P(k + 1) follows, and it is likewise true.
As a result, after demonstrating that P(k) is true for any chosen value of k, we also demonstrated that P(k + 1) is true. According to the induction hypothesis, P(n) is true for all n ∈ Z+
What is mathematical induction ?To demonstrate that the mathematical proposition P(n) holds for all natural integers n = 1, 2, 3, 4,..., one uses the mathematical induction technique. It is frequently referred to as the induction principle in mathematics. We establish that P(1) holds in order to demonstrate a result P(n) using the mathematical induction technique. If P(1) is valid, we can assume that P(k) holds for some natural number k and use this assumption to demonstrate that P(k+1) is valid. The assertion P(n) applies to all natural numbers if and only if P(k+1) holds true.
Let
P(n): 1×2 + 2×3+3×4+........+n(n+1)=[tex]\frac{n(n+1)(n+2)}{3}[/tex]
Putting n =1 we get
P(1):1×2 =2
and
[tex]\frac{1(1+1)(1+2)}{3}=\frac{1(2)(3)}{3} =2[/tex]
Therefore P(1) is true.
Let P(K) is true
So,
P(k): 1×2 + 2×3+3×4+........+k(k+1)=[tex]\frac{k(k+1)(k+2)}{3}[/tex]
Now we have to show that P(k +1 )is also true.
P(k+1): 1×2 + 2×3+3×4+........+k(k+1)+(k+1)(k+2)=[tex]\frac{k(k+1)(k+2)}{3}+ (k+1)(k+2)[/tex]
= (k+1)(k+2)[[tex]\frac{k}{3}+1[/tex]]
=[tex]\frac{(k+1)(k+2)(k+3)}{3}[/tex]
P(k + 1) follows, and it is likewise true.
As a result, after demonstrating that P(k) is true for any chosen value of k, we also demonstrated that P(k + 1) is true. According to the induction hypothesis, P(n) is true for all n ∈ Z+.
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I’m really confused on this.. help!!
Answers:
x = 108
y = 111/5 = 22.2
========================================================
Explanation:
The angles 72 and x form a 180 degree straight line.
72+x = 180
x = 180-72
x = 108
Angles x and 5y-3 are congruent corresponding angles assuming the lines pointing to the northwest are parallel
5y - 3 = x
5y - 3 = 108
5y = 108+3
5y = 111
y = 111/5
y = 22.2
Answer:
x = 108°
y = 22.2°
Step-by-step explanation:
Angles on a straight line sum to 180°.
[tex]\implies 72^{\circ} + x = 180^{\circ}[/tex]
[tex]\implies 72^{\circ} + x - 72^{\circ} = 180^{\circ} - 72^{\circ}[/tex]
[tex]\implies x = 108^{\circ}[/tex]
Corresponding Angles Theorem
When a straight line intersects two parallel straight lines, the angles in the same relative position are congruent.
Assuming the two lines that appear parallel in the given diagram are parallel:
[tex]\implies (5y - 3)^{\circ} = x^{\circ}[/tex]
[tex]\implies 5y - 3 = 108[/tex]
[tex]\implies 5y - 3 + 3 = 108 + 3[/tex]
[tex]\implies 5y = 111[/tex]
[tex]\implies 5y \div 5 = 111 \div 5[/tex]
[tex]\implies y = 22.2^{\circ}[/tex]
Bridget’s rectangular living room was 12ft 8in by 14ft 6in How many of carpet will she need if she was advised to expect 10% waste and she cannot buy part of a square yard?
The first step we need to do is to convert the measures from inches to feet (12 inches is 1 foot):
8 inches = 8/12 feet = 0.667 feet
6 inches = 6/12 feet = 0.5 feet
So the measures are 12.667 ft and 14.5 ft.
Now, to find the area of the rectangular living room, we just need to multiply these measures:
[tex]\text{Area}=12.667\cdot14.5=183.67in^2[/tex]Since we expect a waste of 10%, we need to increase this area by 10%, that is, we multiply it by 110% or 1.1:
[tex]NewArea=Area\cdot1.1=183.67\cdot1.1=202.03in^2[/tex]Since she cannot buy part of a square yard, we need to round up our answer (if we round down, the amount of carpet will not be enough). So rouding up, she will have to buy 203 in2 of carpet.
2. Suppose Georgia wanted to double the
recipe; what would the measurements
be for each ingredient?
Answer:
You can follow these steps to multiply a fraction by a whole number:
Write the whole number as a fraction with a denominator of 1.
Multiply the numerators.
Multiply the denominators.
Simplify. , if needed. If your answer is greater than 1, you may want to write your answer as a mixed number.
4. Suppose one U.S. dollar is equal to 5.8 Egyptian pounds. About
how many Egyptian pounds would you receive for $48.50?
Round to the greatest place value to make it easier to
compute mentally.
5.8
48.50=
X
Multiply.
So, $48.50 is equal to about 9520 Egyptian pounds.
48.50 U.S. dollars = 281 Egyptian pounds
What is a dollar?
The United States dollar (symbol: $; code: USD; often abbreviated US$ or U.S. Dollar to differentiate it from other dollar-denominated currencies; known colloquially as the dollar, U.S. dollar, American dollar, or buck) is the official currency of the United States and numerous other nations. The Coinage Act of 1792 established the United States dollar on equal footing with the Spanish silver dollar, split it into 100 cents, and permitted the minting of coins denominated in dollars and cents. Federal Reserve Notes, generally known as greenbacks owing to their predominately green hue, are the currency of the United States.
Given, 1 U.S. dollar = 5.8 Egyptian pounds
so, 48.50 U.S. dollars = 5.8 x 48.50 = 281.30 Egyptian pounds
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Out of 350 applicants for a job, 163 are male and 56 are male and have a graduate degree. Step 1 of 2 : What is the probability that a randomly chosen applicant has a graduate degree, given that they are male? Express your answer as a fraction or a decimal rounded to four decimal places.
Probability of a randomly chosen male applicant that has a graduate degree is 28/175.
Total number of applicants for a job= 350
Number of male applicants having a graduate degree = 56
We have to calculate the probability of a randomly chosen applicant that has a graduate degree, given that they are male.
We know ,
Probability = ( Total Number of favorable outcomes) / (Total Outcomes)
probability of a randomly chosen applicant that has a graduate degree, given that they are male = (Number of male applicants having a graduate degree)/ (Total number of applicants for a job) = 56/350 = 28/175
So, the probability of a randomly chosen applicant that has a graduate degree, that has a graduate degree, given that they are male is 28/175.
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Suppose you want to model the
difference-4-7. Do you need to add zero pairs? If so, why? How
many should you add? What is the difference?
There are 11 red counters left so the difference is −11.
What is subtraction?To subtract in mathematics is to take something away from a group or a number of objects. The group's total number of items decreases or becomes lower when we subtract from it. The components of a subtraction issue are the minuend, subtrahend, and difference. An arithmetic operation called subtraction simulates the process of deleting items from a collection. The action of subtracting a matrix, vector, or other quantity from another according to predetermined rules in order to find the difference.
Given that,
Add some zero pairs because need to remove 7 yellow counters}$.
Since you need to remove 7 yellow counters, add 7 zero pairs.
When you remove yellow counters, there are 11 red counters left so the difference is:
−4−7=−11
There are 4 red counters so you should add 7 zero pairs in order to eliminate 7 yellow counters. There are 11 red counters left so the difference is −11.
Therefore, there are 11 red counters left so the difference is −11.
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How many halves are there in:
1. 51/2
Answer:
0.25.2
Step-by-step explanation:
1.51/2
So,halves
0.25.2
which is not a correct description of the graph below
ANSWER
The graph of y = sin θ shifted to the left by π/2 units
EXPLANATION
This is the graph of y = sin θ shifted to the left π units. In function notation, this shift is represented by adding π to the variable: y = sin (θ + π).
Also, we know that the graph of y = cos θ is like the graph of y = sin θ but shifted π/2 units to the left. So, if we shift the graph of y = cos θ π/2 units to the left, we get the given graph.
Therefore, the only description that does not match the graph is the graph of y = sin θ shifted to the left by π/2 units.
PQ has a length of 17 units with P(-4,7). If the x- and y-coordinates of Q are both greater than the x- and y-coordinates of P, what are possible integer value coordinates of Q? Explain.
Let PQ be the hypotenuse of a right triangle that also has a horizontal leg and a vertical leg. The hypotenuse then has length 17, and a Pythagorean triple can be used to say the shorter leg has length
and the longer leg has length. If the shorter leg is horizontal, then Q is described by the ordered pair. If the shorter leg is vertical, then Q is described by the ordered pair
Step-by-step explanation:
a distance on a coordinate grid is always the Hypotenuse of a right-angled triangle with the differences of the x coordinates and the y coordinates as the legs.
so, per Pythagoras
distance² = (x diff)² + (y diff)²
we have here a distance 17. 17² = 289.
to get integer coordinates the lengths of the legs must be integer numbers.
what 2 squared integer numbers can be added to get 289 ?
I found after subtracting 4, 9, 16, 25, 36, 49 and finally 64 (the squares of 2, 3, 4, 5, 6, 7, 8) that
289 - 64 = 225, which is also a square number (15²).
so, we get the possible integer coordinates of Q to be
(-4 + 8, 7 + 15) = (4, 22)
in this case the shorter leg has the length 8, and the longer leg the length 15.
the shorter leg is in this case horizontal, because it is the x coordinate difference. and Q is again as described (4, 22).
if we make the shorter leg vertical, that means it is the y coordinate difference. and Q is then
(-4 + 15, 7 + 8) = (11, 15)
Many doctors rely on the use of intravenous medication administration in order to achieve an immediate response of a particular drug's effects. The concentration, C, in mg/L, of a particular medication after being injected into a patient can be given by the function C of t is equal to the quantity negative 8 times t squared plus 56 times t end quantity over the quantity t squared plus 3 times t plus 2 end quantity comma where the time, t, is hours after injection.
Part A: What is the domain of the function C(t) based on the context of the problem? Show all necessary calculations. (5 points)
Part B: Graph the function to determine the greatest concentration of the medication that a patient will have in their body. (5 points)
(a). The domain of the function C(t) based on the context of the problem is t≥0
(b). The greatest concentration of the medication that a patient will have in their body is 7mg/L
Graphs are used to show the relationship between variables, where the variables are represented by a pair of axes.
(a). C(t) = (-8t² +50t)/(t² + 3t +2 )
The domain of the function :
Because the medication concentration C(t) is a function of time (t), the domain is the possible values t can take.
t, cannot take negative values (i.e. it is not possible to have a negative time).
The least possible value of t is 0
So, the domain of the function based on the context is: t≥0
(b).From the graph, the maximum y-value is 7.
Hence, the greatest concentration of the medication is 7mg/L
(a). The domain of the function C(t) based on the context of the problem is t≥0
(b). The greatest concentration of the medication that a patient will have in their body is 7mg/L
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3/11 x 4/9 answered in a fraction form
Answer:
4/33
Hopes this helps!
Have a good day :)
Convention of 1818 helped lessen tensions even further between the United States and Great Britain. They agreed to make __________the boundary between Canada and the United States.
Answer:
The Convention of 1818 was a treaty between the United States and Britain that set the 49th parallel as the boundary between British North America and the US across the West.Forty-Ninth ParallelCutting on the 49th parallel, on the right bank of the Moyie River, looking west, 1860.(courtesy North American Boundary Commission)The Convention of 1818 was a treaty between the United States and Britain that set the 49th parallel of latitude as the boundary between British North America and the US across the West. This remains the boundary today between the two nations.
hope that helps! Have a good day
If UW = 4, what is the length of the radius of circle V?
the radius of of circle V is 2
Explanation:
XZ is the diameter of circle Y (bigger circle)
UW is the diameter of circle V (smaller circle)
We are to find the radius of the smaller circle. To find the radius, we use the formula relating diamter to radius:
Diameter = 2(radius)
UW = 4, diameter =4
4 = 2(radius)
radius = 4/2
radius = 2
Hence, the radius of of circle V is 2
If f(x) = x^5 + 4x - 5, then what is the remainder when f(x) is divided by
x - 3?
If f(x) = x^5 + 4x - 5, is divided by x - 3 the remainder is 220.
What is a remainder ?
In mathematics, the term "remnant" refers to the amount that is "left over" after performing a calculation. In mathematics, the integer that is left over after dividing two integers to produce an integer quotient is known as the residual (integer division). In algebra of polynomials, the "remainder" is the polynomial that is left over after dividing one polynomial by another. The operation that yields such a remainder when a dividend and a divisor are both present is the modulo operation.
How are functions divided?
Simply multiply or divide the numbers at each point where it makes sense to multiply or divide a function. If the functions are specified by formulas, you can simply multiply or divide the formulas (inserting numbers before or after is irrelevant).
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Complete the square to re-write the quadratic function in vertex form:
y=x^2+9x+7
The Vertex form of y= (x+9/2)² - 89/4 hence, vertex=(-9/2 , -89/4),
Given,y=x²+9x-2
Take out the leading coefficient.
y=(x²+9×x)-2
Complete the square.
y = (x² +(2×9/2) × x × (9/2)²) + (-2-(9/2)²)
Express the function in vertex form.
y=(x+9/2)² - 89/4
Hence the vertex form is y=(x+9/2)² - 89/4
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Work out the value of a in the equation below.
x²-18x +81 = (x − a)²
Answer:
x = a/2 + 9/2
Step-by-step explanation:
Solve for x:
x^2 - 18 x + 81 = (x - a)^2
Expand out terms of the right hand side:
x^2 - 18 x + 81 = x^2 - 2 a x + a^2
Subtract x^2 - 2 a x + a^2 from both sides:
2 a x - 18 x - a^2 + 81 = 0
Collect in terms of x:
81 - a^2 + x (2 a - 18) = 0
Factor constant terms from the left hand side:
-(a - 9) (-2 x + a + 9) = 0
Divide both sides by -a + 9:
-2 x + a + 9 = 0
Subtract a + 9 from both sides:
-2 x = -a - 9
Divide both sides by -2:
Answer: x = a/2 + 9/2
9. Extend Your Thinking Georgette bought some craft items. The silk flowers cost three times as much as the ribbon. The ribbon cost two times what the foam cost. The vase cost $12, which was three times as much as the foam. How much did the silk flowers cost?
Given
s: silk flowers
r: ribbon
f: foam
v: vase
Procedure
s = 3r
r = 2f
v = 3f
[tex]\begin{gathered} v=12 \\ 3f=12 \\ f=\frac{12}{3} \\ f=4 \end{gathered}[/tex][tex]\begin{gathered} r=2\cdot f \\ r=2\cdot4 \\ r=8 \end{gathered}[/tex][tex]\begin{gathered} s=3\cdot r \\ s=3\cdot8 \\ s=24 \end{gathered}[/tex]s: silk flowers $24
r: ribbon $ 8
f: foam $ 4
v: vase $12
An elliptical-shaped path surrounds a garden, modeled by quantity x minus 22 end quantity squared over 225 plus quantity y minus 26 end quantity squared over 324 equals 1 comma where all measurements are in feet. What is the maximum distance between any two persons on the path, and what key feature does this represent?
The ellipse has 2 axis, the minor and the major axis,
The major axis is denoted in blue color and the minor axis in purple color. By comparing our equation with the general equation of the ellipse:
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]where the lenght of the major axis is given by
[tex]\text{lenght major axis=2b}[/tex]Fron the given equation, we can note that
[tex]b=\sqrt[]{324}=18[/tex]Therefore, the lenght of the major axis is 36 feet. Therefore, the maximum distance between any two persons is 36 feet corresponding to the major axis, which is option 3 from top to bottom
The midpoint of AB is M(1, 4). If the coordinates of A are (8, 7), what are the
coordinates of B?
[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ A(\stackrel{x_1}{8}~,~\stackrel{y_1}{7})\qquad B(\stackrel{x_2}{x}~,~\stackrel{y_2}{y}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ x +8}{2}~~~ ,~~~ \cfrac{ y +7}{2} \right) ~~ = ~~\stackrel{\textit{\LARGE M}}{(1~~,~~4)} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ x +8}{2}=1\implies x+8=2\implies \boxed{x=-6} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ y +7}{2}=4\implies y+7=8\implies \boxed{y=1}[/tex]
There are 3 consecutive integers that add up to 33. What are the integers?
Answer:
-33, -12, -11, -10Step-by-step explanation:
The three consecutive integers whose sum is -33 are -12, -11 and -10
Brainlist maybe :)?
What is the value of x in the equation below?1/10 (x+30)=-2-15+3x
SOLUTION:
Step 1:
In the question, we are considering the equation:
[tex]\frac{1}{10}(x+30)\text{ =-2 -15 + 3x}[/tex]Step 2:
The details of the solution are as follows:
OCONCLUSION:
The final answer is:
[tex]x\text{ = }\frac{200}{29}[/tex]Expressing the final answer in decimal, we have that:
[tex]x\text{ }\approx\text{ 6 . 90 ( 2 decimal places)}[/tex]PLS HELP
Tavon has a gift card for that loses for each 30-day period it is not used. He has another gift card for that loses for each 30-day period it is not used.
a. Write and solve an equation for the number of 30-day periods until the value of the gift cards will be equal.
b. What will the value of each card be when they have equal value?
Answer:
B
Step-by-step explanation:
x² - 6x - 41= 0 by completing squares
Answer:
x = 3 ± 5[tex]\sqrt{2}[/tex]
Step-by-step explanation:
x² - 6x - 41 = 0 ( add 41 to both sides )
x² - 6x = 41
to complete the square
add ( half the coefficient of the x- term )² to both sides
x² + 2(- 3)x + 9 = 41 + 9
(x - 3)² = 50 ( take square root of both sides )
x - 3 = ± [tex]\sqrt{50}[/tex] = ± 5[tex]\sqrt{2}[/tex] ( add 3 to both sides )
x = 3 ± 5[tex]\sqrt{2}[/tex]
hich is the equation of a line that has a slope of One-half and passes through point (2, –3)?
The equation of line passing that has a slope of One-half and passes through point (2, –3) is y = 0.5x - 4
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Equations are classified based on degree (value of highest exponents) as linear, quadratic, cubic and so on. Variables can be dependent or independent. Dependent variables depend on other variable while an independent variable do not depend.
The standard form for linear equation is:
y = mx + b
Where m is the slope and b is the y intercept
The equation of line passing that has a slope of One-half and passes through point (2, –3) is:
y - (-3) = (1/2)(x - 2)
y + 3 = 0.5x - 1
y = 0.5x - 4
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Write a piecewise function for the given graph.
The graphed piecewise function is conformed of two lines, it can be written as::
y = -x if x < -1y = x + 1 if x ≥ 1How to write the piecewise function?Notice that we have two lines that conform the piecewise function.
The linear equation in the left passes through the points (-8, 8) and (-1, 1), so the slope is:
m = (1 - 8)/(-1 + 8) = -7/7 = -1
Then it is something like:
y = -x + b
To find the value of b we use the point (-1, 1), we will get:
1 = -(-1) + b
1 - 1 = b
0 = b
Then the first line is:
y = -x if x < -1.
The second line passes through points (-1, -3) and (4, 2), then the slope is:
m = (2 + 3)/(4 + 1) = 1
And we can see that it intercepts the y-axis at y = -1, so the linear equation here is:
y = x - 1
Finally, the piecewise function is:
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Solve similar triangles (advanced). khan academy. solve for x
The value of x in the similar triangles is x = 3.375
How to determine the values of x?The triangles represent the given parameter
On the triangle, we have the following corresponding sides
AB to BC and AD to DE
This means that
AB : BC = AD : DE
Express as fractions
So, we have
AB/BC = AD/DE
Substitute the known values in the above equation
x/3 = x + 9/11
This gives
11x = 3x + 27
So, we have
8x =27
Divide
x = 3.375
Hence, the value of x is x = 3.375
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During a search and rescue process after an earthquake, a team rescued 15 people in 5 hours. At this rate, what is the total number of people that the team can rescue in another 2 hours?
Based on the rate that the team is rescuing people per hour, the total number of people that the team can rescue in another two hours is 6 people.
How to find the number of people?First, find the rate at which people are being rescued by the search and rescue team per hour.
The rate of people rescued is:
= Number of people rescued / Number of hours
= 15 / 5
= 3 people per hour
The number of people rescued in 2 hours is:
= Number of hours x People rescued per hour
= 2 x 3
= 6 people
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ANSWER THE FOLLOWING:
According to the given function,
Domain = [tex]\left(-\infty, -4\right) \cup \left(-4, 2\right) \cup \left(2, \infty\right)[/tex]
Vertical Asymptote = 4
Horizontal Asymptote = 0
X - intercept = No x - intercepts
Y - intercept = 1/4
Domain:
The the domain of a function referred as the set of inputs accepted by the function
Given,
The function is,
[tex]f(x)=\frac{x-2}{x^2+2x-8}[/tex]
Here we need to find the domain, vertical asymptote, horizontal asymptote, x - intercept, and y-intercept.
To find the domain and intercepts, we have to plot the graph for the given function.
So, the graph of the function is attached below. It can be drawn using the graphing calculator.
As per the graph,
Domain = [tex]\left(-\infty, -4\right) \cup \left(-4, 2\right) \cup \left(2, \infty\right)[/tex]
And the values of
x - intercept = No x-intercepts
y - intercept = 1/4
Now, the vertical asymptote is calculated as,
The line x=L is a vertical asymptote of the function f(x), if the limit of the function (one-sided) at this point is infinite.
In other words, it means that possible points are points where the denominator equals 0 or doesn't exist.
So, find the points where the denominator equals 0 and check them.
x = -4
Then,
[tex]\lim_{x \to -4^+} \frac{1}{x + 4}=\infty[/tex]
Since the limit is infinite, then x=−4 is a vertical asymptote.
And the horizontal asymptote is,
Line y=L is a horizontal asymptote of the function y=f(x), if either '
[tex]\lim_{x \to \infty} f{\left(x \right)}=L[/tex] or [tex]\lim_{x \to -\infty} f{\left(x \right)}=L[/tex] and L is finite.
When we calculate the limits:
[tex]\lim_{x \to \infty}\left(\frac{x - 2}{x^{2} + 2 x - 8}\right)=0\\\\\lim_{x \to -\infty}\left(\frac{x - 2}{x^{2} + 2 x - 8}\right)=0[/tex]
Thus, the horizontal asymptote is y=0.
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You might observe that the data has an awfully smooth pattern. If you were to view the scatterplot above at the daily level or the monthly level, the points in the scatter diagram would look a lot more messy. It looks a lot less messy because the data is the average CO2 over the entire year. What is the term for this phenomenon that we talked about?
Based on the fact that the data for the average CO₂ over the whole year was smoother than the daily and monthly level, the term for the phenomenon is called Smoothing.
How to define smoothing?Smoothing refers to a process in statistics where the messier points of a data distribution is removed. In other words, the data points that give the data set a varied distribution are removed so that the data set appears smoother.
There are several methods to engage in smoothing but most of them use averaging over a longer period of time. They would for instance, use the averages of a period over a month or weekly and convert it to an annual measure like the scatterplot of CO₂ for the year.
In conclusion, the fact that the average CO₂ scatterplot for the year was smoothly than monthly or daily is due to smoothing.
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Write the equation of the line in slope-intercept form. The coefficients in the equation are b=4 and m=-5
Answer:
y = -5x + 4
Step-by-step explanation:
Slope-intercept form of the equation is:
y = mx + b
In the question, we are given m = -5 and b = 4.
Fill in the numbers for m and b.
The m tells you how steep the line is (the slope) And b tells you where the line crosses the y-axis (y-intercept).