Write a two-column proof.
Given: 7y = 8x – 14; y = 6
Prove: x = 7
Answers
1) [tex]7y=8x-14, y=6[/tex] [given]
2) [tex]7(6)=8x-14[/tex] [substitution]
3) [tex]56=8x[/tex] [addition property of equality]
4) [tex]7=x[/tex] [division property of equality]
5) [tex]x=7[/tex] [symmetric property of equality]
A two columns proof is given in the attached image below as per the concept of solving equations.
A two columns proof is given below:
Statement | Reasons
y = 6 | Given
7y = 8x - 14 | Given
7(6) = 8x - 14 | Substitution (Substitute y with 6 in statement 2)
42 = 8x - 14 | Simplification
42 + 14 = 8x | Addition Property of Equality (Add 14 to both sides)
56 = 8x | Simplification
56/8 = x | Division Property of Equality (Divide both sides by 8)
7 = x | Simplification (Simplify the fraction 56/8)
x = 7 | Symmetric Property of Equality (Rearrange statement 9)
In this two-column proof, we start with the given information that y is equal to 6 (statement 1).
Then, we substitute the value of y into the equation 7y = 8x - 14 (statement 2) to get 42 = 8x - 14 (statement 3).
We simplify the equation to isolate 8x and find that 56 = 8x (statement 6). Dividing both sides of this equation by 8 (statement 7) gives us x = 7 (statement 8).
Finally, we use the symmetric property of equality to rearrange the equation and conclude that x is indeed equal to 7 (statement 9).
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Related Questions
Find so that the distance between (−2,3) and (,1) is √13
Answers
The distance between (-2, 3) and (-5, 1) is √13.
or, the distance between (-2, 3) and (1, 1) is √13.
We know that the length of the line segment connecting any two points represents the distance between them. There is just one line that connects the two points. Therefore, by measuring the length of the line segment that connects the two points, the distance between them can be determined. If (a, b) and (c, d) be two points, then the distance between them is [tex]\sqrt[]{(b - a)^{2} +(d- c)^{2} }[/tex].
Here, one point is (-2, 3).
Let the other point be (x, 1).
Given that the distance is √13.
Now, [tex]\sqrt[]{(x - (-2))^{2} +(1 - 3)^{2} } = \sqrt{13}[/tex]
i.e. [tex]\sqrt[]{(x + 2)^{2} +( - 2)^{2} } =\sqrt{13}[/tex]
i.e. [tex]\sqrt[]{x^{2}+4x +4 +4 }=\sqrt{13}[/tex]
i.e. [tex]x^{2}+4x +8 =13[/tex]
i.e. [tex]x^{2}+4x + 8- 13=0[/tex]
i.e.[tex]x^{2}+4x -5=0[/tex]
i.e. [tex]x^{2} +5x - x -5=0[/tex]
i.e. [tex]x(x+5)-1(x+5)=0[/tex]
i.e. [tex](x+5)(x-1)=0[/tex]
i.e. [tex]x=-5,1[/tex]
So, the point is either (-5, 1) or (1, 1).
Therefore, the required point is either (-5, 1) or (1, 1).
i.e. the distance between (-2, 3) and (-5, 1) is √13.
or, the distance between (-2, 3) and (1, 1) is √13.
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A fair die is rolled 1200 times. Find the approximate probability that the sum falls between
5000 and 6300. Clearly indicate any theorems that you are using / where any rounding has been done.
Your final answer should be accurate to two decimal places.
Answers
Answer:
abcdefghijklmnopqrstuvwxyz
Step-by-step explanation:
these are the alphabets
Find the demand function for the marginal revenue function. Recall that if no items are sold, the revenue is 0.
R'(x) = 599 - 0.21sqrt(x)
What is the integral that is needed to solve the problem and the demand function?
Answers
The integral that is needed to solve the demand function is R(x) = 599x - 0.14[tex]x^{3/2}[/tex]
What is Demand Function?A demand function describes the mathematical relationship between the quantity demanded and one or more determinants of the demand, as the price of the good or service, the price of complementary and substitute goods, disposable income, etc.
Here, given differential equation;
R'(x) = 599 - 0.21[tex]\sqrt{x}[/tex]
we can also write this as;
[tex]\frac{d}{dx}R(x) = 599 - 0.21\sqrt{x}[/tex]
[tex]d R(x) = (599 - 0.21\sqrt{x} ) dx[/tex]
On integrating both sides, we get
[tex]\int\ d R(x) = \int\ (599 - 0.21\sqrt{x} ) dx[/tex]
R(x) = [tex]599x - 0.21 X \frac{2}{3}x^{3/2}[/tex] + C
R(x) = 599x - 0.14[tex]x^{3/2}[/tex] + C ...........(i)
Also given, at x = 0, R(x) = 0, Put these values in equation (i), we get
0 = 0 - 0 + C
C = 0
Put the value of C in equation (i), we get
R(x) = 599x - 0.14[tex]x^{3/2}[/tex]
Thus, the integral that is needed to solve the demand function is R(x) = 599x - 0.14[tex]x^{3/2}[/tex]
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The distance, y, in miles, traveled by a car for a certain amount of time, x, in hours, is shown in the graph below: A graph titled Motion of a Car is shown with Time in hours labeled on x-axis and Distance from Starting Point in miles labeled on y-axis. The scale on the x-axis shows the numbers 1, 2, 3, 4, 5, 6, and the scale on the y-axis shows the numbers 0, 12, 24, 36, 48, 60, 72. There are three straight lines in the graph. The first line joins ordered pair 0, 0 with 1, 12. The second straight line joins 1,12 and 2,12 and the third straight line joins ordered pair 2,12 with the ordered pair 5,36. Which of the following best describes the motion of the car shown? It travels for 1 hour, then stops for 1 hour, and finally travels again for 3 hours. It travels for 2 hours, then stops for 1 hour, and finally travels again for 2 hours. It travels for 1 hour, then stops for 2 hours, and finally travels again for 5 hours. It travels for 2 hours, then stops for 3 hours, and finally travels again for 5 hour
Answers
The best description of the car's motion as shown in the graph is: A. It travels for 1 hour, then stops for 1 hour, and finally travels again for 3 hours.
How to Analyze a Distance-Time Graph?In a distance-time graph, an horizontal line implies no distance was covered within that time frame, meaning there was a stop.
Thus, in the graph given, the stop occurred 1 and 2, which is equivalent to an hour. From 2 to 5 on the x-axis means there was movement for up to 3 hours.
Therefore, the best description is: A. It travels for 1 hour, then stops for 1 hour, and finally travels again for 3 hours.
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can i have the x,y,z for this question? THanks!
Answers
Answer:
x=4, y=-2, p=14
Step-by-step explanation:
First simplify the LHS:
[tex]LHS = (16p^8)^\frac{3}{2} \times (216p^{-3})^{-\frac{2}{3}}[/tex]
[tex]= 64 p^{12} \times \frac{1}{36} p^{2}[/tex]
[tex]=\frac{16}{9}p^{14}[/tex]
[tex]=\frac{2^4}{3^2}p^14\\=2^43^{-2}p^{14}[/tex]
Hence x=4, y=2, p=14
2. Marjorie wants to subdivide a rectangular plot of land measuring 600 m by 720 m
into equal square lots. What is the side length of the largest possible square lot
she can use? Show the prime factorization results to support your answer.
Answers
the largest length of the squares will be 120m.
What is the side length of the largest possible square lot she can use?
The largest possible length will be equal to the greatest common factor between the dimensions of the rectangular plot, so we need to find the GCF between 600 and 720.
If we decompose both numbers, we get:
600 = 2*2*2*3*5*5
720 = 2*2*2*2*3*3*5
Then the greatest common factor is: (2*2*2*3*5) = 120
So the largest length of the squares will be 120m.
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Let [tex]K[/tex] be a circle with centre [tex]O[/tex]. A second circle [tex]L[/tex] passes through [tex]O[/tex] and cuts [tex]K[/tex] at two other points [tex]A[/tex] and [tex]B[/tex]. Let [tex]P[/tex] be a point on arc [tex]AOB[/tex] and extend line segment [tex]BP[/tex] to intersect [tex]K[/tex] at a second point [tex]Q[/tex]. Prove that the bisector of angle [tex]APQ[/tex] passes through [tex]O[/tex].
Answers
The bisector of angle APQ passes through O and this is illustrated below.
How to illustrate the information?From the information given, the center is O. and the circle passes through O and cuts at K.
In this case, it should be noted that the circles are equal according to the SAS test.
Here, AOB + APQ = 180° (Linear pair)
2AOB = 180
AOB = 90.
Therefore, the bisector of angle APQ passes through O.
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Joyce has as much money as George; then they bet 5 cents each and George lost. If, after the bet, George has x cents, how much does Joyce have ?
Answers
Answer:
x+10
Step-by-step explanation:
So they did a bet, and placed 5 cents each. George lost the bet that gives Joyce 5 points plus 5 more that make 10.
After the bet George has: x cents
Before the bet George had: x + 5 cents
Before the bet Joyce had also x + 5 cents
Since Joyce won 5 cents, after the bet she has x + 5 + 5 = x + 10 cents.
Write a proportion for the statement.
40 is to 10 as 32 is to 8.
Answers
Hi! ⋇
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All proportions have this form:
[tex]\sf{\dfrac{a}{b}=\dfrac{c}{d}}[/tex], Where [tex]\sf{\dfrac{a}{b}}[/tex] is equal to [tex]\sf{\dfrac{c}{d}}[/tex].
If [tex]\sf{\dfrac{a}{b}\neq\dfrac{c}{d}}[/tex], it's not a proportion.
_________________________
Here we have two pairs of numbers:
40,10 and 32,8.
Written As a proportion, they look like :
[tex]\sf{\dfrac{40}{10}=\dfrac{32}{8}}[/tex]
Hope this made sense to you :)
[tex]\it{Calligrxphy}[/tex]
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What is the equation of the parabola? coordinate plane with a parabola facing up with vertex at 0 comma 2, the point 0 comma 5 and a horizontal line going through 0 comma negative 1 y = −one twelfthx2 − 2 y = one twelfthx2 − 2 y = −one twelfthx2 2 y = one twelfthx2 2
Answers
The equation of the parabola is y = 2/25x^2 + 2
How to determine the parabola equation?The complete question is in the image
The given parameters are:
Vertex, (h, k) = (0,2)
Point (x,y) = (-5, 4)
The parabola is represented as;
y = a(x - h)^2 + k
Substitute (h, k) = (0,2)
y = a(x - 0)^2 + 2
This gives
y = ax^2 + 2
Substitute (x, y) = (-5,4)
4 = a(-5)^2 + 2
This gives
4 = 25a + 2
Subtract 2 from both sides
25a = 2
Divide by 25
a = 2/25
Substitute a = 2/25 in y = ax^2 + 2
y = 2/25x^2 + 2
Hence, the equation of the parabola is y = 2/25x^2 + 2
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Help me with this please :V
Answers
Answer:
refer to the above attachment
Which composition of transformations maps figure EFGH to figure E"F"G"H"?
a reflection across line k followed by a translation down
a translation down followed by a reflection across line k
a 180° rotation about point G followed by a translation to the right
a translation to the right followed by a 180° rotation about point G
Answers
The composition of transformations maps figure EFGH to figure E"F"G"H" is Option A.a reflection across line k followed by a translation down.
What is the transformations rule that was used here?A transformation is a rule that is used to manipulate the position of a point of geometric figure.
And from the figure, we can see that a translation is used, because of the movement of EFGH to figure E"F"G"H" which occurred in the same direction.
Hence, composition of transformations maps figure EFGH to figure E"F"G"H" is Option A.a reflection across line k followed by a translation down.
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The diameter of a sphere is
a chord that passes through the center of the sphere.
a fixed point equidistant from all points on the surface of the sphere.
a line segment from the center point to a point on the sphere.
a three-dimensional circle in which all points are equidistant from a fixed point.
Answers
The diameter of a sphere is a chord that passes through the centre of the sphere.
We need to complete the given statement.
What is the diameter of the sphere?The diameter of a sphere is the maximum distance between two antipodal points on the surface of the sphere. If is the radius of a circle or sphere, then. The ratio of the circumference of a circle or great circle of a sphere to the diameter.
A chord of a sphere is a segment whose endpoints are on the sphere. A diameter is a chord that contains the centre.
Therefore, the diameter of a sphere is a chord that passes through the centre of the sphere.
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find x and y
pls help
Answers
See diagram
To find X, I know the angles will equal 180
110 + 5x = 180
Subtract 110 from both sides to isolate X
5x = 70, solve for X, divide both sides by 5
X = 14
Do the same for Y
We know the triangle will equal 180
So (5x) is (5x14= 70) and the other angle is 40.
180-40-70 = 70 for the unknown angle in the triangle.
We know the angles in the Y part of the question will equal 180 so
70 + 2y = 180, subtract 70 from both sides to isolate Y
2y = 110, now solve for Y by dividing both sides by 2
Y = 55
Angles on straight line = 180
110 + 5x = 180 degrees
-110
5x. = 70
÷ 5
x. = 14
5 x 14 = 70
Angles in triangle = 180
70 + 40 = 110
So, the missing angle inside the triangle is 70
Now to find y, use angles on straight line = 180
2y + 70 = 180
- 70
2y = 110
÷2
y = 55 degrees
Hope this helps!
The term "freshman 15" refers to the claim that college students typically gain 15lbs during freshman year at college. Assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of 2.9 lb and a standard deviation of 10.4 lb. Find the probability that a randomly selected male college student gains 15 lb or more during their freshman year. What does the result suggest about the claim of the "freshman 15"?
Answers
The probability that a randomly selected male college student gains 15 lb or more during their freshman year is 11.6%
What is Probability ?Probability is defined as the likeliness of an event to happen.
Let X be a random variable that shows the term "freshman 15" that claims that students typically gain 15lb during their freshman year at college.
It is given that
X follows is a normal distribution with a mean of 2.1 lb (μ) and a standard deviation (σ) of 10.8 lb.
Population Mean (μ) = 2.1
Population Standard Deviation (σ) = 10.8
We need to compute Pr(X≥15). The corresponding z-value needed to be computed is:
[tex]\rm Z_{lower} = \dfrac{ X_1 -\mu }{\sigma}\\\\Z_{lower} = \dfrac{ 15-2.1 }{10.8}\\\\\\Z_{lower} = 1.19[/tex]
Then the probability is given as
[tex]\rm Pr(X \geq 16 ) = Pr (\dfrac{X -21}{10.8} \geq \dfrac{15-21}{10.8})\\\\= Pr (Z \geq \dfrac{15-2.1}{10.8}\\\\= Pr (Z\geq 1.19)\\\\ = 0.1162[/tex]
Pr(X≥15)=0.1162. (11.6%)
The probability that a randomly selected male college student gains 15 lb or more during their freshman year is 11.6%
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Tony makes a table to use as a quick reference guide:
Cookies baked (y) is a function of number of batches (x).
Use the table tool to answer the question.
How many cookies will Tony have if he bakes 10 batches more than the maximum number of batches in the table?
Enter the correct answer.
Answers
Tony will have 486 cookies if he bakes 10 batches more.
Take two points from table: (8, 234), (9, 252)
Find slope:
[tex]slope: \dfrac{y_2 - y_1}{x_2- x_1} \ \ where \ (x_1 , \ y_1), ( x_2 , \ y_2) \ are \ points[/tex]
[tex]\rightarrow slope \ (m) : \dfrac{252-234}{9-8} = 18[/tex]
This determines that one batch of cookies has 18 cookies.
So, if he bakes 10 more batches:
(10 × 18) = 180 cookies
Total cookies he then will have:
306 cookies + 180 cookies = 486 cookies.
y = x² + 3
y = x + 5
Answers
Answer:
x^2-x-2=0
Step-by-step explanation:
x^2+3=x+5
x^2-x-2=0
What is the slope of the line that passes through the points (-6, -6) and (-9,-5)? Write your answer in simplest form. What is the slope of the line that passes through the points ( -6 , -6 ) and ( -9 , -5)?
Answers
Answer:
[tex]y=-\frac{1}{3}x -8[/tex]
Step-by-step explanation:
1) Write down the standard form of an equation of a line and the formula to find m.
[tex]y=mx+b[/tex]
[tex]m=\frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex]
2) Find m using the formula then substitute it into [tex]y=mx+b[/tex].
[tex]m=\frac{-5 - (-6) }{-9 - (-6)}\\m=\frac{-5 +6 }{-9 +6}\\m=\frac{1}{-3} \\m=-\frac{1}{3}[/tex]
So, [tex]y=-\frac{1}{3}x +b[/tex].
3) Find b by substituting a pair of coordinates the line goes through.
[tex]-6=-\frac{1}{3}(-6) +b\\-6=2+b\\-6 - 2=b\\-8=b[/tex]
4) Substitute b into [tex]y=-\frac{1}{3}x +b[/tex] to get your final answer.
[tex]y=-\frac{1}{3}x -8[/tex]
Need answer & explanation! Please & thank you!
Answers
Answer: the correct is x=1,
Step-by-step explanation:
Graph each side of the equation. The solution is the x-value of the point of intersection.
x=1
Answer:
1
Step-by-step explanation:
you can plug it in and it gives you 3-1 which results in 2 as the question asks for
Add: - 14 + (-12) + 4
Answers
Answer:
-22
Step-by-step explanation:
-14+(-12）+4
=-14-12+4
=-26+4
=-22
John missed 4 problems on a test but did 75% of them correctly. How many problems
were there in the test?
Answers
Answer:
16
Step-by-step explanation:
he missed 25%
if 4 = 25% then 16 questions total
4*4=16 and 25% * 4 = 100%
Answer: 16 problems
Step-by-step explanation:
First find out how many John got wrong. 100%-75%=25%. Then do cross multiplication: 4*100=25x; 400=25x. Divide 25 on both sides and... BOOM! you get 16 problems.
What's the sum Sn of the first n terms of this given sequence?
1, 1+2, 1+2+3, 1+2+3+4, ...
The kth term is 1+2+3+ ... +k =
(I also attached a picture of the question. Thanks!)
Answers
Answer: [tex]\frac{k(k+1)}{2}[/tex]
Step-by-step explanation:
This is a standard formula.
Pls help! Geometry
What is the total cost of material for bunny costumes?
Answers
Answer: $201.81
Step-by-step explanation:
The total amount of faux fur required is [tex]6(3.1)=18.6[/tex] square yards.
Multiplying this by $10.85 per square yard, we get the total cost is [tex](18.6)(10.85)=\$201.81[/tex]
What is the equation of the line AB, if it passes through (-4,0), and is parallel to the line
-x - 4y = 1?
Answers
Rewriting the equation of the given line in slope-intercept form,
[tex]-x-4y=1\\\\-4y=x+1\\\\4y=-x-1\\\\y=-\frac{1}{4}x-\frac{1}{4}[/tex]
From this, we know the slope of the given line is -1/4.
Since parallel lines have the same slope, we know the line whose equation we want to find has a slope of -1/4.
Substituting into point-slope form,
[tex]y=-\frac{1}{4}(x+4)\\\\\boxed{y=-\frac{1}{4}x-1}[/tex]
Determine if the given ordered triple is a solution of the system.
(4, -5, -4)
4x + 2y + z = 2
5x - 4y - z = 44
3x + y + 4z = -9
Answers
Answer:
It is.
Step-by-step explanation:
(x, y, z) => (4, -5, -4)
substitute 4 in x, -5 in y & -4 in z for all the equations in the system.
4x + 2y + z = 2
4(4) + 2(-5) - 4 = 2
2 = 2
5x - 4y - z = 44
5(4) - 4(-5) - (-4) = 44
44 = 44
3x + y + 4z = -9
3(4) + (-5) + 4(-4) = -9
-9 = -9
All equations therefore agree with the ordered triple.
A system of three linear equations in three variables is inconsistent. how many solutions to the system exist? none one three infinitely many
Answers
None solutions to the system exist.
Given statement: A system of three linear equations in three variables is inconsistent.
Concept: If a, b, c and r are real numbers (and if a, b, and c are not all equal to 0) then ax + by + cz = r is called a linear equation in three variables. (The “three variables” are the x, the y, and the z.) The numbers a, b, and c are called the coefficients of the equation.
We know that if a system has no solution, then it is known as inconsistent.
[If there are one, two, or an infinite number of solutions to the system, then it is called consistent.]
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HELP ME! what is the measure of the supplement of each angle? 165°
Answers
Answer:
15
Step-by-step explanation:
Supplementary angles add to 180 degrees
Let the supplement be x
x+165 = 180
x +165-165= 180-165
x = 15
The supplement is 15
Answer:
Hello! The answer is 15°.
Step-by-step explanation:
Supplementary angles add up to 180°.
To find the measure of each angle, we can now set up an equation because we have the total that supplementary angles add up to:
x + 165 = 180
Now, we can subtract 165 from both sides to get x by itself:
x + 165 = 180
-165 -165
This leaves us with:
x = 15
please help me in this :")
Answers
The numbers of runners that are 50 and above are 3000.
How to find the runners that are 50 and above in the pie chart?Using the pie chart,
If there are 1500 runners that are under 20, therefore,
1500 = 10 / 100 × x
where
x = total number of runners
10x = 150000
x = 15000
Therefore,
the number of runners that are 50 and above = 20 / 100 × 15000 = 3000
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PLEASE HELP
a marine biologist tags 50 fish at lake ness and releases them. five days later, he captures 75 fish and finds that 3 of them are tagged. assuming the population of fish has remained constant over the five days and that this sample is an accurate representation of the portion of the fish in the lake that are tagged, how many fish are in the lake?
Answers
which graph shows the system of inequalities y<-1/3x+1 and y≤2x-3
Answers
The graph of the system of inequalities can be seen below.
Which graph shows the solution for the system?
Here we have the inequalities:
y < -(1/3)*x + 1
y ≤2x-3
To graph the first inequality, we need to graph the line -(1/3)*x + 1 with a dashed line (because the symbol < was used) and shade the region below that line.
For the second inequality we need to graph the line 2x-3 with a solid line this time, and again, shade the region below the line.
The graph of the system is shown in the image below.
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