whenever you have an equation with one variable on both sides, you might end up with one of the following case:
1. You might end up with an inconsistent equation, therefore there is not solution. Example
x = x+3. If you subtract x on both sides, you get 0=3 which is impossible.
2. You might end up having infinite solutions. Example:
x+6 = x+6. If you subtract x on both sides you get 6 =6 which is true regardless of the value of x.
3. You might end up having only one solution. Example:
3x +2 = x +5
If you subtract x on both sides and subtract 2 on both sides, you get
2x = 3.
then if you divide by two on both sides, you find
x = 3/2. So this equation has one unique solution.
Royal Furniture bought a sofa for $820. The sofa had a $1,420 list price. What was the trade discount rate Royal received? (Round your answer to the nearest hundredth percent.)
The received trade discount amount is $600 and percentage is 42.25%.
The trade discount in amount is = Original price - New price
Keep the values in formula to find the trade discount
The trade discount in amount is = 1420 - 820
Performing subtraction on Right Hand Side of the equation
The trade discount amount is = $600
Finding the trade discount in percentage
Trade discount in percentage = discount ÷ original price × 100
Keep the values in formula to find the trade discount in percentage
Trade discount in percentage = (600/1420)× 100
Performing multiplication and division
Trade discount in percentage = 42.25%
The trade discount is $600 and 42.25%.
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I don't understand how to write the equations using this diagram.
Given the segment on the picture, we can write the following 3 equations:
[tex]\begin{gathered} PR=PQ+QR \\ QS=QR+RS \\ PT=PR+RT \end{gathered}[/tex]Suppose you want to have $700,000 for retirement in 30 years. Your account earns 5% interest. How much would you need to deposit in the account each month?$_________________
The formula for a final amount A by depositing a regular amount R each month for n years with a interest rate r is given by:
[tex]A=R\frac{\lbrack(1+\frac{r}{12})^{^{12n}}-1\rbrack}{\frac{r}{12}}[/tex]For A = $700,000, r = 0.05, n = 30, we have:
[tex]\begin{gathered} 700000=R\cdot\frac{\lbrack(1+\frac{0.05}{12})^{12\cdot30}-1\rbrack}{\frac{0.05}{12}} \\ 700000=832.26R \\ R=\text{ \$841.08} \end{gathered}[/tex]Gretchen runs backwards at a speed of -4m/s. Mika continues running at a speed of 3 m/a. If Gretchen dropped her baton-48 meters from her current position how far ahead will Mika be?
The distance by which Mika will be ahead is equal to 36 m from present location.
According to the question Gretchen and Mika are at the same point when Gretchen runs backward at the speed of 4 m/s in opposite direction to pick her baton and Mika runs in the direction of the race at the speed of 3 m/s. We need to find the distance covered by Mika form the current location in the time when Gretchen picks her baton up. Let us assume that the time taken by Gretchen to pick up her baton be equal to t. It is given that 48 meters from her current location she dropped her baton. Since she is running at the speed of 4 m/s then the time will be equal to t = 48/4 = 12 seconds. Now, Gretchen will be able to pick up the baton in 12 seconds. Thus, the distance covered by Mika in 12 seconds will be equal to 3×12 = 36m.
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Complete Question:
Gretchen and Mika are involved in a relay race. Both runners are at the same point on the course when Gretchen realizes that she dropped her baton and has to turn around to retrieve it. Gretchen runs backwards at a speed of −4 m/s, relative to the direction of the race. Mika continues running at a speed of 3 m/s. If Gretchen dropped her baton −48 meters from her current location, relative to the direction of the race, how far ahead will Mika be when Gretchen picks her baton up?
second drop-down options are can not be proven congruent and can be proven congruent
Solution
For this case the answer is:
Side-Angle-Side (SAS)
Since we have two sides congruent and also one angle
Consider the parabola given by the equation:
f
(
x
)
=
−
2
x
2
−
12
x
−
9
Find the following for this parabola:
A) The vertex:
B) The vertical intercept is the point
C) Find the coordinates of the two
x
intercepts of the parabola and write them as a list of points of form (x, y) separated by commas:
It is OK to round your value(s) to to two decimal places.
The features of the given parabola are as follows:
a) Vertex: (3,9).
b) Vertical intercept: y = -9.
c) x-intercepts: (-5.12, 0) and (-0.88, 0).
Features of the quadratic functionThe equation of the parabola is given by:
f(x) = -2x² - 12x - 9.
Hence the coefficients are:
a = -2, b = -12, c = -9.
The formula for the coordinates of the vertex, considering the coefficients of a quadratic equation, is given as follows:
x = -b/2a.y = -(b² - 4ac)/(4a).The coordinates of the vertex are given as follows:
[tex]x_v = -\frac{-12}{2(-4)} = -3[/tex]
[tex]y_v = -\frac{(-12)^2 - 4(-2)(-9)}{4(-2)} = 9[/tex]
Hence (3,9).
The vertical intercept is given as follows:
f(0) = -2(0)² - 12(0) - 9 = -9.
The roots are given as follows:
[tex]x_{1} = \frac{-b + \sqrt{b^2 - 4ac}}{2a} = \frac{12 + \sqrt{(-12)^2 - 4(-2)(-9)}}{2(-2)} = -5.12[/tex][tex]x_{2} = \frac{-b - \sqrt{b^2 - 4ac}}{2a} = \frac{12 - \sqrt{(-12)^2 - 4(-2)(-9)}}{2(-2)} = -0.88[/tex]Hence the points are:
(-5.12, 0) and (-0.88, 0).
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3. John opens a checking account with $100. He writes a check for $110. The back
charges an overdrawn fee of $25. What is his account balance?
John's account balance after considering the withdrawal and the overdrawn fee is -$35
What does it mean for an account to be overdrawn?
An account is overdrawn if the money withdrawn from it is more than the account balance, in this scenario, the check presented whose value is $110 is more than the account balance, which was the account by which the account was opened initially, hence, John's account can be said to be overdrawn.
Account balance=initial balance-amount withdrawn-overdrawn fee
Note overdrawn fee and amount withdrawn would be shown as negative values because they are obligations against the account balance
initial balance=$100
amount withdrawn=$110
overdrawn fee=$25
Account balance=$100-$110-$25
Account balance=-$35
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Points B and C lie on a circle with center O and a radius of 15 units. If the length of arc BC is 21 units, what is m *BOC in radians?
Given:
[tex]\begin{gathered} r=15\text{ }units \\ \hat{BC}=21\pi\text{ }units \end{gathered}[/tex]To find:
The angle BOC.
Explanation:
Using the length of the arc formula,
[tex]\begin{gathered} l=r\theta \\ 21\pi=15\theta \\ \theta=\frac{21\pi}{15} \\ \theta=\frac{7}{5}\pi \end{gathered}[/tex]The angle is
[tex]\angle BOC=\frac{7}{5}\pi[/tex]Final answer:
The angle is
[tex]\angle BOC=\frac{7}{5}\pi[/tex]A box contains different-colored marbles. If P(blue) = P(green) = and P(blue and green) = whic
statement is true?
O The events are independent because P(blue) P(green) = P(blue and green).
Answer:
p(blue and green)
Step-by-step explanation:
the events are independent because p(blue) p(green)=p(blue and green).
Sammy has $125 saved from his summer lifeguarding job. Sammy has to take the bus to get to school this semester. Each week, Sammy uses $10.50 for bus fare. A. Function s represents the remaining amount, in dollars, in his savings account, as a function of the number of weeks, w, since school started. Complete the table: 6 weeks = 10 weeks = w = = is the dollars remaining in savings B. Use function notation to write a rule that defines function s C. Graph function s on a coordinate plane. D. Solve the equation s(w) = 41 and explain what the solution means.
The solution to the equations are
Equation: s = 125 - 10.5wTableWeeks (w) Amount remaining
6 62
10 20
See attachment for graphFunction notation: s(w) = 125 - 10.5ws(w) = 41 means that the amount remaining after 8 weeks is $41How to determine the equation of the scenario?From the question, we have the following parameters
Savings = $125
Bus fare = $10.5
The equation of the amount remaining is represented as
Amount remaining = Savings - Bus fare x Number of weeks
Substitute the known values in the above equation
So, we have the following equation
s = 125 - 10.5w
Where w represents the number of weeks
Also, we have
w = 6 and w = 10
So, we have
s = 125 - 10.5 * 6 = 62
s = 125 - 10.5 * 10 = 20
So, the table is
Weeks (w) Amount remaining
6 62
10 20
The function notationIn (a), we have
s = 125 - 10.5w
Express as a function notation
s(w) = 125 - 10.5w
The graph of the functionSee attachment for the graph
The solution and the interpretation of s(w) = 41We have
s(w) = 125 - 10.5w
This gives
125 - 10.5w = 41
So, we have
10.5w = 84
Evaluate
w = 8
This means that the amount remaining after 8 weeks is $41
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Hi I think I am doing this wrong. If I can get an explanation that would help a bunch!
Investment = $5,000
Price of the product or sale price = $3.49
Production cost = $2.16
Profit = 3.49 - 2.16 = $1.33
We need to sell X items to get our investment back, therefore
X * Profit = Investment
1.33 * X = 5,000
X = 5,000 / 1.33 =3759.4
Therefore you need to sell at least 3760 articles to break even
Additionally, if you want to make a $10,000 profit, then you need to sell those items plus Y more or in other words, you want to sell as many items to get your $5,000 investment plus $10,000
This is...
5,000 + 10,000 = K * (1.33)
15,000 = 1.33 * K
K = 15,000 / 1.33 = 11,278.2 ;
Therefore, you need to sell 11,279 items
Find the value of x:a) 2+3−6=−5b) 2(2+1)=5−1
In order to find the value of x in the expression, we need to isolate the terms with the variable 'x' in one side of the equation, and the variables without 'x' in the other side.
Then, we divide the whole equation by the number multiplying 'x', so we have just 'x' in one side of the equation.
So we have that:
a)
[tex]\begin{gathered} 2+3x-6x=-5x \\ 3x-6x+5x=-2 \\ 2x=-2 \\ x=\frac{-2}{2}=-1 \end{gathered}[/tex]So the value of x is -1.
b)
[tex]\begin{gathered} 2\left(2+1\right)=5-1 \\ 4x+2=5x-1 \\ 4x-5x=-1-2 \\ -x=-3 \\ x=3 \end{gathered}[/tex]So the value of x is 3
y = 3x – 5 y=-x-5Parallel, perpendicular or neither ?
The given equations are-
[tex]\begin{gathered} y=3x-5 \\ y=-x-5 \end{gathered}[/tex]These equations represent straight lines, where the coefficients of x are the slope of each one.
When two lines are parallel, they have the same slope, which is not the case here because the first one has a slope of 3, and the second one has a slope of -1.
Additionally, if two lines are perpendicular, their slopes are totally opposite in number and sign, which is not the case here because 3 is not opposite to -1.
Therefore, the answer is neig
A machine at the candy factory dispensed different numbers of lemon-flavored candies into various bags. Number of lemon-flavored candies Number of bags 29 1 69 4 108 5 4 118 5 162 4 167 2 186 X is the number of lemon-flavored candies that a randomly chosen bag had. What is the expected va of X? Write your answer as a decimal.
In order to calculate the expected value of x, we just need to divide the total number of lemon-flavored candies by the total number of bags, so we have:
[tex]\begin{gathered} E(x)=\frac{29\cdot1+69\cdot4+108\cdot5+118\cdot4+162\cdot5+167\cdot4+186\cdot2}{1+4+5+4+5+4+2} \\ E(x)=\frac{3167}{25} \\ E(x)=126.68 \end{gathered}[/tex]So the expected value of x is 126.68.
The Pederson family needed to cut expenses on their cell and landline phone service, cable television service, and Internet service. The best prices they found for basicservice, using different service providers, are listed below. They found a company with a package deal with all 4 services for $143.66. What is the percentage decrease bypurchasing the package deal? Cell Phone Service: $34.15 per month Landline Phone Service: $31.36 per month Cable TV Service: $56.25 per month Internet Service:$41.48 per month9.7%10.9%11.4%12.0%None of these choices are correct.
From the information given, the original costs per month were
Cell phone service $34.15
Landline phone service = $31.36
Cable TV service = $56.25
Internet service = 41.48
The total cost per month would be
34.15 + 31.36 + 56.25 + 41.48 = 163.24
For the new company that they found, the total cost is $1431
Hi, can you help me to solve this problem, please !!!
the y intercept of the curve of the function is 0.
y intercept is the point at which the curve of the function intersects the y axis.
here,
the curve intersects the Y axis at the origin.
coordinates of the origin are:
(0, 0)
So, the y intercept will be:
y = 0
Therefore, we get that, the y intercept of the curve of the function is 0.
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The rent in an urban neighborhood is normally distributed. The mean is $650 per month with a standarddeviation of $50. Estimate the percent of apartment residents who pay from $600 to $750 per month.Show Your Work
Solution
The z-score, also referred to as standard score, z-value, and normal score, among other things, is a dimensionless quantity that is used to indicate the signed, fractional, number of standard deviations by which an event is above the mean value being measured. Values above the mean have positive z-scores, while values below the mean have negative z-scores.
Mean is $650
Standard deviation = $50
[tex]\begin{gathered} \mu=650 \\ \sigma=50 \end{gathered}[/tex][tex]Z=\frac{x-\mu}{\sigma}=\frac{600-650}{50}=-\frac{50}{50}=-1[/tex][tex]Z=\frac{x-\mu}{\sigma}=\frac{750-650}{50}=\frac{100}{50}=2[/tex][tex]P(-1Therefore the estimated percent of apartment residents = 81.86%
A barrel of oil is filled at a constant rate of 7.5 gal/min. The barrel had 12 gallons before filling began. Write the equation in standard form to model the linear situation
Lin knows that angle A is congruent to angle D. She claims that triangles ABC and DEF must be similar. Convince Lin that the triangles are not similar.
The triangles are not similar.
Similar triangles are what?Triangles that are comparable to one another are those that have an equal number of corresponding sides and corresponding angles.
If the respective sides of two triangles are proportional and the corresponding angles are congruent, then the triangles are said to be comparable. To put it another way, similar triangles have the same shape but may differ in size. In addition, if the corresponding sides of the triangles are identical in length, the triangles are congruent.
Here, the ratio of the sides are not similar
AB/AC= 4/10 = 2/5
DE/DF= 8/19
Hence, 2/5≠8/19
therefore, the triangles are not similar.
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Which inequality is equivalent to this one?y-85-2y-8+82-2+ 8y-8+8 <-2+8y-8+25-2+8y-8+25-2+2
Explanation
[tex]y+8=2[/tex]solve for y :
The subtraction property of equality tells us that if we subtract from one side of an equation, we also must subtract from the other side of the equation to keep the equation the same
hence,
subtract 8 in both sides
[tex]\begin{gathered} y+8=2 \\ y+8-8=2-8 \\ y=-6 \end{gathered}[/tex]so, the answer is
y=-6
I hope this helps you
In a triangle ABC, a line BP is drawn from B so that P lies on the side AC. For the triangle, AP=27mm, BP=30mm, BC=82mm and the angle APB=76∘ apply. Determine the area of triangle ABC
SOLUTION
The diagram for this is shown below
From the diagram above, considering triangle BPC, let us find angle C
Using sine rule, we have
[tex]\begin{gathered} \frac{sinC\degree}{30}=\frac{sin104\degree}{82} \\ sinC=\frac{30\times sin104\degree}{82} \\ C=sin^{-1}\frac{30sin104}{82} \\ C=20.79\degree \end{gathered}[/tex]From the same triangle BPC to get angle B, we have
[tex]\begin{gathered} B+P+C=180\degree \\ B+104+20.79=180 \\ B+124.79=180 \\ B=55.21\degree \end{gathered}[/tex]From the same triangle BPC, using sine rule to get the side PC, which I called x, we have
[tex]\begin{gathered} \frac{sinB}{x}=\frac{sinP}{82} \\ \frac{sin55.21\degree}{x}=\frac{sin104\degree}{82} \\ x=\frac{sin55.21\times82}{sin104\degree} \\ x=69.404mm \end{gathered}[/tex]This makes the side AC to become
[tex]27+69.404=96.404mm[/tex]So, to get the area of the triangle ABC, we have that
[tex]Area=\frac{1}{2}|AC|\times|BC|\times sinC[/tex]Applying we have
[tex]\begin{gathered} Area=\frac{1}{2}|96.404|\times|82|\times sin20.79\degree \\ =1402.94mm^2 \end{gathered}[/tex]Hence the answer is approximately 1402.94 square-millimeters to the nearest hundredth
write the equation of the line that goes through points (1, 1) and (3, 7)
Given the points:
(x1, y1) ==> (1, 1)
(x2, y2) ==> (3, 7)
To find an equation of the line that goes through the points, first find the slope using the slope formula below:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]Substitue values into the formula and solve for the slope, m:
[tex]\begin{gathered} m=\frac{7-1}{3-1} \\ \\ m=\frac{6}{2} \\ \\ m=3 \end{gathered}[/tex]The slope of the line, m is = 3.
Now, use the point slope form:
(y - y1) = m(x - x1)
Substitute 1 for y1, 1 for x1, and 3 for m:
(y - 1) = 3(x - 1)
Let's rewrite the equation to slope intercept form: y = mx + b
Where m is the slope and b is the y-intercept
y - 1 = 3(x) + 3(-1)
y - 1 = 3x - 3
Add 1 to both sides:
y - 1 + 1 = 3x - 3 + 1
y + 0 = 3x - 2
y = 3x - 2
Therefore, the equation of the line in slope intercept form is:
y = 3x - 2
ANSWER:
y = 3x - 2
1 point2. Consider the inequality below. Which value of x is a solution to theinequality?*2x + 39O x=-10х+13x = -60 x = 0O x = 6
We will solve as follows:
[tex]\frac{2x+3}{9}>\frac{x}{3}+1\Rightarrow\frac{2}{9}x+\frac{1}{3}>\frac{x}{3}+1[/tex][tex]\Rightarrow\frac{2}{9}x-\frac{x}{3}>1-\frac{1}{3}\Rightarrow-\frac{x}{9}>\frac{2}{3}[/tex][tex]\Rightarrow x<-6[/tex]So, the value that is a solution for the inequality is:
[tex]x=-10[/tex][Option A]
Purchases of streamed music can be modeled by f (t) = 3.7 + 0.84 ln(t +1), where f (t) is measured in billions of dollars and t is measured in years, with t=0 corresponding to 2012.
a. What was the amount spent by consumers on streamed music in 2014? Use the correct units.
b. How fast was the amount spent by consumers on streamed music changing in 2014?
Use the correct units.
If the purchases of streamed music can be modeled by function f(t) = 3.7+0.84 ln(t+1)
Part a
The amount spent by consumers on streamed music in 2014 is 4.62 billion
Part b
The amount spent by consumers on streamed music changing in 2014 at the rate of 0.28 billions per year
The purchases of streamed music model
f(t) = 3.7+0.84 ln(t+1)
Where f(t) is measure in billion of dollar
t is the measured in years
Part a
We have to find the amount spent by consumer on streamed music in 2014.
t = 0 corresponding to the year 2012
Then 2014 corresponding to t = 2
Substitute the value of t in the equation
f(2) = 3.7+0.84 ln(2+1)
f(2) = 3.7+0.922
f(2) = 4.62 billion
Part b
Differentiate the model with respect to x
f'(t) = 0.84×[tex]\frac{1}{t+1}[/tex]
t = 2
Substitute the value in the equation
f'(2) = 0.84×[tex]\frac{1}{2+1}[/tex]
f'(2) = 0.28 billions per year
Hence, if the purchases of streamed music can be modeled by f(t) = 3.7+0.84 ln(t+1)
Part a
The amount spent by consumers on streamed music in 2014 is 4.62 billion
Part b
The amount spent by consumers on streamed music changing in 2014 at the rate of 0.28 billions per year
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need help with this choices:x coordinatey coordinatey axisx axis not equal toequal toinput output
First, let's remember that the zeros of a function are the points where the graph intersects the x-axis, that is, those points where an x-coordinate intersects the x-axis since its output is y=0.
Therefore, the following paragraph should be filled in as follows:
You find a zero of a linear function from a graph identifying the x coordinate of any point of intersection with the x-axis, from an equation by setting the output equal to 0, and from a table by identifying the input values when the output is 0.
Condense the left-hand side into a single logarithm.log(4) + log(2) + log(y) = log(____)
Condensation of left-hand side of the given expression into a single logarithm is :log(4) + log(2) + log(y) = log(8y)
How to condense log?To evaluate logarithmic expressions, methods for condensing logarithms in order to rewrite multiple logarithmic terms into one can be used. it is a useful tool for the simplification of logarithmic terms. To condense logarithms we use the rules of logarithms: the product rule, the quotient rule and the power rule.
According to the product laws of logarithm,
log x+log y = log (xy)
For the given expression,
log(4) + log(2) + log(y) = log(____)
So, after applying the product rule:
log(4) + log(2) + log(y) = log(4×2×y)
log(4) + log(2) + log(y) = log(8y)
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What is the perimeter of a rectangle with a length of:
3+√3
and a width of:
4+ √5
Answer:
7√8
Step-by-step explanation:
3+√3+4√5
3+4+√3+√5
7+√8
Fig. A is a scaled image of fig. B. Fig. A maps to fig. B with a scaled factor of 1.75. What’s the value of x?
Given:
• Length of top base of figure A = 3
,• Length of top base of figure B = x
,• Scale factor = 1.75
Given that figure A is a scale image of figure B, let's find the value of x.
To find the value of x since figure A is a scale image of figure B, multiply the corre
Suppose the population of a small town increases from 12,334 to 16,049. What is the percentincrease in the population, rounded to the nearest tenth of a percent?
First, we compute the population increase:
[tex]16049-12334=3715.[/tex]Now we divided the above result by the initial population:
[tex]\frac{3715}{16049}\approx0.23147.[/tex]Finally, we multiply by 100 and round to the nearest tenth of percent:
[tex]100\cdot0.23147=23.147\approx23.1.[/tex]Answer: 23.1%
can someone help me with this problem a triangle has a base of 2 meters and a height of 8 meters what is the area in square feet round your answer to the nearest tenth
Answer:
8.
Step-by-step explanation:
A little trick is to multiply 2 and 8, then divide by two because a triangle is half a square.
2 x 8 = 16
16 divided by 2 is 8.
Answer = 8