Answer:
No solutions
Step-by-step explanation:
3x^2=y
3(-2)^2=1
Follow P.E.M.D.A.S.
3(4)=1
12=1
What is the following simplified product? Assume x>0
2 square root 8x^3(3 square root 10x^4-x square root 5x^2
Answer:
[tex] 24x^3\sqrt{5x} - 4x^3\sqrt{10x} [/tex]
Step-by-step explanation:
The product [tex]2\sqrt{8x^3} (3\sqrt{10x^4} - x\sqrt{5x^2})[/tex] can be simplified as follows:
Step 1: Use the distributive property of multiplication
[tex]2\sqrt{8x^3}(3\sqrt{10x^4)} - 2\sqrt{8x^3}(x\sqrt{5x^2})[/tex]
[tex] 2*3\sqrt{8x^3*10x^4} - 2*x\sqrt{8x^3*5x^2} [/tex]
[tex] 6\sqrt{80x^7} - 2x\sqrt{40x^5} [/tex]
Step 2: simplify further
[tex] 6\sqrt{16*5*x^3*x^3*x} - 2x\sqrt{4*10*x^4*x} [/tex]
[tex] 6*4*x^3\sqrt{5*x} - 2x*2*x^2\sqrt{10*x} [/tex]
[tex] 24x^3\sqrt{5x} - 4x^3\sqrt{10x} [/tex]
Example 2: The GPAs of 20 students are listed.
Make a stem-and-leaf plot for this data.
1.8 2.9 0.9 4.0 3.3
2.4 2.3 1.6 1.6 4.0
1.7 0.5 3.6 3.4 1.9
4.0 2.1 1.9 1.1 0.5
How do I make a stem and leaf plot for these numbers?
Answer/Step-by-step Explanation:
To create a stem-and-leaf plot for the GPAs of the 20 students that were listed in the above question, take the following steps:
Step 1: for easy plotting, write down the GPAs in an ordered manner, that is, from the smallest value to the largest.
0.5, 0.5, 0.9, 1.1, 1.6, 1.6, 1.7, 1.8, 1.9, 1.9, 2.1, 2.3, 2.4, 2.9, 3.3, 3.4, 3.6, 4.0, 4.0, 4.0
Step 2: divide each set of data into a stem and a leaf. For example, for a GPA, 0.5, 0 would be the stem, while 5 would be the leaf.
For a GPA listed, 2.9, 2 is the stem, 9 is the leaf. Same applies to all the other GPAs.
Step 3: All stems should be written in ascending order, vertically, from the smallest to the largest. From the listed GPAs, you would observe that the highest stem value would be 4, while the lowest would be 0.
Therefore, write down your stems vertically, in ascending order as shown below:
0 |
1 |
2 |
3 |
4 |
Step 4: All leaves should be written also in ascending order to their corresponding stem, from the smallest to the largest, as shown below:
Stem | Leaf
0 | 5 5 9
1 | 1 6 6 7 8 9 9
2 | 1 3 4 9
3 | 3 4 6
4 | 0 0 0
Thus,
0 | 5 5 9 represents => 0.5, 0.5, 0.9
*See attachment below for the stem-and-leaf plot.
YOU WILL GET 30 POINTS AND BRAINLIEST IF YOU GET THIS CORRECT AND ANSWER THIS IN 5 MIN!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
A car manufacturer is reducing the number of incidents with the transmission by issuing a voluntary recall. During week 3 of the recall, the manufacturer fixed 391 cars. In week 13, the manufacturer fixed 361 cars. Assume that the reduction in the number of cars each week is linear. Write an equation in function form to show the number of cars seen each week by the mechanic. f(x) = 3x + 400 f(x) = 3x + 391 f(x) = −3x + 391 f(x) = −3x + 400
Answer:
f(x)= -3x + 400
Step-by-step explanation:
[tex]\frac{x-x_{1} }{x_{2}-x_{1} } = \frac{y-y_{1} }{y_{2}-y_{1} }[/tex]
[tex]\frac{x-3}{13-3} =\frac{y-391}{361-391}[/tex]
-3 ( x-3 ) = (y - 391 )
-3x + 400
Answer:
he is correct
Step-by-step explanation:
find the slope for (-4,-2)(-3,-6)
Answer:
The slope is -4.
Step-by-step explanation:
The values -2 and -6 are 4 values apart.
The values -4 and -3 are 1 value apart.
Since the second coordinate is lower than the first one, the slope of this is negative.
4 / 1 = 1
Negating 1 gets us -1.
Hope this helped!
Answer:
[tex] \frac{y}{x} = \frac{ - 4}{1} = - 4[/tex]
Step-by-step explanation:
[tex]x = ( - 3) - ( - 4) = 1[/tex]
[tex]y = ( - 6) - ( - 2) = - 4[/tex]
|x–5|=|x+5| If you answer this question before 2:35 pm on July 28, 2020, I will give 10 points!!
Answer:
[tex]\boxed{x=0}[/tex]
Step-by-step explanation:
[tex]|x-5|=|x+5|[/tex]
Solve absolute value.
There are two possibilities.
First possibility:
[tex]x-5=x+5\\0=10[/tex]
No solution.
Second possibility:
[tex]x-5=-(x + 5)\\x-5=-x-5\\2x=0\\x=0[/tex]
The value of y varies inversely as the square of x, and y = 16, when I = 3.
Find the value of x when y = 1.
Answer:
x = 12Step-by-step explanation:
The statement
The value of y varies inversely as the square of x is written as
[tex]y = \frac{k}{ {x}^{2} } [/tex]
where k is the constant of proportionality
To find the value of x when y = 1 first find the formula for the variation
y = 16 x = 3
k = yx²
k = 16(3)²
k = 16 × 9
k = 144
The formula for the variation is
[tex]y = \frac{144}{ {x}^{2} } [/tex]
when y = 1
We have
[tex]1 = \frac{144}{ {x}^{2} } [/tex]
Cross multiply
x² = 144
Find the square root of both sides
We have the final answer as
x = 12Hope this helps you
Question 5 of 13, Step 1 of 1
3/15
Correct
2
The Chandlers are moving across the country. Mr. Chandler leaves 2.5 hours before Mrs. Chandler. If he averages 75 mph and she averages 85 mph, how many hours
will it take Mrs. Chandler to catch up to Mr. Chandler?
Answer:
It will take Mrs Chandler 18 hours 45 minutes or 18.75 hours to catch up Mr Chandler
Step-by-step explanation:
What we want to know here is that at how many hours will they have traveled same distance.
Let the total time taken by Mrs Chandler to catch up be x hours
Since Mr Chandler left 2.5 hours earlier , then the total time taken by him would be x + 2.5 hours
Now, we know that distance = speed * time
Since it’s same distance covered;
For Mr Chandler, his distance is calculated as 75(x + 2.5)
For Mrs Chandler, her distance is calculated as 85x
We equate both since they are equal;
75(x + 2.5) = 85x
75x + 187.5 = 85x
85x -75x = 187.5
10x = 187.5
x = 18.75 hours or 18 hours 45 minutes
If a transversal is perpendicular to one of two parallel lines, then it's ________ to the other line. Question 16 options: A) perpendicular B) congruent C) parallel D) supplementary
Answer: Perpendicular.
Step-by-step explanation:
Suppose that you have two perpendicular lines:
Remember that a line is something like:
y = a*x +b
and two lines are parallel if they have the same slope (a) but a different y-intercept(b)
Then our lines can be:
y1 = a*x + b1
y2 = a*x + b2.
Now, if we have a line:
y = a*x + b
Then a perpendicular line will have a slope equal to -(1/a):
yp = (-1/a)*x + c
So this only depends on the slope, and we know that our two parallel lines have the same slope. So if we construct a transversal line that is perpendicular to one of our lines, it also must be perpendicular to the other line.
Answer:
A
Step-by-step explanation:
A paint manufacturer has a uniform annual demand for 16,000 cans of automobile primer. It costs $4 to store one can of paint for one year and $500 to set up the plan for production of the primer. Let x be the number of cans of paint produced during each production run, and let y be the number of production runs. Then the setup cost is 500y and the storage cost is 2.c, so the total storage and setup cost is C = 500y +2.c. Furthermore, .cy = 16,000 to account for the annual demand. How many times a year should the company produce this primer in order to minimize the total storage and setup costs?
A. The company should have 6 production runs each year.
B. The company should have 8 production runs each year.
C. The company should have 10 production runs each year.
D. The company should have 11 production runs each year.
Answer:
B. The company should have 8 production runs each year.
Step-by-step explanation:
From the given information:
A paint manufacturer has a uniform annual demand for 16,000 cans of automobile primer
It costs $4 to store one can of paint for one year
$500 to set up the plan for production of the primer
Let x be the number of cans of paint produced during each production run
Let y be the number of production runs.
If the total storage and setup cost is C = 500y + 2c
and cy = 16000
Then c = 16000/y
From;
C = 500y + 2c
Replacing c with 16000/y, we have;
C = 500y + 2(16000/y)
C = 500y + 32000/y
in order to minimize the total storage and setup costs [tex]C_{min} = C[/tex]
Therefore [tex]\dfrac{dc}{dy}=0[/tex]
⇒ 500 - 32000/y² =0
y² = 32000/500
y² = 320/5
y² = 64
y = (√64)
y = 8
Therefore; The company should have 8 production runs each year in order to minimize the total storage and setup costs
The summer has ended and it’s time to drain the swimming pool. 20 minutes after pulling the plug, there is still 45 000L of water in the pool. The pool is empty after 70 minutes. Calculate the rate that the water is draining out of the pool. (Hint: remember this line is sloping down to the right) (3 marks) Rate=0-45000/70-20 Rate = (0 - 45000) / 70-20 Rate = -45000 / 50 Rate=-900/minute b) Calculate how much water was in the pool initially (at time 0). (2 marks)
Answer:
there were 63,000L of water initially in the pool
Step-by-step explanation:
do 900L times the first 20 minutes to find out how much water was drained during the first 20 minutes and you get 18,000L drained
then, add 18,000L plus the rest of the 45,000L of water drained from the pool to get 63,000L of water intially in the pool
Solve the proportion below.
X =
A. 24
B. 49
c. 27
D. 6
Answer:
A. 24
Step-by-step explanation:
4/9 = x/54
x= 54*4/9 ===== multiplying both sides by 54
x= 24
Answer is 24, choice A is correct one
81^x^2=27^x solve for x
Step-by-step explanation:
81^x² = 27^x
(3^4)^x² = (3^3)^x
3^(4x²) = 3^(3x)
4x² = 3x
4x² − 3x = 0
x (4x − 3) = 0
x = 0 or ¾
The graphed line shown below is y = negative 4 x minus 12. On a coordinate plane, a line goes through (negative 3, 0) and (negative 2, negative 4). Which equation, when graphed with the given equation, will form a system that has no solution? y = 4 x + 12 y = negative 4 x y = negative 12 y = negative 4 (x + 3)
Answer:
y = -4x or the second option on edge.
This is because after you form it into the given equation, it equals y = -4x.
In order to clarify, edge also states that's the answer.
Answer:
2nd option
Step-by-step explanation:
Below are some of the scores on a math quiz given last week,
{82, 73, 74, 78, 46, 73}
What will happen to the mean of the quiz scores if the outlier is removed?
A
The mean will decrease.
OB
The mean will increase
C
There is not enough information given.
OD
The mean will not change.
Answer:
B: The mean will increase
Step-by-step explanation: The outlier is 46, which is way below all the other numbers, which is the definition of an outlier. If we remove a really low number from the set, then the mean(average) will increase.
6th grade math, help me please.
Answer:
a) [tex]\frac{2}{3} \,\frac{lb}{bread}[/tex]
b) [tex]1\frac{1}{4} \,\frac{in}{domino}[/tex]
Step-by-step explanation:
Part a:
every 4 lbs of flour, she makes 6 loaves of bread. this as a rate in simplest fraction form is:
[tex]\frac{4}{6} \,\frac{lb}{bread} = \frac{2}{3} \,\frac{lb}{bread}[/tex]
Part b:
every 10 inches , 8 dominoes can be placed. then the rate can be written as:
[tex]\frac{10}{8} \,\frac{in}{domino} = \frac{5}{4} \,\frac{in}{domino} =1\frac{1}{4} \,\frac{in}{domino}[/tex]
A) The perimeter of a rectangle is the sum of the lengths of its four sides. Write an expression for the perimeter of the rectangle and then evaluate when x=1/2 foot? B) The area of a rectangle is the product of its length and width. Write an expression for the area of the rectangle and then evaluate when x=1/2 feet?
Answer:
Below
Step-by-step explanation:
The length of this triangle is 3x+1 and the width is x.
The perimeter P is:
P= 2(3x+1)+2*x
P= 6x+2+2x
P= 8x+2
Let's evaluate it when x=1/2
●1/2 =0.5
P= 8*0.5+2 =4+2= 6 ft
●●●●●●●●●●●●●●●●●●●●●●●●
The area A is:
A = (3x+1)*x
A= 3x^2 +x
Let's evaluate it when x=0.5 feet
A= 3*0.5^2 +0.5
A= 3*0.25+0.5
A= 0.75 +0.5
A= 1.25 ft^2
I BEG OF YOU HELPPPP Twice last month, Judy Carter rented a car in Fresno, California, and traveled around the Southwest on business. The car rental agency rents its cars for a daily fee, plus an additional charge per mile driven. Judy recalls that her first trip lasted 4 days, she drove 440 miles, and the rental cost her $286. On her second business trip she drove 190 miles in 3 days, and paid $165.50 for the rental. Find the daily fee and the mileage charge.
Answer:
the daily fee =33 dollars
and the mileage charge.=0.35
Step-by-step explanation:
let d: be daily fee and m for mileage
cost of rental =(d*number of days)+ (m*number of mileage)
her first trip: 4d+440m=286
her second trip: 3d+190m=165.5
solve by addition and elimination
4d+440m=286 ⇒ multiply by 3 ⇒12d +1320m=(3)286
3d+190m=165.5⇒ multiply by 4⇒12d+190(4)m=4(165.5)
12d+1320m=858
12d+760m=662
subtract two equation to eliminate d
12d+1320m-12d-760m=858-662
560m=196
m=7/20=0.35 for on mileage
d: 4d+440m=286
4d=286-440(0.35)
d=(286-154)/4 33 dollars
Solve the system by substitution. x−5y=13 4x−3y=1 Enter your answer as an ordered pair (x,y).
Answer:
(-2,-3)
Step-by-step explanation:
Well in the system,
x−5y=13
4x−3y=1
We need to find x or y in either equation.
Let's do x - 5y = 13 for x.
+5y to both sides
x = 5y + 13
Now we substitute 5y + 13 for y in 4x - 3y = 1.
4(5y + 13) - 3y = 1
20y + 52 - 3y = 1
17y + 52 = 1
-52 to both sides
17y = -51
Divide all by 17
y = -3
Now we can substitute -3 for y in 4x - 3y = 1.
4x - 3(-3) = 1
4x + 9 = 1
-9 to both sides
4x = -8
Divide 4 to both sides
x = -2
Thus,
the solution is (-2,-3).
Hope this helps :)
Answer:
( - 2 , - 3 )Step-by-step explanation:
x - 5y = 13
4x - 3y = 1
Solve the equation for x
[tex]x - 5y = 13[/tex]
Move '5y' to R.H.S and change it's sign
[tex]x = 13 + 5y[/tex]
Substitute the given value of X into the equation
4x - 3y = 1
[tex]4(13 + 5y) - 3y = 1[/tex]
Solve the equation for y
distribute 4 through the parentheses
[tex]52 + 20y - 3y = 1[/tex]
Collect like terms
[tex]52 + 17y = 1[/tex]
Move constant to R.H.S and change it's sign
[tex]17y = 1 - 52[/tex]
Calculate
[tex]17y = - 51[/tex]
Divide both sides of the equation by 17
[tex] \frac{17y}{17} = \frac{ - 51}{17} [/tex]
Calculate
[tex]y = - 3[/tex]
Now, substitute the given value of y into the equation
x = 13 + 5y
[tex]x = 13 + 5 \times ( - 3)[/tex]
Solve the equation for x
Multiply the numbers
[tex] = 13 - 15[/tex]
Calculate the difference
[tex] = - 2[/tex]
The possible solution of the system is the ordered pair
( x , y )
( x , y ) = ( - 2 , - 3 )
-----------------------------------------------------------------------
Check if the given ordered pair is the solution of the system of equation
[tex] - 2 - 5 \times ( - 3) = 15[/tex]
[tex]4 \times ( - 2) - 3 \times ( - 3) = 1[/tex]
Simplify the equalities
[tex]13 = 13[/tex]
[tex]1 = 1[/tex]
Since all of the equalities are true, the ordered pair is the solution of the system
( x , y ) = ( - 2 , - 3 )Hope this helps..
Best regards!!
Create a circle such that its center is point a and b is a point on the circle
Step-by-step explanation:
The center of a circle is the point in the circle which is equidistant to all the edges of thr circle. The point a is the center, while point b is an arbitrary point in the circle. Find attachment for the diagram.
Answer:
i think that this question is wrong
Step-by-step explanation:
A line passes through the points (6, 10) and (4, -2). What is the equation of the line
Answer:
y = 6x - 26
Step-by-step explanation:
1. find slope: (y₂ - y₁) / (x₂ - x₁)
(-2 - 10) / (4 - 6) = -12 / -2 = 6
basic equation: y = 6x + b
2. plug in (x,y) value using one set of coordinates.
10 = 6(6) + b
10 = 36 + b
b = 10 - 36
b = -26
3. plug b in to find full equation.
y = 6x -26
Answer:
y = -1/6 x + 11
Step-by-step explanation:
In order to write an equation of a line you need slope (m) and y-intercept (b) or where the graph grosses the y- axis. since you are given two points (6, 10) and (4, -2). Slope when given two points is (y - y) / (x - x)
so (-2 - 10) / (6 - 4) = 6 / - 1 =- 6
use the equation y = mx + b and substitute either point (6, 10) or (4, -2) as a replacement for x and y respectively. (I chose (6, 10) because they are positive numbers. Substituting x = 6 and y = 10 and m = -6 into y = mx + b
10 = -6(6) + b
10 = -36 + b
b = 46 (add -36 to both sides)
so our equation: y = -6x + 46 :-)
-2x(x+3)-(x+1)(x-2)=
Answer:
-3x^2 -5x +2
Step-by-step explanation:
-2x(x+3)-(x+1)(x-2)=
Distribute
-2x^2 -6x -(x+1)(x-2)
Foil
-2x^2 -6x -(x^2 -2x +x -2)
Combine like terms
-2x^2 -6x -(x^2 -x -2)
Distribute the minus sign
-2x^2 -6x -x^2 +x +2
Combine like terms
-2x^2 -x^2 -6x +x +2
-3x^2 -5x +2
Answer:
[tex]\huge\boxed{-2x(x+3)-(x+1)(x-2)=-3x^2-5x+2}[/tex]
Step-by-step explanation:
[tex]-2x(x+3)-(x+1)(x-2)[/tex]
Use the distributive property: a(b + c) = ab + ac
and FOIL: (a + b)(c + d) = ac + ad + bc + bd
[tex]=(-2x)(x)+(-2x)(3)-\bigg[(x)(x)+(x)(-2)+(1)(x)+(1)(-2)\bigg]\\\\=-2x^2-6x-\bigg(x^2-2x+x-2\bigg)=-2x^2-6x-x^2-(-2x)-x-(-2)\\\\=-2x^2-6x-x^2+2x-x+2[/tex]
Combine like terms:
[tex]=(-2x^2-x^2)+(-6x+2x-x)+2=-3x^2+(-5x)+2\\\\=-3x^2-5x+2[/tex]
WILL GIVE BRAINLIEST IF CORRECT
A 10 ft ladder is propped up against a building at an angle of 39°. How far up the wall does the ladder go?
Answer:
6.3 ft
Step-by-step explanation:
The ladder lying on the wall with an elevation of 39° forms a right angled triangle.
Hypotenuse = length of ladder = 10 ft
Opposite = x = ?
θ = 39°
Use trigonometric ratio formula to find, x, which is how far the ladder goes up the wall.
The trigonometric ratio formula to use is:
sin(θ) = opposite/hypotenuse
Sin(39) = x/10
Multiply both sides by 10
10*sin(39) = x
x = 10*sin(39)
x = 6.29 ≈ 6.3 ft (to nearest tenth)
Answer:
6.3 ft
Step-by-step explanation:
did the quiz got it right
Each of the following linear equations defines y as a function of x for all integers x from 1 to 100. For which of the following equations is the standard deviation of the y-values corresponding to all the x-values the greatest?
a) y = x/3
b) y = x/2+40
c) y = x
d) y = 2x + 50
e) y = 3x − 20
Answer:
Option E
Step-by-step explanation:
y = x /3
let x = 1, 2, 3
y = 0.333, 0.667, 1
y = x/2 + 40
let x = 1, 2, 3
y = 40.5, 41, 41.5
y = x
let x = 1, 2, 3
y = 1, 2, 3
y = 2x + 50
let x = 1, 2, 3
y = 52, 54, 56
y = 3x - 20
let x = 1, 2, 3
y = -17, -14, -11
The standard deviation is the spread of data, the data that is most spread is option E.
a. Find the probability of randomly selecting a student who spent the money, given that the student was given four quarters. The probability is . (Round to three decimal places as needed.) b. Find the probability of randomly selecting a student who kept the money, given that the student was given four quarters. The probability is . (Round to three decimal places as needed.) c. What do the preceding results suggest?
Answer:
Hello your questions is incomplete attached below is the missing part of the question
answer : A ) 0.647 , (B) 0.353, (C) students are more likely to spend the money than to have kept it
Step-by-step explanation:
from the attached table below
Given data :
Total number of students = Number of students who spent money + number of students who kept money
Total number of students = (33+13 ) + (18 + 27 ) = 91
p(Number of students given four quarters) = (33 +18 ) / 91 = 51/91
p( number of students who spent money ) = ( 33 +13 ) / 91 = 46/91
p( number of students who saved money ) = (18 +27 ) / 91 = 45 /91
p( number of students who spent money and given four quarters ) = 33/91
p( number of students who saved money and given four quarters ) = 18/91
A) The probability of randomly selecting a student who spent the money and also given four quarters
= p ( 33/91 | 51/91 )
= 33/91 * 91/51
= 33/51 = 0.647
B ) The probability of randomly selecting a student who kept the money and given that the student was given four quarters
= p ( 18/91 | 51/91 )
= 18/91 * 91/51
= 18 /51 = 0.353
C) students are more likely to spend the money than to have kept it
Write the first 4 terms of the sequence defined by the given rule f(n)=n2 -1
Answer:
0, 3, 8, 15Step-by-step explanation:
Substitute n = 1, n = 2, n = 3 and n = 4 to the equation f(n) = n² - 1:
f(1) = 1² - 1 = 1 - 1 = 0
f(2) = 2² - 1 = 4 - 1 = 3
f(3) = 3² - 1 = 9 - 1 = 8
f(4) = 4² - 1 = 16 - 1 = 15
A student is using the elimination method to solve the system of equations below. What is the best first
step?
4x - 5y = 2
2x + y = -3
Answer:
The best first step would be to multiply the second equation by -2
Step-by-step explanation:
The best first step would be to multiply the second equation by -2
then you would have the following
[tex]\ \ \ \ \ \ 4x - 5y = 2 \\-2*(2x)+ (-2)*y = (-2)*3[/tex]
and when you multiply it is easy to eliminate because you will get
[tex]4x - 5y = 2 \\-4x -2y = 6[/tex]
and if you sum the equations you get
-7y = 8
so that is a single variable equation which is easier to solve.
Describe how to simplify the expression
3^-6
3^-4
Answer:
For 3^-6 = 1/3^6 = 726 and for 3^-4 = 1/3^4 = 81
Step-by-step explanation:
ARE THE NUMBERS SEPARATED OR THEY ARE TO BE JOIN TOGETHER,IF THERE ARE SEPARATED THE SOLUTION IS GOING TO BE...
3^-6
using law of indices x^-a = 1/x^a
So 3^-6 = 1/3^6
Now find the power of 3^6
which is the same as 3×3×3×3×3×3 = 729
therefore 1/3^6 = 729
For 3^-4
using the above method
3^-4 is same as 1/3^4
which is equal to
finding the nominator
3×3×3×3 = 81
so the answer is 81
3+x=8 What would like match this answer
Answer:
x = 5
Step-by-step explanation:
x = 8 - 3
Thus, x = 5
13. If 6 times the 6th term of an A.P. is equal to
13 times the 13th term, prove that 19th term
of this A.P. is zero.
please give the answer as fast as you can
please
Answer:
see explanation
Step-by-step explanation:
The n th term of an AP is
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
Given
6(a₁ + 5d) = 13(a₁ + 12d) ← distribute parenthesis on both sides
6a₁ + 30d = 13a₁ + 156d ( subtract 13a₁ from both sides )
- 7a₁ + 30d = 156d ( subtract 30d from both sides )
- 7a₁ = 126d ( divide both sides by - 7 )
a₁ = - 18d
Now
a₁₉ = a₁ + 18d = - 18d + 18d = 0 ← as required
Find the x-coordinates of the two points on the curve
y=x-1/x at which the tangent is parallel to the straight line 4y= x + 8. (4 marks)
Answer: x = {-2, 2}
Step-by-step explanation:
Tangent means it is touching. Find the intersection of the two equations.
Solve the linear equation for y, then set the two equations equal to each other.
[tex]4y=x+8\qquad \rightarrow \qquad y=\dfrac{x+8}{4}[/tex]
[tex]\dfrac{x-1}{x}=\dfrac{x+8}{4}\\\\\\\text{Cross multiply and solve for x:}\\4(x-1)=x(x+8)\\4x-4=x^2+8x\\.\qquad 0=x^2+4x+4\\.\qquad 0=(x+2)^2\\.\qquad 0=x+2\\.\qquad x=-2[/tex]
To find the next point that is parallel to the linear equation and tangent to the curve, we need to use the linear equation with slope (m) = [tex]\dfrac{1}{4}[/tex] and unknown b.
Let's try b = 0, then the equation of the linear equation is: [tex]y=\dfrac{1}{4}x[/tex]
Set the equations equal to each other and solve for x:
[tex]\dfrac{x-1}{x}=\dfrac{x}{4}\\\\\\4(x-1)=x^2\\4x-4=x^2\\.\qquad 0=x^2-4x+4\\.\qquad 0=(x-2)^2\\.\qquad 0=x-2\\.\qquad x=2[/tex]
This works!!! If it didn't work, we would have tried other values for b until we arrived at a solution.