EXPLANATION
Given the operation:
28------> rounded to 30
56-------> rounded to 50
76 ------> rounded to 80
Now, we can mentally calculate that:
3x5= 15 so 30x50 = 1500 (two zeros)
15x8 = 120 so,
1500x80 = 120,000
The answer is 144,000.
Question 13 of 19 Use the elimination method to solve the system of equations. Choose the correct ordered pair x+y=9 x-y=7 O A. (8,1) B. (12,-3) C. (16, -7) D. (7.0) SUBMIT
The solution for the system of equations is x = 8, y = 1
Explanation:Given the pair of equations:
x + y = 9 .....................................................(1)
x - y = 7 ......................................................(2)
To solve by elimintation, add (1) and (2) to eliminate y
2x = 16
x = 16/2 = 8 ............................................(3)
Subtract (2) from (1) to eliminate x
2y = 2
y = 2/2 = 1 ................................................(4)
(3) and (4) form the solution of the equation.
x = 8, y = 1
A triangle has a 30 degree angle and two sides that are each 6em in length. Select ALL the statements below that are TRUE1. The triangle might be an equilateral triangle (having all the same sides and angles) 2. One of the angles in triangle must be 120 3. The length of the third side must be 11cm or smaller4. One of the angles in the triangle might be 50
The value of x can be determined as,
[tex]\begin{gathered} x+30+30=180 \\ x=120 \end{gathered}[/tex]In triangle ABC,
[tex]\begin{gathered} BCThus, option (2) is the correct solution.(-25) + (42) + (-62) + (20) =
Answer
The answer = -25
Explanation
As long as we note that
(+) × (-) = (-)
(-) × (+) = (-)
We can easily solve this,
(-25) + (42) + (-62) + (20)
= -25 + 42 - 62 + 20
= 17 - 62 + 20
= -45 + 20
= -25
Hope this Helps!!!
Identify the quadrant in which the point (−3,2) is located.Question 19 options:Quadrant IQuadrant IIQuadrant IIIQuadrant IV
The given point is (-3,2). It is required to identify the quadrant in which the point is located.
Notice that the x-coordinate of the point is negative, while the y-coordinate is positive.
This implies that the point is located in the second quadrant.
Plot the point on the coordinate plane:
The answer is quadrant II.
determine if the statement is true or false if it is false explain why must be changing the statements and make it true if it's true explain why you believe to be true the common difference for a geometric sequence given 5:1 -1 negative 3 negative 5 negative 7 is 2
The common difference can be determined by subtracting the first term with the second term, second term with the third term, and so forth.
In this case we have:
[tex]1-(-1)=1+1=2[/tex][tex]-1-(-3)=-1+3=2[/tex][tex]-3-(-5)=-3+5=2[/tex]In this way we can determine that the common difference of the sequence is 2, so the statement is true.
what is the lcm of 6 and 8
to determine the lcm of 6 and 8, express these numbers as the product of prime numbers:
6 = 2x3
8 = 2x2x2
the same factors determine the lcm. In this case, the factors
From the diagram below, if side AB is 48 cm., side DE would be ______.
The triangle ABC is similar to the triangle DCE.
Hence, we need to find a proportion to find side DE:
If side AB = 48, it will represent the double value of the side DE.
Hence, DE = 48/2 = 24
The correct answer is option b.
Solve using substitution x=-9-4x+4y=20
The solution of the simultaneous equation using substitution is
(x ,y) = (-9,-4) .
The given system of equation are:
x=-9............................................1
-4x+4y=20 ...............................2
Now we will simplify the equation 2
or, -x + y = 5
now we will substitute the value of x in the equation 2
or, -(-9) + y = 5
or, 9+ y= 5
or, y = -4
Two or more algebra equations that share variables, such as x and y, are said to be simultaneous equations. Since the equations are resolved simultaneously, they are known as simultaneous equations. Each equation represents a straight line.
These equations alone could have an endless number of solutions. There are several ways to solve the simultaneous linear equations.
The simultaneous equations can be solved using one of four methods:
substitutioneliminationaugmented matrix method.Graphical methodHence the values of x and y are -9 and -4 respectively.
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Solve the system of equations: 2x + 3y = 8 and 3x - 3y = 12.
1) In this problem, let's solve this given system of equations by the method of Elimination.
We can start by adding both equations simultaneously:
[tex]\begin{gathered} 2x+3y=8 \\ 3x-3y=12 \\ --------- \\ 5x=20 \\ \\ \frac{5x}{5}=\frac{20}{5} \\ \\ x=4 \end{gathered}[/tex]Now, that we know the quantity of x, let's plug it into any of those equations. Most of the time, we opt to plug it into the simpler equation.
[tex]\begin{gathered} 2x+3y=8 \\ 2(4)+3y=8 \\ 8+3y=8 \\ -8+8+3y=8-8 \\ 3y=0 \\ \frac{3y}{3}=\frac{0}{3} \\ \\ y=0 \end{gathered}[/tex]2) Thus, the answer is (4,0)
A stick is 10 1/5 inches in length. A carpenter will cut it into shorter pieces, each 1 2/15 inches in length. How many pieces will the stick be cut into?
Answer:
9 pieces
Explanation:
From the question, we're told that the length of the stick is 10 1/5 inches, let's convert the mixed fraction into an improper fraction;
[tex]10\frac{1}{5}=\frac{51}{5}[/tex]Also, we're told that the stick was cut into shorter pieces of length 1 2/15 inches each. Converting 1 2/15 into an improper fraction, we'll have;
[tex]1\frac{2}{15}=\frac{17}{15}[/tex]To determine the number of pieces that the stick will be cut into, we'll need to divide 51/5 by 17/15;
[tex]\frac{\frac{51}{5}}{\frac{17}{15}}=\frac{51}{5}\ast\frac{15}{17}=\frac{15}{1}\ast\frac{3}{17}=\frac{153}{17}=9[/tex]heyy could you help me out I have been stuck in this problem for a long time I sent a pic of the problem by the way
Given two triangles ABC and DEF
They have the following:
1. AC = DF
2. 3. AB = DE
4.
so, if we take 1, 2 and 3
the triangles are congruent using SAS
And if we take 1 , 2 and 4
the triangle are congruent using ASA
So, the answer is the options: D and E
Solve a system of two linear inequalities graphically. Graph the solution set of the second linear inequality. Type of boundary line? Two points on the boundary line? Region you wish to be shaded?
ANSWER
EXPLANATION
The second inequality is y > -5x + 10. To graph this inequality we have to draw a dashed line y = -5x + 10 and since the inequality represents the values of y greater than the line, the shaded area is the one above the line.
Two points on the line are the y-intercept (0, 10) and the x-intercept (2, 0).
According to the model, how many marriage licenses were issued in 2006? Round your answer to the nearest hundred.
ANSWER:
C. 124,900
STEP-BY-STEP EXPLANATION:
We have that the following function is the one that models the situation:
[tex]y=3.4905\left(x\right)^2-17674\left(x\right)+21533000[/tex]We evaluate when x = 2006, like this
[tex]\begin{gathered} y=3.4905\left(2006\right)^2-17674\left(2006\right)+21533000 \\ \\ y=4024036\cdot\:3.4905-35454044+21533000 \\ \\ y=14045897.658-13921044 \\ \\ y=124853.658\cong124900 \end{gathered}[/tex]Therefore, the correct answer is C. 124,900
The product of two consecutive positive odd numbers is 323. Find the smaller of the two numbers. The small number is _
To answer this question, we need to know that we can represent, algebraically, two consecutive positive odd numbers as follows:
[tex]2n+1,2n+3[/tex]Then, if we have that the product of both consecutive positive odd numbers is 323, then:
[tex](2n+1)(2n+3)=323[/tex]Now, we will need to expand the formula as follows:
[tex](2n+1)(2n+3)=2n\cdot2n+2n\cdot3+(1)(2n)+1\cdot3[/tex]We applied the FOIL method to expand the expression. Then, we have:
[tex]4n^2+6n+2n+3=4n^2+8n+3[/tex]Now, we have:
[tex]4n^2+8n+3=323[/tex][tex]4n^2+8n+3-323=0\Rightarrow4n^2+8n-320=0_{}[/tex]We have here a polynomial (a quadratic equation) that we can solve using the quadratic formula:
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a},ax^2+bx+c=0[/tex]Then, we have that:
• a = 4
,• b = 8
,• c = -320
Then
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}\Rightarrow x=\frac{-8\pm\sqrt[]{8^2-4(4)(-320)}}{2\cdot4}[/tex][tex]\Rightarrow x=\frac{-8\pm\sqrt[]{64^{}-4(4)(-320)}}{2\cdot4}\Rightarrow x=\frac{-8\pm\sqrt[]{64+5120}}{8}[/tex][tex]x=\frac{-8\pm\sqrt[]{5184}}{8}\Rightarrow x=\frac{-8\pm72}{8}[/tex]Then, the solutions are:
[tex]x=\frac{-8+72}{8}\Rightarrow x=\frac{64}{8}\Rightarrow x=8[/tex][tex]x=\frac{-8-72}{8}\Rightarrow x=-\frac{80}{8}\Rightarrow x=-10[/tex]Therefore, we have two solutions for n, n = 8 or n = -10.
If we substitute the value of n = 8 in the original equations, we have:
[tex](2n+1)(2n+3)=323\Rightarrow(2\cdot8+1)(2\cdot8+3)=323[/tex][tex](16+1)(16+3)=323\Rightarrow17\cdot19=323[/tex]If we use the negative value for the solution, we obtain:
[tex](2(-10)+1)(2(-10)+3)=323\Rightarrow(-20+1)(-20+3)=323[/tex][tex]-19\cdot-17=323[/tex]Since these two numbers are negative, we have that the appropriate solution is n = 8.
Therefore, we have that the smaller of the two numbers is 17:
[tex]17\cdot19=323[/tex]The numbers 17 and 19 are consecutive positive odd numbers.
In summary, we have that the smaller number is 17.
i need help on this question
a) Based on the naming of the triangle, the line segment WR is congruent with the line segment PL.
[tex]\bar{WR}\cong\bar{PL}[/tex]b) Based on the naming of the triangle, the third point for triangle BGT is T. The third point for the other triangle, DSN, is N. Hence, the angle for T is congruent with angle N.
[tex]\angle T\cong\angle N[/tex]c) The name of the triangle RHK is rearranged as KRH. This means that name of the triangle WVO can also be rearranged as OWV and is congruent with the triangle KRH.
[tex]\Delta KRH\cong\Delta OWV[/tex]What is the equation in slope-intercept form of a line that has a slope of −1/2
and passes through the point (−2, 7)?
The equation in slope-intercept form of a line that has a slope of −1/2
and passes through the point (−2, 7) is y = (-1/2)x + 6.
Given:
slope-intercept form of a line that has a slope of −1/2 and passes through the point (−2, 7).
slope m = -1/2
substitute m and (-2,7) in standard form y = mx + c
7 = -1/2*-2 + c
7 = 2/2 + c
7 = 1 +c
c = 7 - 1
c = 6
substitute c and m value
y = mx+c
y = (-1/2)x + 6
Therefore The equation in slope-intercept form of a line that has a slope of −1/2 and passes through the point (−2, 7) is y = (-1/2)x + 6.
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The are of a square is 49m2. What is its side length?
We know that the area of a square is 49 square meters.
The area of a square is defined by
[tex]A=l^2[/tex]Where l is the length of each side.
Replacing the given are, we have
[tex]\begin{gathered} 49=l^2 \\ l=\sqrt[]{49} \\ l=7 \end{gathered}[/tex]Therefore, the side length is 7 meters long.What Is 3×10³=3×10×10×10=
3 x 10 ^3 = 3 x 10 x 10 x10 = 3 x 100 = 300
A football team is losing by 14 points near the end of a game. The team scores two touchdowns (worth 6 points each) before the end of the game. After each touchdown, the coach must decide whether to go for 1 point with a kick (which is successful 99% of the time) or 2 points with a run or pass (which is successful 45% of the time). If the team goes for 1 point after each touchdown, what is the probability that the coach’s team wins? loses? ties? If the team goes for 2 points after each touchdown, what is the probability that the coach’s team wins? loses? ties?
From the question, there are some scenarios we need to cater for:
- The team is down 14 points.
- It is a given that the team scores 2 touchdowns whatever the case. This means the team has 12 points in the bag.
- That leaves 2 points to overturn the loss, or draw or lose
If the team wins:
The team can only win if they score their 2 points runs twice. i.e. An increase in 4 points from the two plays would overturn the score and the team would lead the game by 2 points.
The question asks us to find the probability of the team winning if the team goes for 2 points after each 6-point touchdown.
We can solve this as:
[tex]\begin{gathered} \text{Probability of scoring 2 =} \\ P(2)=45\text{ \%=0.45} \\ \\ \therefore\text{Probability of scoring 2 the first time AND Probability of scoring 2 the second time=} \\ P(2)\times P(2)=0.45\times0.45=0.2025 \\ \\ \therefore\text{probability of the team winning by going for 2 points twice=} \\ 0.2025\times100\text{ \%} \\ =20.25\text{ \%} \end{gathered}[/tex]If the team loses:
If the team loses, there are some scenarios to take into consideration:
1. If the team tries 1 point plays and succeeds one time and failing the other time
2. If the team tries 1 point plays and fails twice.
3. If the team tries 2 point plays and they fail twice. (i.e. if they succeed even once, they can draw the match
find bounds on the real zeros of the polynomial functionf(x)= 17x^4 + 17x^3 - x^2 - 68x - 68
In order to identify bounds on the real zeros of this polynomial, first we need to find tendencies about the signal of f(x)
We know that the two term with highest degree is being multiplied by a positive coefficient. Therefore, we can initially conclude the f(x) tends to positive infinite as x grows either positive or negative.
We can check that, for x = -2, the first term is 272, and the remaining thermis togheter are given by:
[tex]17\cdot(-2)^3-(-2)^2-68\cdot(-2)-68=-72[/tex]Then for x < -2, and for also for x > 2, we can state for sure that f(x) remains always positive.
Then, any possible roots must lies in the intervel (-2,2)
e can also chegck that, for x = 1, f(x) = -103, and, for x = -1, f(x) = -1.
Therefore, f(x) must have a root between
Suppose that the functions and s are defined for all real numbers x as follows r(x) = 4x; s(x) = 3x ^ 2 Write the expressions for (s + r)(x) and (sr)(x) and evaluate (s - r)(2) .
Solution
Given
[tex]\begin{gathered} r(x)=4x \\ \\ s(x)=3x^2 \end{gathered}[/tex]Then
[tex]\begin{gathered} (s+r)(x)=s(x)+r(x)=4x+3x^2 \\ \\ \Rightarrow(s+r)(x)=4x+3x^2 \end{gathered}[/tex][tex](s\cdot r)(x)=s(r(x))=s(4x)=3(4x)^2=3\times16x^2=48x^2[/tex][tex](s-r)(2)=s(2)-r(2)=3(2)^2-4(2)=12-8=4[/tex]EXERCISE Carlos is jogging at a constant speed. He starts a timer when he is 12 feet from his starting position. After 3 seconds, Carlos is 21 feet from his starting position. Write a linear equation to represent the distance d of Carlos from his starting position t seconds after starting the timer.
The linear equation that represent the distance d of Carlos from his starting position is d=3t+12 where d denotes the distance and t denotes the time.
What is the meaning of speed?
The speed at which an object's location changes in any direction. The distance travelled in relation to the time it took to travel that distance is how speed is defined.
Given that when Carlos is 12 feet from his starting position, starts a timer.
He is 21 feet from his starting position after 3 s.
He covers (21 - 12) = 9 feet in 3 second.
The speed of an object is the distance that covers in unit time.
The Carlos's speed is 9/3 = 3 feet/s.
After t seconds, he covers (3×t) = 3t feet.
The distance of his from his starting position after t seconds is (3t + 12) feet.
The linear equation is d = 3t + 12.
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Hi I’m I don’t understand a variable of what this is saying.
given expression to simplify,
[tex]q^{-3}r^0s^{-1}\cdot\: q^3r^{-9}s^0[/tex][tex]\begin{gathered} q^{-3}r^0s^{-1}\cdot\: q^3r^{-9}s^0 \\ q^{-3}\cdot q^3=1 \\ \: =r^0s^{-1}r^{-9}s^0 \\ =1\cdot\frac{1}{r^9}s^{-1}s^0 \\ =1\cdot\frac{1}{r^9}\cdot\frac{1}{s} \\ =\frac{1}{r^9}\cdot\frac{1}{s} \\ =\frac{1}{r^9s} \\ =r^{-9}s^{-1} \end{gathered}[/tex]The following system is graphed below: x - y = 2 -X = -y - 1 14 Which of the following best describes the system?
Since the graphs are parallel to each other, there is no solution to the system. If a system has no solution, it is said to be an inconsistent system. So, the given system is inconsistent.
What is the value of m?
In this case, we have two similar triangles (the angles are congruents).
So, the next relations holds:
[tex]\frac{m}{18}=\frac{4}{6}[/tex]So, we only need to solve for m:
[tex]m=\frac{18\cdot4}{6}=12[/tex]Dominick weighs 30 pounds more than his sister. Together they weigh 330 pounds. How much does Dominick weigh?
Domik weights 30 pounds more than his sister
Together they weigh 330 pounds
Let "x" represent the weight of Dominik's sister, then his weight can be expressed as "x+30"
And their total weight can be calculated as:
[tex]x+(x+30)=330[/tex]From this expression you can calculate the value of x:
[tex]\begin{gathered} 2x+30=330 \\ 2x+30-30=330-30 \\ 2x=300 \\ \frac{2x}{2}=\frac{300}{2} \\ x=150 \end{gathered}[/tex]x=150 pounds
x+30=150+30=180 pounds
Dominik weights 180pounds and his sister 150 pounds
A skydiver jumps out of a plane form a certain height. The graph below shows their height h in meters after t seconds. What is the skydiver’s initial height?
The x-intercept and the slope Set y=0 giving 0=10x+3000 x=300010=300 seconds, or precisely 5 minutes.
How to you interpret the x-intercept and the slope?Y is described as being "above ground" in height. He will get into problems if he descends below ground!
In regard to time x, the right hand side (RHS) has negative 10x, which decreases 3000. Consequently, the starting height must be 3000 feet. Moreover, it is the y-intercept.
The number of feet that the parachutist will descend in x seconds must be the -10x. It is also the graph's gradient or slope.
The graph's intersection with the x-axis indicates that y=0 and y is the height above the earth.
The point when the item is about to strike the ground is therefore the x-intercept.
The rate of descent, or slope, remains constant as time (x) passes.
The rate of fall is actually a extremely rounded value In metric form, I believe the speed to be 9.81 meters per second per second, to two decimal places.
It is approximately 32.2 feet per second per second in imperial form.
to determine how long it will take to reach the ground.
Set y=0 giving 0=10x+3000 x=300010=300 seconds, or precisely 5 minutes.
The complete question is : A skydiver parachutes to the ground. The height y (in feet) of the skydiver after x seconds is y=−10x+3000, how do you graph and Interpret the x-intercept and the slope?
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Answer 5 mins
becasue i solved in. my phone and my teacher said its right
Module 17 - Distribution of Sample Proportions (6 of 6 discussion 3)20 20 unread replies. 20 20 replies.Learn by DoingSome features of this activity may not work well on a cell phone or tablet. We highly recommend that you complete this activity on a computer.ContextRecall the use of data from the National Health Survey to estimate behaviors such as alcohol consumption, cigarette smoking, and hours of sleep for all U.S. adults. In the 2005-2007 report, they estimated that 30% of all current smokers started smoking before the age of 16.PromptSuppose that we randomly select 100 U.S. adults who are smokers and find that 25% of this sample started smoking before the age of 16. In this random sample, the sample proportion (25%) differs from the estimated population proportion (30%) by 5%. In other words, there is a 5% error in the sample proportion; this sample under-estimates the population proportion by 5%.Is this much error surprising? To answer this question, find the probability that a sample proportion will over- or under-estimate the parameter by more than 5%.Show your work by checking normality conditions, calculating a z-score, explaining how the area under the normal curve is used to answer the question, and stating a conclusion in the context of this problem. Module 17 - Distribution of Sample Proportions (6 of 6 discussion 3)20 20 unread replies. 20 20 replies.Learn by DoingSome features of this activity may not work well on a cell phone or tablet. We highly recommend that you complete this activity on a computer.ContextRecall the use of data from the National Health Survey to estimate behaviors such as alcohol consumption, cigarette smoking, and hours of sleep for all U.S. adults. In the 2005-2007 report, they estimated that 30% of all current smokers started smoking before the age of 16.PromptSuppose that we randomly select 100 U.S. adults who are smokers and find that 25% of this sample started smoking before the age of 16. In this random sample, the sample proportion (25%) differs from the estimated population proportion (30%) by 5%. In other words, there is a 5% error in the sample proportion; this sample under-estimates the population proportion by 5%.Is this much error surprising? To answer this question, find the probability that a sample proportion will over- or under-estimate the parameter by more than 5%.Show your work by checking normality conditions, calculating a z-score, explaining how the area under the normal curve is used to answer the question, and stating a conclusion in the context of this problem. m the National Health Survey to estimate behaviors such as alcohol consumption, cigarette smoking, and hours of sleep for all U.S. adults. In the 2005-2007 report, they estimated that 30% of all current smokers started smoking before the age of 16.PromptSuppose that we randomly select 100 U.S. adults who are smokers and find that 25% of this sample started smoking before the age of 16. In this random sample, the sample proportion (25%) differs from the estimated population proportion (30%) by 5%. In other words, there is a 5% error in the sample proportion; this sample under-estimates the population proportion by 5%.Is this much error surprising? To answer this question, find the probability that a sample proportion will over- or under-estimate the parameter by more than 5%.Show your work by checking normality conditions, calculating a z-score, explaining how the area under the normal curve is used to answer the question, and stating a conclusion in the context of this problem. Content by the Open Learning Initiative (Links to an external site.) and licensed under CC BY (Links to an external site.).Search entries or author
Given:
Sample size = 100
p = 30% = 0.30
p' = 25% = 0.25
Let's find the probability that a sample proportion will over- or under-estimate the parameter by more than 5%.
Here, the error is:
Errror = |p' - p| = |0.25 - 0.30| = |-0.05| = 0.05
This error is not surprising.
Now, apply the formula:
[tex]\begin{gathered} \sigma p^{\prime}=\sqrt{\frac{p(1-p)}{n}} \\ \\ \sigma p^{\prime}=\sqrt{\frac{0.3(1-0.3)}{100}} \\ \\ \sigma p^{\prime}=\sqrt{\frac{0.3(0.7)}{100}}=\sqrt{\frac{0.21}{100}}=\sqrt{0.0021}=0.0458 \end{gathered}[/tex]Now, to find the probability that a sample proportion will be over or underestimate more than 5% will be:
[tex]\begin{gathered} p(p^{\prime}<0.3-0.05)+p(p^{\prime}>0.3+0.05) \\ \\ p(p^{\prime}<0.25)+p(p^{\prime}>0.35) \\ \\ z=\frac{p^{\prime}-\mu p^{\prime}}{\sigma p} \\ \\ Where:\mu p^{\prime}=0.3 \\ \end{gathered}[/tex]Hence, we have:
[tex]\begin{gathered} p(z<\frac{0.25-0.3}{0.0458})+p(z>\frac{0.35-0.3}{0.0458}) \\ \\ p(z<\frac{-0.05}{0.0458})+p(z>\frac{0.05}{0.0458}) \\ \\ p(z<-1.09)+p(z>1.09) \end{gathered}[/tex]Using the standard normal distribution table, we have:
NORMSDIST(-1.09) =0.1379
NORMSDIST(1.09) = 0.8621
Hence, we have:
p(z<-1.09) = 0.1379
p(z>1.09) = 1 - 0.8621 = 0.1379
p(z<-1.09) + p(z>1.09) = 0.1379 + 0.1379 = 0.2758
Therefore, the probability is 0.2758.
ANSWER:
0.2758
The linear functions f(x) and g(x) are represented on the graph, where g(x) is a transformation of f(x):A graph with two linear functions; f of x passes through 0, negative 1 and 5, 14, and g of x passes through negative 6, negative 1 and negative 1, 14.Part A: Describe two types of transformations that can be used to transform f(x) to g(x). Part B: Solve for k in each type of transformation. Part C: Write an equation for each type of transformation that can be used to transform f(x) to g(x).
Let's write the equation for every function:
Since f(x) passes through (0,-1) and (5,14), the equation will be given by:
[tex]\begin{gathered} (x1,y1)=(0,-1) \\ (x2,y2)=(5,14) \\ m=\frac{14-(-1)}{5-0}=\frac{15}{5} \\ m=3 \end{gathered}[/tex]Using the point-slope equation:
[tex]\begin{gathered} y-y1=m(x-x1) \\ y-(-1)=3(x-0) \\ y+1=3x \\ y=3x-1 \\ f(x)=3x-1 \end{gathered}[/tex]Using the same procedure for g(x):
[tex]\begin{gathered} (x1,y1)=(-6,-1) \\ (x2,y2)=(-1,14) \\ m=\frac{14-(-1)}{-1-(-6)}=\frac{15}{5}^{} \\ m=3 \\ \end{gathered}[/tex][tex]\begin{gathered} y-y1=m(x-x1) \\ y-(-1)=3(x-(-6)) \\ y+1=3x+18 \\ y=3x+17 \\ g(x)=3x+17 \end{gathered}[/tex]Now we have our functions:
[tex]\begin{gathered} f(x)=3x-1 \\ g(x)=3x+17 \end{gathered}[/tex]Part A:
We can translated f(x) k units up in order to get g(x) ( A vertical translation)
[tex]f(x)=3x-1+k[/tex]Or, we can translated f(x) k units to the left in order to get g(x) ( A horizontal translation)
[tex]f(x)=3(x+k)-1[/tex]We can see a graph of the functions:
Where the red function is f(x) and the blue function is g(x).
Part B:
Since both functions must be equal:
[tex]\begin{gathered} f(x)=g(x) \\ so\colon \\ 3x-1+k=3x+17 \end{gathered}[/tex]Solve for k:
[tex]\begin{gathered} k=3x+17-3x+1 \\ k=18 \end{gathered}[/tex]-----------------------
For the other case, let's use the same procedure:
[tex]\begin{gathered} 3(x+k)-1=3x+17 \\ 3x+3k-1=3x+17 \\ 3k=3x+17-3x+1 \\ k=\frac{18}{3} \\ k=6 \end{gathered}[/tex]Part C:
For the vertical translation:
[tex]3x-1+18[/tex]For the horizontal translation:
[tex]3(x+6)-1[/tex]Could you please help me with questions 36 & 37?
We have to determine if the functions are linear or not.
We can do this by rearranging the equations in this form:
[tex]y=mx+b[/tex]where m and b are constants.
NOTE: There are many ways to prove that a function is linear, but this is the easiest for this question.
36.
[tex]\begin{gathered} x+\frac{1}{y}=7 \\ \frac{1}{y}=-x+7 \end{gathered}[/tex]As this function can not be written in the form y=mx+b, then it is not linear.
37.
[tex]\begin{gathered} \frac{x}{2}=10+\frac{2y}{3} \\ \frac{x}{2}-10=\frac{2y}{3} \\ \frac{2y}{3}=\frac{1}{2}x-10 \\ y=\frac{3}{2}(\frac{1}{2}x-10) \\ y=\frac{3}{4}x-15 \end{gathered}[/tex]This function is now in the form y=mx+b, where m=3/4 and b=-15. Then, this function is a linear function.
Answer:
36. Non-linear.
37. Linear.