign, y-coordinate stays the same)

R'(3,6) (x-coordinate changes sign, y-coordinate stays the same)

B'(-4,5) (x-coordinate changes sign, y-coordinate stays the same)

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ΔABC is similar to [tex]~[/tex]ΔADE. BC= 162 and m∠ACB≈ 67.86°

Two triangles are similar if they have the same ratio of corresponding sides and equal pair of corresponding angles.

In  ΔABC and [tex]~[/tex]ΔADE,

m∠ADE=m∠ABC=90⁰

m∠EAD=m∠CAB...(common angle)

Thus, ΔABC is similar to [tex]~[/tex]ΔADE, by AA test of similarity.

Corresponding angles of both the triangles are equal, and

Corresponding sides of both the triangles are in proportion to each other.

In  ΔABC and [tex]~[/tex]ΔADE,

To find the value of x, we can cross-multiply the given equation and solve for x:

AE/AC=DE/BC

230/315 = 120/BC

Multiplying both sides by BC, we get:

230BC/315 = 120

Multiplying both sides by 315, we get:

230BC = 120 * 315

Dividing both sides by 230, we get:

BC= (120 * 315) / 230

Simplifying, we get:

BC= 162

The cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse.

In ΔABC

Cos ∠ACB=BC/AC

Cos ∠ACB=162/315

We can use the inverse cosine function (cos⁻¹) to find the measure of angle ACB:

∠ACB= cos⁻¹(162/315)

Using a calculator, we get:

∠ACB ≈ 67.86°

Therefore, the measure of angle ACB is approximately 67.86 degrees.

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The selling price of the oranges was 472.5 dollars.

The profit is equal to the difference between the selling price and the original cost price.

We have been given that a basket containing 150 oranges costs 450 find the selling price of an orange if it is to be sold at a profit of 5%

Let's start with the profit.

Let's convert 5% to decimal form to find out.

[tex]\implies 5\% = \dfrac{5}{100} = 0.05[/tex]

Then double that by the starting price of 450.

[tex]450\times 0.05 = 22.5[/tex]

As a result, the profit is $22.5.

Now, add the profit to the beginning cost to get the selling price.

[tex]450 + 22.5 = 472.5[/tex]

Therefore, the selling price of the oranges was 472.5 dollars.

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Answer:

4.  4/13

5.  7/13

6.  4/13

Step-by-step explanation:

A standard deck has 52 cards, 4 suits, each suit has cards Ace through 10 and the face cards, Jack, Queen, King.

If the idea of suits confuses you, think about Uno where the cards are blue, yellow, green, and red. Only here the suits are Diamonds, Clubs, Hearts, and Spades

To calculate compound probability use the following formula:

P(A ∪ B) = P(A) + P(B) - P(A | B)

This says that the probability of A or B occuring is the probability of A, plus the probability of B, minus the probability of them both occuring at the same time.

4)  Randomly selecting a diamond or a seven.

There are 13 diamonds in a deck and 4 sevens, one from each suit. The joint event for this probabaility is randomly drawing a 7 of diamonds. There is only one of these in the deck.

P(Diamond ∪ 7) = P( Diamond ) + P( 7 ) - P( Diamond | 7)

                         = 13/52 + 4/52 - 1/52

                         = 16/52 = 4/13

5)  Randomly selecting a red card or a queen.

There are 26 red cards in the deck, from diamonds and hearts suits, 13 from each. There are 4 queens in the deck, one from each suit. The joint event is a queen of a diamonds or hearts, which there are 2 of in the deck.

P( Red ∪ Queen ) = P( Red ) + P( Queen ) - P( Red | Queen)

                             = 26/52 + 4/52 - 2/52

                             = 28/52 = 7/13

6) Randomly selecting a three or a face card.

There are 4 threes in a deck of cards, one from each suit. There are 12 face cards in a deck (Jack, Queen, and King with three from each suit). There is no joint event because you cannot draw a three and a face card. So this time we subtract 0 because the events are mutually exclusive.

P( 3 | Face ) = P( 3 ) + P( Face ) - P( 3 | Face )

                   = 4/52 + 12/52 - 0/52

                   = 16/52 = 4/13

The calculated  half-life of the radioactive substance is 18.905 years.

% remaining after each year: 100 - 3.1 = 96.9

Solve for t in

1/2 = (0.969)t

t = log(1/2) / log(0.969) ≅ 18.905 years

t1/2 = 18.905 years

The time it takes for a radioactive element's atoms to divide in half from their initial number is known as the half life of the element or substance.

nuclear decay Radioactive decay is the process by which the nucleus of a heavy radioactive element breaks down and releases alpha, beta, and gamma rays.

Radioactivity law says that when a radioactive element breaks down or decays, a new element will either emit alpha particles two places below the original element in the periodic table or beta particles two places above the original element.

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To solve the equation a (a^2 - 16) (a - 2) = 0, we can use the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.

So, we need to find the values of a that make each factor equal to zero.

a = 0 is a solution, since any number times zero is zero.

a^2 - 16 = 0 can be factored as (a - 4)(a + 4) = 0. So, a = 4 or a = -4 are solutions.

a - 2 = 0 gives us a = 2 as a solution.

Therefore, the possible solutions to the equation are a = 0, a = 4, a = -4, and a = 2.

So, the answers to select from are: 0, 6, 4, 16, 1, 2, -4, 8. Among these options, the possible solutions to the equation are 0, 4, -4, and 2.

Answer:

Step-by-step explanation:

The volume of a square-based cuboid is given by the formula:

V = l × w × h

where l is the length, w is the width, and h is the height of the cuboid.

If the length is 4 cm, the width is also 4 cm (since it is a square-based cuboid), and the height is 10 cm, we can substitute these values into the formula and calculate the volume:

V = 4 cm × 4 cm × 10 cm

V = 160 cubic centimeters (cm³)

Therefore, the volume of the square-based cuboid is 160 cubic centimeters.

Answer:

Step-by-step explanation:

Unfortunately, I cannot see the diagram you are referring to. However, I can provide you with the formula for calculating the volume of a square-based cuboid.

The volume of a square-based cuboid is calculated by multiplying the length, width, and height of the cuboid together. In other words:

Volume = length x width x height

So if the length of the cuboid is 4 cm, the width is also 4 cm (since it is a square-based cuboid), and the height is 10 cm, the volume can be calculated as:

Volume = 4 cm x 4 cm x 10 cm

Volume = 160 cubic centimeters

Therefore, the volume of the cuboid is 160 cubic centimeters (cc), or cm^3.

Answer:

Exact SA = 166.4π cm²

Approximate SA = 522.8 cm²

Step-by-step explanation:

SA = 2πr(r + h)

r = d/2 = 8 cm / 2 = 4 cm

h = 16.8 cm

SA = 2π(4 cm)(4 cm + 16.8 cm)

SA = 2π(4 cm)(20.8 cm)

SA = 166.4π cm²

SA = 522.8 cm²

Answer:

2.936

Step-by-step explanation:

32.297 / 11= 2.936090909

Estimate = 2.936

Hope this helps :)

Answer:

Step-by-step explanation:

[tex]FV = (PMT * \frac{(1+i)^{n} -1}{i} )* (1 + i)[/tex]

FV = Future Value of the annuity

PMT = Amount of each annuity payment

i = Interest rate per period

n = Number of periods for which annuity will last

FV = ?

PMT = $4,500

i = 5.2% = 5.2/100 = 0.052

n = 3

[tex]FV = (4500 * \frac{(1+0.052)^{3} -1}{0.052} )* (1 + 0.052)[/tex]

[tex]FV = 14,953.30[/tex]

You will have $14,953.30 in 3 years.

Answer: b I believe.

Step-by-step explanation:

4,500 times 5.2% = 4,754 add that to the amount of money you have already. giving you b.

The required remainder and the quotient is 86 and 4 respectively.

None of the options are correct.

To determine the quotient and remainder by dividing 1534 ÷ 134.

In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication and division,

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,

134 ]  1534  [4

     -448

  --------------

      060

Remainder = 60

Quotient = 11

Thus, the required remainder and quotient is 60 and 11 respectively.

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The statement describes the relationships between x and y in these two equations is B. In y = 10x, the value of y is 10 times the value of x, and in y = x + 10, the value of y is 10 more than the value of x.

An equation simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario.

In this case, the statement describes the relationships between x and y in these two equations is that in y = 10x, the value of y is 10 times the value of x, and in y = x + 10, the value of y is 10 more than the value of x.

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We can factor the left side of the equation and use the zero product property to find possible values of a. We get a=-3 and a=2. By substituting these values back into the original equation, we can find the corresponding values of b which are -18 and 12, respectively.

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations and understand relationships between variables. It involves using letters and symbols to represent numbers and quantities, and manipulating equations to solve for unknown variables. Algebra is used in various fields of study, including science, engineering, economics, and finance.

To solve for the possible values of a and b given the equation:

We can use algebraic manipulation to rewrite the equation in terms of one variable.

Starting with:

we can simplify the left side of the equation by factoring out a common factor of (a + 3):

Now, we can use the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero.

Therefore, we have two possibilities:

(a + 3) = 0

If a + 3 = 0, then a = -3. Substituting this value of a back into the original equation, we get:

(2a - 3) = 0

If 2a - 3 = 0, then a = 3/2. Substituting this value of a back into the original equation, we get:

Now we have found the possible values of a. To find the corresponding values of b, we can substitute each value of a into either of the original equations and solve for b. Let's use the first equation:

When a = -3, we have:

Simplifying, we get:

-3b = -54

Dividing both sides by -3, we get:

b = 18

Therefore, when a = -3, b = 18.

When a = 2, we have:

Simplifying, we get:

2b = 12

Dividing both sides by 2, we get:

b = 6

Therefore, when a = 2, b = 6.

Thus, the possible values of a are -3 and 2, and the corresponding values of b are -18 and 12, respectively.

Therefore,We can factor the left side of the equation and use the zero product property to find possible values of a. We get a=-3 and a=2. By substituting these values back into the original equation, we can find the corresponding values of b which are -18 and 12, respectively

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In conclusion  the solution to the system of equations is x = 1 and y = 2.

How to justify each step?

Here are the steps to solve the system of equations:

5x - y = 3

x + 2y = 5

To eliminate y, we can multiply the second equation by 2:

2(x + 2y = 5)

2x + 4y = 10

Now we can add the first equation and the new second equation together, to eliminate y:

5x - y = 3

2x + 4y = 10

7x = 13

To solve for x, we can divide both sides by 7:

7x / 7 = 13 / 7

Simplifying, we get:

x = 1

Now we can use substitution to find y. We can substitute x = 1 into the first equation:

5x - y = 3

5(1) - y = 3

Solving for y, we get:

-y = -2

To solve for y, we can divide both sides by -1:

-y / -1 = -2 / -1

Simplifying, we get:

y = 2

So, the solution to the system of equations is x = 1 and y = 2.

The steps in solving the system of equations are justified using various properties of equality, as follows:

Step 1 uses the multiplication property of equality, which states that multiplying both sides of an equation by the same nonzero number does not change the solution set of the equation.

Step 2 uses the addition property of equality, which states that adding the same quantity to both sides of an equation does not change the solution set of the equation.

Step 3 uses the division property of equality, which states that dividing both sides of an equation by the same nonzero number does not change the solution set of the equation.

Step 4 uses the substitution property of equality, which states that if two expressions are equal, then one can be substituted for the other in any equation or expression without changing the solution set.

Step 5 uses the subtraction property of equality, which states that subtracting the same quantity from both sides of an equation does not change the solution set of the equation.

Step 6 uses the substitution property of equality again.

Step 7 uses the division property of equality again.

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