Answers
To find the horizontal tangent line, we have to conduct the first derivative test. The first derivative test sets the derivative of the function equal to 0.
Let's start out with finding the derivative of the function:
It seems like we will have to use Quotient Rule.
[tex]f^{\prime}(x)=\frac{(x^2-7x+1)(x+9)^{\prime}+(x^2-7x+1)^{\prime}(x+9)}{(x^2-7x+1)^2}[/tex]Now we can use basic derivative rules and properties to find the final derivative:
[tex]f^{\prime}(x)=\frac{(x^2-7x+1)+(2x-7)(x+9)}{(x^2-7x+1)^2}[/tex]Setting the derivative equal to 0, we can begin the first derivative test:
[tex]0=\frac{(x^{2}-7x+1)+(2x-7)(x+9)}{(x^{2}-7x+1)^{2}}[/tex]We can simplify the derivative before solving (expand and combine like terms):
[tex]0=\frac{-(x^2+18x-64)}{(x^2-7x+1)^2}[/tex]From here, we need to determine our values of x that will make the derivative equal to 0 or undefined. In this case, since we are not trying to find relative maxima or minima, we just need to find where the entire derivative is equal to 0. We can simply just graph the numerator and find its roots:
We can see that the values of x will be -21.0416 and 3.04159 will be the values that make the derivative equal to 0.
Also recall what the derivative is. The derivative is the slope of the tangent line. In order to find the horizontal tangent line, we know that horizontal tangent lines' slope is equal to 0.
Therefore, we can use either value of x to now find our equation.
Recall what the point-slope form of a line is:
[tex]y-y_1=m(x-x_1)[/tex]Since we know that the derivative slope must equal 0, then we know that m = 0. We now need to find a point (x, y). Substitute one of the x values into the original equation to find an actual point on the function (remember not to round intermediate values on your calculator!):
[tex]\begin{gathered} f(3.04159)=\frac{3.04159+9}{3.04159^2-7(3.04159)+1} \\ f(3.04159)=-1.09074 \end{gathered}[/tex]Now we have a point (x, y) (I will put it in the actual radical form here):
[tex]\bigg(\sqrt{145}-9,\frac{-2\sqrt{145}}{45}-\frac{5}{9}\bigg)[/tex]Substitute our variables into point-slope form:
[tex]y-\bigg(\frac{-2\sqrt{145}}{45}-\frac{5}{9}\bigg)=0\bigg(x-(\sqrt{145}-9)\bigg)[/tex]Simplify and rewrite to obtain our final answer:
[tex]y=\frac{2\sqrt{145}}{45}+\frac{5}{9}[/tex]
Related Questions
What is the range of the function f(x) = |x – 5| – 3?
R: {f(x) ∈ ℝ | f(x) > 2}
R: {f(x) ∈ ℝ | f(x) < 2}
R: {f(x) ∈ ℝ | f(x) ≥ –3}
R: {f(x) ∈ ℝ | f(x) ≤ –3}
Answers
Reason:
The smallest |x-5| can get is 0 since the result of an absolute value is never negative (because it represents distance).
So the smallest f(x) = |x-5|-3 can get is 0-3 = -3
f(x) = -3 or f(x) > -3 are the only two options for the output f(x).
Condense that into [tex]f(\text{x}) \ge -3[/tex] to arrive at choice C as the final answer.
The fancy notation of f(x) ∈ ℝ translates to "f(x) is a real number".
If you were to graph this using a tool like Desmos, then notice how the lowest point occurs when y = -3
See the diagram below.
Answer:
R: {f(x) ∈ ℝ | f(x) ≥ –3}
Step-by-step explanation:
I got it right on the test.
How to simplify:
[tex]\frac{5}{2\sqrt{5} }[/tex]
My textbook says the answer is [tex]\frac{\sqrt{5} }{2}[/tex], but I don't know to get there.
Someone pls help
Answers
hmmm first of all, let's keep in mind that any expression multiplied by 1 is always the same expression
[tex]12\cdot 1=12\qquad 13567\cdot 1=13567\qquad spaghetti\cdot 1=spaghetti[/tex]
and so on.
now, hmm 1 can be expressed as a rational, 1/1 = 1, but but but, we can really express it as
[tex]\cfrac{9999}{9999}\implies \cfrac{7857}{7357}\implies \cfrac{\sqrt[11]{1375}}{\sqrt[11]{1375}}\implies \cfrac{spaghetti}{spaghetti}\implies \cfrac{lettuce}{lettuce}\implies \text{\Huge 1}[/tex]
so let's do the so-called rationalizing of the denominator, we can also do it with the numerator btw, but we only need it for the denominator right now, by multiplying for 1 :), 1 is Da Bomb baby!!
[tex]\cfrac{5}{2\sqrt{5}}\cdot \text{\LARGE 1}\implies \cfrac{5}{2\sqrt{5}}\cdot \cfrac{\sqrt{5}}{\sqrt{5}}\implies \cfrac{5\sqrt{5}}{2\sqrt{5^2}}\implies \cfrac{5\sqrt{5}}{2(5)}\implies \cfrac{5\sqrt{5}}{10}\implies {\Large \begin{array}{llll} \cfrac{\sqrt{5}}{2} \end{array}}[/tex]
An agency charges $100 per person for a trip to a concert if 50 people travel in a group. But for each person above the 50, the amount charged for each traveler will be reduced by $3. If x represents the number of people above the 50, write the agency's revenue R as a function of x. (Assume that x is greater than zero.)
R(x)
Answers
The revenue of the the agency is given by the function
R(x) = 5000 - 50 x - 3x² .
A function is a relationship between an independent variable and a dependent variable which defines all points on a cartesian plane which satisfies the given condition.
The set of the independent variables is called a domain and the set of the dependent variable of the function is called a range.Each data in range is related to a particular data in the domain by the function.Now in the given Agency it is given that the charge of each tourist is $100 . Now for every tourist above 50 the charge is reduced by 3 dollars.
So for x people above 50 charge will be ( 100-3 x ) for all members.
The total revenue can be measured by the function
R(x) = ( 50 + x ) ( 100 - 3x )
or , R(x) = 5000 - 150 x + 100 x - 3x²
or, R(x) = 5000 - 50 x - 3x²
Hence the revenue function is given as : R(x) = 5000 - 50 x - 3x²
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help video
Brooklyn is trying to pick out an outfit for the first day of school. She can choose from
6 pairs of pants, 8 t-shirts, 3 sweaters or hoodies, and 5 pairs of shoes. How many
different outfits does Brooklyn have to choose from?
Answer:
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attempt 1 out of 2
Answers
Answer:
There are 16 ways in which Danny can mix and match all his outfits. You can determine this by multiplying. 4 x 2 x 2. Danny has 4 shirts. He can wear each with two different pants, so 4 x 2 giving 8 different shirt/pant combinations. Now each combination can be worn with two different pairs of shoes, so 8 x 2 giving 16 total combinations of shirt, pants and shoes.
Step-by-step explanation:
How many outfits does that make? First he has to decide on a pair of pants and he has 4 different choices. For each of those choices he has a choice of 7 ...
you would times each one for example 6 x8 6x3 6x5 and 8x6 8x3 8x5 and 3 x6 and so forth
Which fraction is in simplest form? A. 11 15 B. 12 18 C. 9 21 D. 14 24
Answers
Answer : 11 15
Reasoning : The fraction is in simplest form
9 21 is able to be simplified to 3 7,
12 18 is able to be simplified to 2 3
14 24 is able to be simplified to 7 12
Write the sum. 54+63 as the product of a number and relatively prime addends
Answers
54 + 63 = 13 * 9 is the expression obtained as the product of a number and relatively prime addends.
A prime number is a number that can be divided only by 1 and itself.
54+63 (given)
We now have to find the sum of the given numbers as the product of relatively prime numbers.
In order to find the numbers we will find first sum and then factor
Sum=54+63=117
Factors of 117 are 13 and 9
117 = 13 * 9
13 and 9 are relatively prime because the factor of 13 and 9 is 1.
Therefore, 54 + 63 = 13 * 9 is the expression obtained as the product of a number and relatively prime addends.
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a=3 and b=6 and c=5 what is ab+bc?
Answers
Answer:
you have the values of a, b and c
all you gotta do is just plug them in so
ab+bc is (3)(6)+(6)(5)
which equals 48.
Answer:
48
Step-by-step explanation:
since a=3, b=6 and c=5, you would replace the letters in ab+bc with their numbers, making it (3)(6)+(6)(5). then you would multiply 3 x 6 which would be 18, and 6 x 5, which would be 30, making your expression 18+30. finally you would add them together, making it 48.
Please help I’ll mark you as brainliest if correct!!
Answers
Answer:
14, 9, 16
Step-by-step explanation:
from 3 to 10, you add 7
from 10 to 5, you minus 5
from 5 to 12, you add 7
from 12 to 7, you minus 5
so the pattern is add 7, minus 5
7 plus 7 is 14
14 minus 5 is 9
9 plus 7 is 16
You have a rectangular prism whose cross section is a square with side length
3 cm. You need to know its length, but you do not have a ruler. Luckily, you have a calibrated container of water nearby. The container has marks that measure cubic centimeters of liquid. It appears to have 70 cubic centimeters of water in it, but when you submerge the prism into the water, the water level rises to the 115-cubic-centimeter mark. Determine how long your prism is.
Answers
Answer:
side length = 5cm
Step-by-step explanation:
The volume of the prism is:
V = l × w × h
But we only know the width and height which are both 3cm. See image.
We put it in the calibrated container (container with markings on it) We find out the volume is 45. Because 115 - 70 is 45.
Divide to find the missing length. See image.
Vol
= 3 × 3 × sidelength
V = 9s
45 = 9s
5 = s
see image.
The graph of y = f (x) is given. Graph the indicated function.Graph y = f (-x) - 4
Answers
Note the - sign in the evaluation of f(x), which changes the function directly, and the term -4 indicates a displacement vertically (down), but the graph goes up because of the sign -, therefore, the only option with these conditions is option b.
The correct answer is option b.
how many 5 digit numbers can be created using the digits 2,3,4,7,8 and 9 without repeating any digits in that five digit number
Answers
Answer:
there are 720 three digit numbers that can be created from the digits provided given that no digit may be repeated.
Step-by-step explanation:
ANSWER FAST: 4 thumb drives and 1 compact disk have a total capacity of 18 gigabytes. 3 compact disks and 4 thumb drives have a total capacity of 22 gigabytes. find the capacity 1 thumb drive and the capacity of 1 compact disk.
Answers
Capacity of 1 thumb drives = 4 gigabytes.
Capacity of 1 compact disk = 2 gigabytes.
What is linear expression?
A linear expression is an algebraic statement where each term is either a constant or a variable raised to the first power.
Let capacity of a thumb dives = x
And, Capacity of a compact disk = y
Then, For first condition;
''4 thumb drives and 1 compact disk have a total capacity of 18 gigabytes.''
We can formulate;
4x + y = 18 .... (i)
And, For second condition;
'3 compact disks and 4 thumb drives have a total capacity of 22 gigabytes.'
We can formulate;
4x + 3y = 22 .... (ii)
Solve the equation (i) and (ii) we get;
Subtract (ii) by equation (i) we get;
2y = 4
y = 2
And, 4x + 2 = 18
4x = 16
x = 4
So, Capacity of 1 thumb drives = 4 gigabytes.
Capacity of 1 compact disk = 2 gigabytes.
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Which pair of angels are alternate exterior angles
○ <1and <2
○ <1and<7
○ <2and<7
○ <2and<5
Answers
∠2 and ∠7 are a pair of angels are alternate exterior angles .
A transversal that crosses two parallel lines results in alternate exterior angles.
They form two pairs of alternate outside angles since they are situated "outside" the two parallel lines but on different sides of the transversal (four total angles).When a third line, a transversal, separates two lines, the expression "alternating exterior angles" is frequently used. The alternate exterior angles theorem states that pairs of alternate exterior angles are congruent if k and l are parallel. A: One of two internal angles inside the two intersecting lines, also known as an alternate internal angle. B: One of two angles with distinct vertices that are on opposite sides of a transversal at the intersection of two other lines.So from the given figure the pairs of external alternate angles are:
∠1 and ∠8
∠2 and ∠7
Hence of the given options , ∠2 and ∠7 is the required external alternate angles .
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Adam is baking 4 pizzas. Each pizza is a rectangle. It takes Adam 35 minutes to bake the pizzas. He divides each pizza into the equal-size square slices shown.
Answers
Answer:
Step-by-step explanation:
Incomplete question.
the following table lists the ages and number of players per age group that are on a volleyball team. let A = (volleyball players who are younger than 8 years old) how many players are in the complement of A?
Answers
The number of elements in the complement of a set A is the total number of elements in the universal set minus the number of elements in the set A.
It is given that A={volleyball players who are younger than 8 years old}.
From the table, volleyball players who are younger than the age of 8, are players who are 7 years old.
Hence, the number of players in A is 5.
Add the number of players for each age to get the total number of players:
[tex]5+3+7+4=19\text{ players}[/tex]It follows that the number of players in the complement of A is:
[tex]19-5=14\text{ players}[/tex]Hence, the number of players in the comp
when sarah planted a tree in her yard it was 24 inches tall it grew at a rate of 2.5 inches per month the tree is now 41.5 inches tall how many months has it been since sarah planted the tree
Answers
In linear equations, 103.75 inches per month tall has it been since sarah planted the tree.
What are instances of linear equations?
Ax+By=C is the usual form for two-variable linear equations. A standard form linear equation is, for instance, 2x+3y=5.When an equation is given in this format, finding both intercepts is rather simple (x and y).When attempting to solve systems involving two linear equations, this form is also quite helpful.Sarah planted a tree in her yard = 24 inches tall
grew at a rate =2.5 inches per month
When tree is now 41.5 inches tall = 41 . 5 * 2.5
= 103.75
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5 is to 7 as 15 is to x.
Answers
Answer:
x = 21
Step-by-step explanation:
To get your answer, you can turn these to fractions: 5/7 = 15/x
5 * 3 = 15
7 * 3 = x
x = 21
Find the x intercepts of the graph of the function.
g(x)=1/4(x-6)(x-3). Hint x intercepts happen when g(x)=0.
Answers
The x intercepts of the function given in the question will be equal to x = 6 and x = 3.
A function may be defined as an entity in which for one value of input there is only one value of output. The input value is the value of x variable, or the independent variable and the output value is the value of y variable or the dependent variable. For example, f(x) = y = x + 2, in this function for the value of input x = 1 the value of output is y = 3. The x intercept of a function may be defined as the point on the graph at which the line of function crosses the x axis. For finding the x intercept the y value or the value becomes zero. We are given the function g(x) = (1/4) (x - 6) (x - 3)
Now, g(x) = 0
(1/4) (x - 6) (x - 3) = 0
=> (x - 6) (x - 3) = 0
=> x = 6 and x = 3
Therefore, x intercepts are x = 6 and x = 3.
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Write an equation for the graph in slope-intercept form. Do not use spaces in the equation
Answers
Answer:
I think this is correct.
Step-by-step explanation:
y=-3x+10
5. Which of the following is a rational number that can be written as a decimal in which one or more nonzero digits repeat?
Answers
The rational number that can be written as a decimal in which one or more nonzero digits repeat is given by:
D. 5/9.
Which numbers compose the set of rational numbers?The set of rational numbers is composed by numbers that can be represented by fractions, such as numbers that have no decimal parts, or numbers in which the decimal parts are terminating or repeating.
For this problem, we have a focus on numbers that have repeating decimal part. The most known case is the division by 9, as when an integer that is not a multiple of 9 is divided by 9, the result is a decimal with a repeating decimal part.
To find the equivalent decimal to 5/9, we divide the numerator of 5 by the denominator of 9, hence:
5/9 = 0.55555...
The decimal digit 5 repeats infinitely, hence option D is correct.
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Heather, Ahmad, and Jim have a total of $68 in their wallets. Heather has $8 less than Jim. Ahmad has 2 times what Jim has. How much does each have?
Answers
Answer:
Step-by-step explanation:
1. Set x equal to the amount of money Jim has
Jim's money = x2. Create equations of the amount of money the other people have based on x
Heather's money = x - 8Ahmad's money = 2x3. Now, we set all of these equations as equal to 68
[tex](x)+(x-8)+(2x)=68[/tex]4. Combine the like terms, which means that the previous equation will simplify to:
[tex]4x-8=68[/tex]5. Add 8 to both sides
[tex]8+4x-8=68+8[/tex]This simplifies to: [tex]4x=76[/tex]6. Divide 76 by 4
[tex]\frac{4x}{4} =\frac{76}{4}[/tex]This equals: x = 197. Plug the value of x into the previous equations:
Jim's money: $19Heather's money: [tex]19-8=11[/tex] -> $11Ahmad's money: [tex]2*19=36[/tex] -> $36Final Answer:
Jim has $19Heather has $11Ahmad has $36Write a problem that could be solved by finding 5/8 divided by 2/5.
Answers
Answer:107/56 is the answer
19. Write an ordered pair for the point in Quadrant II, that is 8 units away from the point Q
(0.75 -3.25) on a coordinate plane.
Answers
An ordered pair of point in Quadrant II is 8 units away from this point, the resulting point will be (-7.25, 4.25)
In this question, we need to find an ordered pair for the point in Quadrant II, that is 8 units away from the point Q(0.75 -3.25) on a coordinate plane.
If we have a coordinate point (x, y) in Quadrant ||, if the coordinate is 8 units away, the resulting coordinate will be expressed as (x - 8, y + 8)
Given the point Q(0.75, -3.25) on the coordinate the plane, if a point in Quadrant II is 8 units away from this point, the resulting point will be (0.75 - 8, -3.25 + 8) which is (-7.25, 4.25)
Therefore, an ordered pair a point in Quadrant II is 8 units away from this point, the resulting point will be (-7.25, 4.25)
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I dont understand any of these functions please help on all!
Answers
1.The slope -intercept form of the function is y = 7x
2.The slope -intercept form of the function is y = x - 8
3.The slope -intercept form of the function is y = -2x +9
4.The slope -intercept form of the function is [tex]y = \frac{1}{3}x+9[/tex]
5.The slope -intercept form of the function is y = 0.5x - 1.5
6.The slope -intercept form of the function is y = 3x - 12
How is the slope -intercept form calculated?
1. The slope -intercept form of the equation of the function is y = 7x
Slope = m
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\\\m =\frac{0+7}{0+1} \\\\m= 7[/tex]
Substituting point (-1,-7) and m in the slope-intercept form of the equation,
y = mx+ c ----(1)
-7 = 7(-1) + c
c = 7 -7
c = 0
Substituting m and c in (1)
y = 7x
2. The slope -intercept form of the equation of the function is y = x - 8
Slope = m
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\\\m =\frac{-8+17}{0+9} \\\\m= \frac{9}{9} =1[/tex]
Substituting point (-9,-17) and m in the slope-intercept form of the equation,
y = mx+ c ----(1)
-17 = (-9) + c
c = -17 +9
c = -8
Substituting m and c in (1)
y = x - 8
3. The slope -intercept form of the equation of the function is y = -2x +9
Slope = m
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\\\m =\frac{5-9}{2-0} \\\\m= \frac{-4}{2} =-2[/tex]
Substituting point (0,9) and m in the slope-intercept form of the equation,
y = mx+ c ----(1)
9 = -2 (0) + c
c = 9
Substituting m and c in (1)
y = -2x +9
4.The slope -intercept form of the equation of the function is [tex]y = \frac{1}{3}x+9[/tex]
Slope = m
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\\\m =\frac{8-7}{-3+6} \\\\m= \frac{1}{3}[/tex]
Substituting point (0,9) and m in the slope-intercept form of the equation,
y = mx+ c ----(1)
9 = 1/3 (0) + c
c = 9
Substituting m and c in (1)
[tex]y = \frac{1}{3}x+9[/tex]
5. The slope -intercept form of the equation of the function is
y = 0.5x - 1.5
Slope = m
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\\\m =\frac{1-0.5}{5-4} \\\\m= \frac{0.5}{1}= 0.5[/tex]
Substituting point (5,1) and m in the slope-intercept form of the equation,
y = mx+ c ----(1)
1 =0.5 (5) + c
1 -2.5 = c
c = - 1.5
Substituting m and c in (1)
y = 0.5x - 1.5
6. The slope -intercept form of the equation of the function is y = 3x - 12
Slope = m
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\\\m =\frac{9-3}{7-5} \\\\m= \frac{6}{2}\\\\m= 3[/tex]
Substituting point (5,3) and m in the slope-intercept form of the equation,
y = mx+ c ----(1)
3 =3(5) + c
3 -15 = c
c = - 12
Substituting m and c in (1)
y = 3x - 12
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Chelsy wants to buy a new pair of jeans in one store the jeans are selling for 1/5 off the original price of $70 in second store the same jeans are selling for 1/3 off the original price $81 chelsy wants to buy the cheaper pair of jeans in which store should chelsy buy the jeans and how much will they cost
Answers
iN linear equations, store 1 jeans is cheaper than store 2.
What are instances of linear equations?
Ax+By=C is the usual form for two-variable linear equations. A standard form linear equation is, for instance, 2x+3y=5.When an equation is given in this format, finding both intercepts is rather simple (x and y).When attempting to solve systems involving two linear equations, this form is also quite helpful.one store the jeans are selling for 1/5 off the original price of $70
$70 * 1/5 = 14
second store the same jeans are selling for 1/3 off the original price $81
$81 * 1/3 = 27
So, store 1 jeans is cheaper than store 2.
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How to find the slope of a graph? 9 and 10
Answers
9. To draw the rise, you can follow these steps:
- Identify a point on the line. In this case, this can be:
[tex](-3,1)[/tex]- Draw a vertical line that rises from that point. See the picture below:
10. To draw the line that represents the run, you need to Draw a horizontal line that goes from the endpoint of the line that represents the rise and extend it until it touches the function. See the picture below:
Therefore, the answers are (See the picture):
Describe how the change affects the volume of the prism or pyramid. Multiplying the height by 3/2.
Answers
Answer:
1512 cm²
Volume is also multiplied by [tex]\dfrac{3}{2}[/tex]
Step-by-step explanation:
The volume of a pyramid is given by the expression
[tex]V = \dfrac{1}{3} l\cdot w\cdot h[/tex]
where l = base length, w = base width and h = height
The volume of this pyramid
[tex]= \dfrac{18\cdot 12\cdot 14}{3} = 1008\; cm^3[/tex]
If we multiply the height by [tex]\dfrac{3}{2}[/tex] then new height
= [tex]14 \cdot \dfrac{3}{2} = 21 cm[/tex]
New Volume [tex]= \dfrac{18\cdot 12\cdot 21}{3} = 1512\; cm^3[/tex]
[tex]\dfrac{\text{NewVolume}}{\text{Old Volume}} = \dfrac{1512}{1008} = \dfrac{3}{2}[/tex]
So volume is also multiplied by 3/2
I’m struggling finding the frequency, can anyone help me?
Answers
Frequency is how often something appears. Count the number of “leaves” you have for each “stem” in your stem-and-leaf plot.
Help, write an equation in slope intercept form for slope -3 and y intercept -2
Answers
that’s how you write it i think
Hi :)
[tex]\star\sim\star\sim\star\sim\star\sim\star\sim[/tex]
Q-u-e-s-t-i-o-n:what is an equation with slope -3 and y int. of -2A-n-s-w-e-r:[tex]\sf y=-3x-2[/tex]
- we just insert the number for slope before x in the y=mx+b formula and then write the y-intercept
[tex]\star\sim\star\sim\star\sim\star\sim\star\sim[/tex]
Which decimal is equivalent to39?
Answers
Answer: is 7
Step-by-step explanation: Its 7 cuz 14 weeks