Answers
The graphs are:
1) not one-to-one
2) one-to-one
3) not one-to-one
Are the graphs one-to-one?A graph represents an one-to-one function only if.
We can't draw a vertical line that touches the graph twice or more.
We can't draw an horizontal line that touches the graph twice or more.
For example, for the first graph, we can see that there are horizontal lines like y = -5 that touch the graph two times.
Similarly forthe third graph, the line y = 0 touches two points-
So these two are not one-to-one.,
For the remaining graph, the middle one, no line touches the graph more than once, so it is one-to-one.
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Related Questions
I need help please can somebody help me please
Answers
The values of the matrices operations are [tex]4G + 2F = \left[\begin{array}{ccccc}34&-4&-42&-18&48\\-42&-8&16&30&8\\24&32&4&34&34&-12&-32&-12&-42&-20\end{array}\right][/tex] and [tex]4D - 3B= \left[\begin{array}{ccccc}31&6&34&-14&-57\\-19&-27&-18&18&27\\47&14&-1&-40&9&10&12&-19&-15&-58&-14&-23&10&-8&10\end{array}\right][/tex]
Evaluating the matrices operationsFrom the question, we have the following parameters that can be used in our computation:
[tex]G = \left[\begin{array}{ccccc}8&-5&-7&-1&10\\-6&-7&1&9&2\\4&6&3&7&5&-4&-3&0&-10&-9\end{array}\right][/tex]
Also, we have
[tex]F = \left[\begin{array}{ccccc}1&8&-2&-5&9\\-9&10&6&-3&0\\4&5&-4&3&7&2&-10&-6&-1&-8\end{array}\right][/tex]
Using the above as a guide, we have
[tex]4G + 2F = 4\left[\begin{array}{ccccc}8&-5&-7&-1&10\\-6&-7&1&9&2\\4&6&3&7&5&-4&-3&0&-10&-9\end{array}\right] + 2 \left[\begin{array}{ccccc}1&8&-2&-5&9\\-9&10&6&-3&0\\4&5&-4&3&7&2&-10&-6&-1&-8\end{array}\right][/tex]
Evaluate the sum
[tex]4G + 2F = \left[\begin{array}{ccccc}34&-4&-42&-18&48\\-42&-8&16&30&8\\24&32&4&34&34&-12&-32&-12&-42&-20\end{array}\right][/tex]
Next, we have
[tex]D = \left[\begin{array}{ccccc}7&-6&3&-8&-9\\2&-9&-6&6&9\\8&2&5&-10&-3&10&-3&-4&3&-7&1&-2&4&-5&-2\end{array}\right][/tex]
Also, we have
[tex]B = \left[\begin{array}{ccccc}-1&-10&-8&-6&7\\9&-3&-2&2&3\\-5&-2&7&0&-7&10&-8&1&9&10&6&5&2&-4&-9\end{array}\right][/tex]
The matrix expression is then represented as
[tex]4D - 3B= 4\left[\begin{array}{ccccc}7&-6&3&-8&-9\\2&-9&-6&6&9\\8&2&5&-10&-3&10&-3&-4&3&-7&1&-2&4&-5&-2\end{array}\right] - 3\left[\begin{array}{ccccc}-1&-10&-8&-6&7\\9&-3&-2&2&3\\-5&-2&7&0&-7&10&-8&1&9&10&6&5&2&-4&-9\end{array}\right][/tex]
Evaluate
[tex]4D - 3B= \left[\begin{array}{ccccc}31&6&34&-14&-57\\-19&-27&-18&18&27\\47&14&-1&-40&9&10&12&-19&-15&-58&-14&-23&10&-8&10\end{array}\right][/tex]
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Triangle QRS is rotated 180° about the origin.
On a coordinate plane, triangle Q R S has points (negative 4, 1), (negative 4, 4), (2, 1).
What are the coordinates of point S’?
Answers
Therefore, the coordinates of [tex]S'[/tex] are [tex](-2, -1)[/tex] .
What are coordinates?A series of numbers called coordinates is used to describe a point's location in a two- or three-dimensional space.
Typically, coordinates in a two-dimensional space are expressed as a pair of integers (x, y), where x denotes the point's horizontal location (also known as the abscissa) and y denotes its vertical position (also known as the ordinate).
Coordinates are commonly expressed as a triplet of numbers (x, y, z) in a three-dimensional space, where x, y, and z stand for the locations along the x, y, and z axes, respectively.
In order to describe the location of objects or points in space, coordinates are often employed in a variety of disciplines, including mathematics, physics, engineering, and computer graphics.
To rotate a point 180° about the origin, we need to negate the x-coordinate and the y-coordinate of the point.
So to find the coordinates of S', we can apply this rule to the coordinates of S:
x-coordinate of [tex]S' = -x-[/tex] coordinate of [tex]S = -(2) = -2[/tex]
y-coordinate of [tex]S' = -y-[/tex] coordinate of [tex]S = -(1) = -1[/tex]
Therefore, the coordinates of S' are [tex](-2, -1).[/tex]
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Correct answer gets brainliest!!!!
Answers
Answer: 5
Step-by-step explanation:
They are the points
There are 5 points so answer is 5
Which of the following numbers is shown plotted on the number line?
0
0 2.75 0 2 0-3 0-2 0-15
Answers
Answer:
-3 3/8
Step-by-step explanation:
The only number we're given on the number line is 0, but we can assume that each tick on the number line represents 1 units. We see that between the ticks that are 4 and 3 spaces away from 0.Since the point is to the left of 0, we know the number must be negative.The only answer that would make sense is -3 3/8, since the tick 4 places away from 0 is -4 and the tick 3 units away from 0 is -3 and -3 3/8 lies between -3 and -4help!!!
in the fridge below, m WXZ =72, abs m 1 is 6 degrees more than m 2. find m2
Answers
Step-by-step explanation:
m ∠WXY = 72°
m ∠ 1 = m ∠ 2 + 6°
m ∠WXY = m ∠ 1 + m ∠ 2
72° = (m ∠ 2 + 6°) + m ∠ 2
72° = 2 × (m ∠ 2) + 6°
2 (m ∠ 2) = 72° - 6°
2 (m ∠ 2) = 66°
(m ∠ 2) = 33°
#CMIIW1. Rachel has $60 to spend
on lunches this week. On
Monday she spent $5, and
on Tuesday she spent $7.
Which inequality can be
used to find, x, the amount
of money she can spend
the rest of the week?
Answers
Find the length of the third side. if necessary, round to the nearest tenth of 1 and 3
Answers
Using the Pythagorean theorem we know that the third side is 15 units.
What is the Pythagorean theorem?The Pythagorean theorem, sometimes known as Pythagoras' theorem, is a basic relationship between a right triangle's three sides in Euclidean geometry.
The Pythagorean Theorem states that the squares on the hypotenuse (the side across from the right angle) of a right triangle, or, in standard algebraic notation, a² + b², are equal to the squares on the legs.
To determine c, use the Pythagorean Theorem as follows:
a² + b² = c²
9² + 12² = c²
81 + 144 = c²
225 = c
c² = 225
c = √(225)
c = 15
Therefore, using the Pythagorean theorem we know that the third side is 15 units.
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Correct question:
Find the length of the third side. If necessary, round to the nearest tenth.
12 and 9
A good rule of thumb is to save 10 percent of what you earn, and have at least three months' worth of living expenses saved up in case of an emergency.
False
True
Answers
Answer:
False
Step-by-step explanation:
you have to save at least 20 percent
Why does income for a family tend to go up when two single persons marry?
A. Newly married couples tend to have two earners.
B. Married people work longer hours than single people.
C. Two people can live as cheaply as one.
D. The Federal government's marriage tax favors married people.
Answers
Answer:
A.) Newly married couples tend to have two earners.
Step-by-step explanation:
When newly married couples first start off marriage, they tend to both work to make stable income since they recently had to work to make money for only their selves, which is why they both tend to have two earners.
HELP its a matric or matrix problem I NEED IT SOLVED AS QUICK AS POSSIBLE PLEASE
Answers
Using the given matrix, the value of -10(T) is
[tex]-10(T) = \left[\begin{array}{cc}-230&380\\-170&60\end{array}\right][/tex]
Multiplication of matricesFrom the question, we are to carry out the matrix multiplication
From the given information,
The given matrices is:
[tex]T =\left[\begin{array}{cc}23&-38\\17&-6\end{array}\right][/tex]
Now, we are to determine -10(T)
To achieve this, we will multiply each of the element in the given matrix by -10
That is,
-10 ×23 = -230
-10 × -38 = 380
-10 × 17 = -170
-10 × -6 = 60
Thus,
The result of the multiplication will be:
[tex]\left[\begin{array}{cc}-230&380\\-170&60\end{array}\right][/tex]
Hence, the value of -10(T) is
[tex]-10(T) = \left[\begin{array}{cc}-230&380\\-170&60\end{array}\right][/tex]
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If polygon A is a parallelogram, and polygon B is a rectangle, which of the
B
following is true?
X
35°
W
Polygon A
35 Z
A
90°
90%
PolygonB
90°
90
C. Polygon A and B are parallelograms so neither can be
circumscribed by a circle.
D
A. Polygon B represents the only parallelogram that a circle can
circumscribe because opposite angles are supplementary.
B. Both polygon A and B are quadrilaterals, therefore a circle can
circumscribe either one.
D. Polygon A can be circumscribed by a circle because opposite
sides must be supplementary.
Answers
Answer:
The answer is A
Step-by-step explanation:
Trust me
what are the answers to these questions?
Answers
The height and radius that minimize the amount of material needed to manufacture the can are both approximately 6.39 cm.
The total surface area of the can is therefore:
A = 2πr² + 2πrh
We know that the volume of the can is 810 cm³, which is given by:
V = πr²h
We can solve this equation for h to get:
h = V/(πr²)
Substituting this expression for height h into the equation for the surface area, we get:
A = 2πr² + 2πr(V/(πr²))
Simplifying, we get:
A = 2πr² + 2V/r
Now we have an equation for the surface area of the can in terms of the radius, r.
To minimize the surface area, we need to take the derivative of this equation with respect to r, set it equal to zero, and solve for r.
dA/dr = 4πr - 2V/r² = 0
Solving for radius r, we get:
[tex]r = (810/\pi)^1^/^3[/tex]
r=∛810/3.14
r=6.35 cm
Now find h:
h = 810/πr²
h=810/3.14×6.35²
h=810/126.6
h=6.39 cm
Hence, the height and radius that minimize the amount of material needed to manufacture the can are both approximately 6.39 cm.
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Using the discriminant
Answers
Answer:
n = 20/3
Step-by-step explanation:
Theory about the discriminant, and how solutions are acquired via the quadratic formula
One variable quadratic equations in the form [tex]0=ax^2+bx+c[/tex] have exactly one solution when the discriminant [tex](b^2-4ac)=0[/tex].
This is because for quadratic equations of the original form, the two solutions are given by the quadratic formula:
[tex]x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
which is equivalent to [tex]x=\dfrac{-b}{2a}\pm \dfrac{\sqrt{b^2-4ac}}{2a}[/tex].
The [tex]x=\dfrac{-b}{2a}[/tex] is the axis of symmetry, and the [tex]\dfrac{\sqrt{b^2-4ac}}{2a}[/tex] part is how much the solutions deviate from the axis of symmetry. If the deviation is zero, then the two solutions are both ON the axis of symmetry, and are the same number, giving exactly one solution.
Addressing the given equation
Putting it into standard form
Given the given equation...
[tex](n-4)u^2+6=8u[/tex]
subtract 8u from both sides to obtain an equation that looks like the standard form equal to zero...
[tex](n-4)u^2-8u+6=0[/tex]
The variable here is "u" instead of "x", so the solutions to this equation would be [tex]u=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex], where [tex]a=n-4[/tex], [tex]b=-8[/tex], and [tex]c=6[/tex].
Identifying the discriminant
The discriminant then, with values of a, b, and c, substituted, becomes:
[tex]Discriminant=(-8)^2-4(6)(n-4)[/tex]
If we want the discriminant to equal zero (so that there is exactly one solution for "u"), substitute zero on the left side of the equation, and solve for n.
Solving for n
[tex]0=(-8)^2-4(6)(n-4)[/tex]
[tex]0=64-4(6)(n-4)[/tex]
[tex]0=64-24(n-4)[/tex]
Add -24(n-4) to both sides...
[tex]24(n-4)=64[/tex]
Divide both sides by 24...
[tex]n-4=\dfrac{64}{24}[/tex]
Reduce the right hand side...
[tex]n-4=\dfrac{8}{3}[/tex]
Add 4 to both sides...
[tex]n=\dfrac{8}{3}+4[/tex]
Find a common denominator...
[tex]n=\dfrac{8}{3}+\dfrac{12}{3}[/tex]
Find a common denominator...
[tex]n=\dfrac{20}{3}[/tex]
So, if [tex]n=\dfrac{20}{3}[/tex], then the equation [tex](n-4)u^2+6=8u[/tex] has exactly one solution. Furthermore, this is the only value of "n" for which the equation has exactly one solution, because it is the only value of "n" for which the discriminant is zero.
Law if sines math question
Answers
Question: What are the two possible angles of a triangle with sides of 100 and 75 with a 40° as marked in the triangles above? Solution: Use the law of sines formula where the 75 length is opposite of 40°, and 100 is opposite of θ. Answer: The two possible angles for this triangle is 58.97° and 121.03°.
there are 23 students in mr wilson's classroom. mr wilson wants to determaine the probabillities of male and female students playingon a school sports team. he surveyed his students and determained
Answers
The probability of boys and girls playing on school sport team is x/23 and y/23 respectively.
What is probability?A probability is a number that reflects the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1.
This means that the addition of probability of boys and probability of girls playing in a school tem will be 1. Represent the number of boys by x and the number of girls by y.
Probability = sample space /Total outcome
Since the total number of students is 23
Therefore, the probability of boys in the team = x/23 and,
Probability of girls = y/23
therefore x/23 +y/23 = 1
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find x, y, and z
will give brainly
Answers
Answers:
[tex]z=12\sqrt{3} \\y=12\\x=12\sqrt2[/tex]
Read the paragraph and choose a sentence that describes it best. Because of the association with TERFs, many feminists have stopped identifying with radical feminism. Though some of their views may be similar to the original tenets of radical feminism, many feminists no longer associate with the term because they are trans-inclusive. TERF is not just transphobic feminism; it is a violent international movement that often compromises its feminist stances to partner with conservatives, with a goal to endanger and get rid of trans people, especially transfeminine people.
Answers
Explanation:
The paragraph explains how TERFs have alienated many feminists from radical feminism and why TERF is a harmful and anti-trans movement.
Some other possible sentences are:
The paragraph discusses the impact of TERFs on radical feminism and their transphobic and violent agenda.The paragraph analyzes why many feminists have distanced themselves from radical feminism due to TERFs and how TERFs are not true feminists but transphobes.A woman earned wages of 33,900 receive 2600 and interest from savings account and contributed 3600 to tax deferred retirement plan she was entitled to a personal exemption of 3500 and her deductions totaling 5440 find her gross income adjusted income and taxable income
Answers
The woman's gross income, adjusted income and taxable income are:
gross income - $ 36, 500adjusted income - $ 32, 900taxable income -$ 23, 960How to find the types of income ?The gross income to the woman is:
= Wages + interest
= 33, 900 + 2, 600
= $ 36, 500
The adjusted gross income :
= Gross income - Retirement plan contribution
= 36, 500 - 3, 600
= $ 32, 900
The taxable income ;
= AGI - Personal exemption - Deductions
= 32, 900 - 3, 500 - 5, 440
= $ 23, 960
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The wolf population in a certain region was monitored over several years for a project aimed at
boosting the number of wolves. The number of wolves can be modeled by the function
f(x)=41x +7 where x is the number of years since 2005. Using algebraic techniques, find the
value of x so that f(x) = 89.
Answers
Answer:
x=2
Step-by-step explanation:
f(x) is 89
f(x)=41x+7
89=41x+7
41x+7=89
41x=89-7
41x=81
x=81/41
x=2
How many kilograms are
1 kg =
100 dag
45.4 kg
119.5 kg
54.5 kg
264.3 kg
there 5,450 dekagrams?
Answers
The number of kilogram that are there in there 5,450 dekagrams is: C. 54.5 kg.
What is kilograms?The kilogram is widely used as a unit of mass in everyday life, as well as in science, engineering, and industry. One kilogram is equal to 1,000 grams, and it is the base unit for other mass-related units such as milligrams, centigrams, and decigrams.
To convert 5,450 dag to kg, we need to divide by 100, since 1 kg = 100 dag:
5,450 dag ÷ 100 = 54.5 kg
Therefore, the answer is (c) 54.5 kg.
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The correct question is:
How many kilograms are there in there 5,450 dekagrams?
1 kg = 100 dag
a. 45.4 kg
b. 119.5 kg
c. 54.5 kg
d. 264.3 kg
In Tasheena's Anthropology class Quizzes are worth 15% of the final grade, Exams are worth 55%, Projects are worth 25%, and Attendance is worth 5%.
At mid-semester Tasheena scored 117 out of 150 points on quizzes, 74, 86, and 91 on the first three exams,each worth 100 points. She got extra credit on her project with a score of 29 out of 25 possible points, and she had perfect attendance to class. Compute Tasheena's grade percentage in the class so far.
Answers
Tasheena's grade percentage in the class so far is approximately 71.00935%.
How do you find percentages?The percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100.
To compute Tasheena's grade percentage in the class so far, we can calculate the weighted average of her scores based on the weightage of each component of the final grade.
Given:
Quizzes: 15% weightage
Exams: 55% weightage
Projects: 25% weightage
Attendance: 5% weightage
Tasheena's scores:
Quizzes: 117 out of 150 points
Exams: 74, 86, and 91 on the first three exams (each worth 100 points)
Projects: 29 out of 25 possible points
Attendance: Perfect attendance
Let's calculate Tasheena's grade percentage:
Quizzes:
Tasheena's quiz score percentage = (117 / 150) * 100 = 78%
Exams:
Tasheena's exam average = (74 + 86 + 91) / 3 = 83.67
Tasheena's exam score percentage = (83.67 / 100) * 55 = 46.017%
Projects:
Tasheena's project score percentage = (29 / 25) * 100 = 116%
Attendance:
Tasheena's attendance score percentage = 100% (since she had perfect attendance)
Now, let's calculate the weighted average of Tasheena's scores:
Weighted average = (Quizzes weightage * Quiz score percentage) + (Exams weightage * Exam score percentage) + (Projects weightage * Project score percentage) + (Attendance weightage * Attendance score percentage)
= (15% * 78%) + (55% * 46.017%) + (25% * 116%) + (5% * 100%)
= 11.7% + 25.30935% + 29% + 5%
= 71.00935%
Hence, Tasheena's grade percentage in the class so far is approximately 71.00935%.
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One factor of this polynomial is (x + 8).
2) + 522 - 11x + 104
Use synthetic division to find the other factor of the polynomial.
Answers
Answer:
We can use synthetic division to divide the polynomial 2x^3 + 522x^2 - 11x + 104 by x + 8, since x + 8 is a factor of the polynomial:
-8 | 2 522 -11 104
| -16 -412 2584
|--------------------
2 506 -423 2688
The numbers in the bottom row of the synthetic division represent the coefficients of the quotient polynomial, in order of decreasing degree. So the quotient polynomial is:
2x^2 + 506x - 423
Therefore, the other factor of the polynomial is 2x^2 + 506x - 423.
what principal will have majority value of $10,000 at 8.25% p.a in three months?
Answers
__________________________________
Simple Interest: A = P × i ÷ 100 × n = 10,000 × 8.25 ÷ 100 × 3= 247.5= 10,000 + 247.5= $10,247.5 Compound Interest: A = P (1 + i ÷ 100) n = 10,000 (1 + 8.25 ÷ 100) ³= $12,684.80The Compound Interest Principal Will Have Majority Value In 3 Months__________________________________
Rafael finished the Boston Marathon ( 26.2 miles) in 4.75 hours. At this pace, how long would it take him to run a 50 mile marathon in Washington D.C?
Answers
The time taken by the runner is equal to 9.064 hours.
How to determine the time taken by a runner
In this problem we find the case of runner taking a time (t), in hours, to run a certain distance (d), in miles, at constant speed (v), in miles per hour. The kinematic equation is now introduced:
v = d / t
Now we proceed to determine the missing time:
(26.2 mi) / (4.75 hours) = (50 mi) / t
t = 9.064 h
The runner takes a time of 9.064 hours to run a distance of 50 miles.
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A department manager recently launched a new initiative so the members of her department can more easily present innovative ideas they have. To get feedback on the initiative, she emails a link to all department members for an anonymous online survey.
Which statement is true?
Answers
The statement that best describes the potential bias in this scenario is option B: The survey results are most at risk for department members not being able to answer honestly.
The reason for this is that department members may be afraid to share their true thoughts and opinions, especially if they believe their ideas may be seen as too innovative or risky. They may be concerned about how their managers or colleagues will perceive them if they express ideas that go against the status quo or challenge established norms.
This type of bias is known as social desirability bias, which refers to the tendency of respondents to answer questions in a way that is socially acceptable or desirable, rather than in a way that reflects their true beliefs or behaviors.
In this case, department members may feel pressure to conform to the expectations of their superiors or colleagues, even if it means not expressing their true opinions or ideas.
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Complete Question
A department manager recently launched a new initiative so the members of her department can more easily present innovative ideas they have. To get feedback on the initiative, she emails a link to all department members for an anonymous online survey.
Which statement is true?
A. The survey results are most at risk for nonresponse bias.
B. The survey results are most at risk for department members not being able to answer honestly.
C. The survey results are not at risk of any bias.
D. The survey results are most at risk for under coverage.
Please help me with this
Answers
An example of a function that models a linear relationship between two quantities, x and y is y = mx + b
How to explain the functionWe need to use the equation of a straight line, which is commonly expressed in slope-intercept form as:
y = mx + b
In this function, x represents the independent variable, m is the slope of the line, and b is the y-intercept. To use this function, we simply plug in the values of x, m, and b that correspond to the specific relationship we are modeling.
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Make a rap about why triangular pyramids are the best shape. No cursing. It must be appropriate and at least 1 minute long.
Answers
The rap about why triangular pyramids are the best shape is given below.
Why Triangular Pyramids are the best shape lyricsVerse 1:
Listen up, y'all, I'm here to spit some truth
About a shape that's often overlooked, but it's the truth
It's the triangular pyramid, and let me tell you why
It's the best shape out there, and that's no lie
Chorus:
Triangular pyramids, they're the way to go
Strong and sturdy, they're the best, you know
With their pointed tips and angled sides
They're the perfect shape, no need to hide
Verse 2:
When it comes to architecture, they're the top choice
Their shape is so versatile, you can't deny its voice
From ancient pyramids to modern buildings tall
The triangular pyramid can withstand it all
Chorus:
Triangular pyramids, they're the way to go
Strong and sturdy, they're the best, you know
With their pointed tips and angled sides
They're the perfect shape, no need to hide
Verse 3:
And let's not forget their mathematical prowess
Their angles add up to 180, no need to guess
With their faces all congruent, they're a sight to see
The triangular pyramid, it's the shape for me
Chorus:
Triangular pyramids, they're the way to go
Strong and sturdy, they're the best, you know
With their pointed tips and angled sides
They're the perfect shape, no need to hide
Outro:
So there you have it, folks, the triangular pyramid
It's the best shape out there, there's no need to forbid
From its strength to its geometry
The triangular pyramid, it's the shape for me!
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The slope of the line whose equation is 5x - 3y = 4 is 4/5 5/3 -3/5
Answers
The slope of the relation with the equation 5x - 3y = 4 is 5/3
Calculating the slope of the relationTo find the slope of the line whose equation is 5x - 3y = 4, we need to rearrange the equation to be in slope-intercept form, y = mx + b, where m is the slope of the line.
First, we'll subtract 5x from both sides:
-3y = -5x + 4
Next, we'll divide both sides by -3 to isolate y:
y = (5/3)x - 4/3
Now we can see that the slope of the line is 5/3.
So the correct answer is: 5/3
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PLEASE HELP ME!!!!!
Write the expression of the graph the function represents.
Answers
The expression of the function which is represented by the given graph is given by,
xy = -8
The given figure is a rectangular hyperbola.
So the equation of the given graphed curve be, xy = c, where c is a constant.
We can clearly see that the curve passes through the point with coordinates (-2, 4). So it must satisfy the equation of the curve.
Substituting the point (-2, 4) in the equation of the curve xy = c we get,
(-2)*4 = c
c = -8
Hence the expression for the graph the function represents is given by,
xy = -8.
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Name the property used to produce an equivalent expression for 2(6+y)