Lesson 14 Introduction
Solutions of Linear Equations
Use What You Know
You've learned how to solve linear equations and how to check your solution. In this
lesson, you'll learn that not every linear equation has just one solution. Take a look
at this problem.
Jason and his friend Amy are arguing. Jason says that a linear equation always has just one
solution. Amy says that some linear equations have more than one solution. Who's right?
Amy asked Jason to explore solutions to the following equation
2x + 1 + x = 3x - 2)+7
Use the math you already know to solve this problem.
Remember that a solution to an equation is a number that makes the equation true.
To check to see if a number is a solution to this equation, replace x with its value.
a. Is 6 a solution to the equation? Show your work
b. Is -2 a solution to the equation? Show your work. 2X
c. Is 0 a solution to the equation? Show your work.

The rectangular coordinates of points (4, -180°), (0, -4), (-4,0), (0,4), and (4,0) can be found from the polar coordinates of the points. In polar coordinates, we use the distance r and the angle θ. We can convert these into rectangular coordinates using the following equations:x = r cos θy = r sin θ1.

For point (4, -180°):r = 4, θ = -180°x = 4 cos (-180°) = -4y = 4 sin (-180°) = 02. For point (0, -4):r = 0, θ = -4x = 0 cos (-4) = 0y = 0 sin (-4) = 03. For point (-4,0):r = 4, θ = 0°x = 4 cos (0°) = 4y = 4 sin (0°) = 04. For point (0,4):r = 4, θ = 90°x = 4 cos (90°) = 0y = 4 sin (90°) = 45. For point (4,0):r = 4, θ = 0°x = 4 cos (0°) = 4y = 4 sin (0°) = 0

The rectangular coordinates of the points are:(-4, 0), (0, -4), (4, 0), (0, 4), and (4, 0).

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Answer:

all 3 options are true : A, B, C

Step-by-step explanation:

warning : it has come to my attention that some testing systems have an incorrect answer stored as right answer for this problem.

they say that A and C are correct.

but I am going to show you that if A and C are correct, then also B must be correct.

therefore, my given answer above is the actual correct answer (no matter what the test systems say).

originally the information about the alignment of the point F in relation to point E was missing.

therefore, I considered both options :

1. F is on the same vertical line as E.

2. F is not on the same vertical line as E.

because of optical reasons (and the - incomplete - expected correct answers of A and C confirm that) I used the 1. assumption for the provided answer :

the vertical line of EF is like a mirror between the left and the right half of the picture.

A is mirrored across the vertical line resulting in B. and vice versa.

the same for C and D.

this leads to the effect that all 3 given congruence relationships are true.

if we consider assumption 2, none of the 3 answer options could be true.

but if the assumptions are true, then all 3 options have to be true.

now, for the "why" :

remember what congruence means :

both shapes, after turning and rotating, can be laid on top of each other, and nothing "sticks out", they are covering each other perfectly.

for that to be possible, both shapes must have the same basic structure (like number of sides and vertices), both shapes must have the same side lengths and also equally sized angles.

so, when EF is a mirror, then each side is an exact copy of the other, just left/right being turned.

therefore, yes absolutely, CAD is congruent with CBD. and ACB is congruent to ADB.

but do you notice something ?

both mentioned triangles on the left side contain the side AC, and both triangles in the right side contain the side BD.

now, if the triangles are congruent, that means that each of the 3 sides must have an equally long corresponding side in the other triangle.

therefore, AC must be equal to BD.

and that means that AC is congruent to BD.

because lines have no other congruent criteria - only the lengths must be identical.

There can be 144 unique sandwiches created.

To determine the Possibility of unique sandwiches that can be created, we multiply the number of options for each category: 3 types of bread, 6 meats, and 8 toppings.

To calculate the total number of unique sandwich combinations, we multiply these numbers together:

\(3 (bread\ options) * 6 (meat\ options) *\ 8 (topping\ options) = 144.\)

Therefore, it is possible to create 144 distinct sandwiches by selecting one item from each category, considering the given options.

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Answer:  4 pencil

Step-by-step explanation:

A square is always a rhombus!

The values of a and b are 4 and -3

To solve for a, we make use of the function f(x).

x    f(x)

2     4

0     5

2     a

3     7

Remove the x values 0 and 3

x    f(x)

2     4

2     a

The above table implies that:

f(2) = 4 and f(2) = a

Substitute 4 for f(2) in f(2) = a

4 = a

Rewrite as:

a = 4

To solve for b, we make use of the function g(x).

x    g(x)

2     b

0     0

2     -3

3     -4

Remove the x values 0 and 3

x    g(x)

2     b

2     -3

The above table implies that:

f(2) = b and f(2) = -3

Substitute -3 for f(2) in f(2) = b

-3 = b

Rewrite as:

b = -3

Hence, the values of a and b are 4 and -3

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Complete question

Use the Symmetry of a Function to Find Coordinates

f is an even function             g is an odd function

x    f(x)                                        x             g(x)

2     4                                          2              b

0     5                                          0              0

2     a                                          2              -3

3     7                                          3              -4

Find a and b

Answer:

4, 3

Step-by-step explanation:

Answer:

B) 6

Step-by-step explanation:

Firstly, we need to know what the question is asking for.

"How many miles does she need to run each hour to complete the run" is asking for a speed in miles per hour.

miles / hour = speed in mph

18 miles / 3 hours = 18/3 mph

18/3 simplifies to 6

Lisa needs to run 6 mph

Answer:

The answer is c

Step-by-step explanation:

Source: Trust me bro

Answer:

None

Step-by-step explanation:

It would be easier to ask your teacher instead of getting wrong answers from random people. Ever think of that :0

Answer:

1695579

Step-by-step explanation:

The minimum percentage of commuters in Tulsa within 2 standard deviations of the mean is approximately 95%.  within 1.5 standard deviations of the mean is approximately 68%.the minimum percentage of commuters who have commute times between 3.9 minutes and 39.1 minutes is approximately 100%.

To solve this problem, we'll use the properties of the normal distribution and the empirical rule.

Given:

Mean (μ) = 21.5 minutes

Standard deviation (σ) = 4.4 minutes

a. To find the minimum percentage of commuters in Tulsa with a commute time within 2 standard deviations of the mean, we can use the empirical rule. According to the empirical rule, approximately 95% of the data falls within 2 standard deviations of the mean in a normal distribution.

So, the minimum percentage of commuters in Tulsa within 2 standard deviations of the mean is approximately 95%.

b. To find the minimum percentage of commuters in Tulsa with a commute time within 1.5 standard deviations of the mean, we again use the empirical rule. According to the empirical rule, approximately 68% of the data falls within 1 standard deviation of the mean in a normal distribution.

Since 1 standard deviation is equal to 4.4 minutes, 1.5 standard deviations would be 1.5 * 4.4 = 6.6 minutes.

Therefore, the minimum percentage of commuters in Tulsa within 1.5 standard deviations of the mean is approximately 68%.

The commute times within 1.5 standard deviations of the mean are between μ - 1.5σ and μ + 1.5σ. Plugging in the values, we get:

μ - 1.5σ = 21.5 - 1.5 * 4.4 = 21.5 - 6.6 = 14.9 minutes

μ + 1.5σ = 21.5 + 1.5 * 4.4 = 21.5 + 6.6 = 28.1 minutes

So, the commute times within 1.5 standard deviations of the mean are between 14.9 minutes and 28.1 minutes.

c. To find the minimum percentage of commuters with commute times between 3.9 minutes and 39.1 minutes, we can calculate the z-scores for these values and use the z-table.

The z-score formula is:

z = (x - μ) / σ

For 3.9 minutes:

z1 = (3.9 - 21.5) / 4.4 = -4.0

For 39.1 minutes:

z2 = (39.1 - 21.5) / 4.4 = 3.99 (approximately)

Using the z-table, we can find the area under the curve between these two z-scores. Subtracting the area from 0.5 (to account for both tails), we can find the minimum percentage.

Looking up z1 = -4.0, we find that the area to the left is practically 0, so we'll approximate it as 0.

Looking up z2 = 3.99, we find that the area to the left is practically 1, so we'll approximate it as 1.

The area between -4.0 and 3.99 is essentially 1 - 0 = 1.

Therefore, the minimum percentage of commuters who have commute times between 3.9 minutes and 39.1 minutes is approximately 100%.

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Answer:

Yes, A,B, and F is correct

Step-by-step explanation:

-- Gage Millar, Algebra 2 Tutor

The values of displacement, velocity, and acceleration are 2.68 m, 2.24 m/s, and -18.07 m/s2 respectively.

We know that the amplitude, A = 2 + T; the angular velocity, ω = 4 + U rad/s; and time, t = 1s. Here, the value of T = 1 and the value of U = 4 (as mentioned in the question).

Harmonic motion is a motion that repeats itself after a certain period of time.

Harmonic motion is caused by the restoring force that is proportional to the displacement from equilibrium.

The three types of harmonic motions are as follows: Free harmonic motion: When an object is set to oscillate, and there is no external force acting on it, the motion is known as free harmonic motion.

Damped harmonic motion: When an external force is acting on a system, and that force opposes the system's motion, it is called damped harmonic motion.

Forced harmonic motion: When an external periodic force is applied to a system, it is known as forced harmonic motion.Vectorial formVibrations of harmonic motion can be represented in a vectorial form.

A simple harmonic motion is a projection of uniform circular motion in a straight line.

The displacement, velocity, and acceleration of a particle in simple harmonic motion are all vector quantities, and their magnitudes and directions can be determined using a coordinate system.

Let's now calculate the values of displacement, velocity, and acceleration.

Displacement, s = A sin (ωt)

Here, A = 2 + 1 = 3 (since T = 1)and, ω = 4 + 4 = 8 (since U = 4)

So, s = 3 sin (8 x 1) = 2.68 m (approx)

Velocity, v = Aω cos(ωt)

Here, v = 3 x 8 cos (8 x 1) = 2.24 m/s (approx)

Acceleration, a = -Aω2 sin(ωt)

Here, a = -3 x 82 sin(8 x 1) = -18.07 m/s2 (approx)

Thus, the values of displacement, velocity, and acceleration are 2.68 m, 2.24 m/s, and -18.07 m/s2 respectively.

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By understanding the proportions of each section with respect to the entire hexagon we have:  

The first row represents the proportion of each section with respect to the entire hexagon, which is shown below:

{1, 1/2, 1/3, 1/6}

The elements of the second row are obtained by mutiplying all the elements of the first row by 6:

{6, 3, 2, 1}

The elements of the third row are calculated by multiplying all the elements of the first row by 3:

{3, 3/2, 1, 1/2}

And the elements of the fourth row are determined by multiplying all the elements of the first row by 2:

{2, 1, 2/3, 1/3}

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Standard deviation is a measure of variation.

The degree of variance or dispersion in a set of data values is measured statistically using the standard deviation. It reveals how far the data values differ from the data set's average. Data points tend to be close to the mean when the standard deviation is low, and are dispersed over a larger range when the standard deviation is high.

While measurements of the centre, such as mean, median, and mode, are used to describe a data set's central tendency. The median is the midway value when a data set is ordered, and the mode is the value that appears in a data set the most frequently. The mean is the sum of the data divided by the number of observations.

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The bell tolls 6 times between 11:45 AM and 3:10 PM.

Determine how many times the bell tolls between 11:45 AM and 3:10 PM, follow these steps:

1. Calculate the time interval between 11:45 AM and 3:10 PM:
3:10 PM - 11:45 AM = 3 hours and 25 minutes
2. Find the next bell toll after 11:45 AM:
Since the bell tolls at half past the hour, the next bell toll is at 12:30 PM.
3. Calculate the time interval between 12:30 PM and 3:10 PM:
3:10 PM - 12:30 PM = 2 hours and 40 minutes
4. Divide the time interval by the bell's tolling interval:
2 hours and 40 minutes = 160 minutes
160 minutes / 30 minutes (bell's tolling interval) = 5.33
5. Since the bell can't toll a fraction of a time, round down the result:
5.33 rounds down to 5
6. Add 1 to the rounded result to include the bell toll at 12:30 PM:
5 + 1 = 6.

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Answer:

- 0.006173 m/s²

Step-by-step explanation:

Given :

Initial Velocity, u = 65km/hr = (65*1000) / 3600 = 18.056 m/s

Final velocity, v = 45 km/ hr = (45*1000) / 3600 = 12.5 m/s

Time at which change in velocity occurred = 15 minutes = 15 * 60 = 900 s

Acceleration = Change in velocity with time

Acceleration = (v - u) / t

Acceleration = (12.5 - 18.056) / 900

Acceleration (deceleration) = - 5.556 / 900

Deceleration = - 0.006173 m/s²

The coordinates of the centroid are the average values of the \(x\)- and \(y\)-coordinates of the points \((x,y)\) that belong to the region. Let \(R\) denote the bounded region. These averages are given by the integral expressions

\(\dfrac{\displaystyle \iint_R x \, dA}{\displaystyle \iint_R dA} \text{ and } \dfrac{\displaystyle \iint_R y \, dA}{\displaystyle \iint_R dA}\)

The denominator is just the area of \(R\), given by

\(\displaystyle \iint_R dA = \int_0^8 \int_{\min(8\sin(2x), 8\cos(2x))}^{\max(8\sin(2x),8\cos(2x))} dy \, dx \\\\ ~~~~~~~~ = \int_0^8 |8\sin(2x) - 8\cos(2x)| \, dx \\\\ ~~~~~~~~ = 8\sqrt2 \int_0^8 \left|\sin\left(2x-\frac\pi4\right)\right| \, dx\)

where we rewrite the integrand using the identities

\(\sin(\alpha + \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\)

Now, if

\(8(\cos(2x) - \sin(2x)) = R \sin(2x + \alpha) = R \sin(2x) \cos(\alpha) + R \cos(2x) \sin(\alpha)\)

with \(R>0\), then

\(\begin{cases} R\cos(\alpha) = 8 \\ R\sin(\alpha) = -8 \end{cases} \implies \begin{cases}R^2 = 128 \\ \tan(\alpha) = -1\end{cases} \implies R=8\sqrt2\text{ and } \alpha = -\dfrac\pi4\)

Find where this simpler sine curve crosses the \(x\)-axis.

\(\sin\left(2x - \dfrac\pi4\right) = 0\)

\(2x - \dfrac\pi4 = n\pi\)

\(2x = \dfrac\pi4 + n\pi\)

\(x = \dfrac\pi8 + \dfrac{n\pi}2\)

In the interval [0, 8], this happens a total of 5 times at

\(x \in \left\{\dfrac\pi8, \dfrac{5\pi}8, \dfrac{9\pi}8, \dfrac{13\pi}8, \dfrac{17\pi}8\right\}\)

See the attached plots, which demonstrates the area between the two curves is the same as the area between the simpler sine wave and the \(x\)-axis.

By symmetry, the areas of the middle four regions (the completely filled "lobes") are the same, so the area integral reduces to

\(\displaystyle \iint_R dA \\\\ ~~~~ = 8\sqrt2 \left(-\int_0^{\pi/8} \sin\left(2x-\frac\pi4\right) \, dx + 4 \int_{\pi/8}^{5\pi/8} \sin\left(2x-\frac\pi4\right) \, dx \right. \\\\ ~~~~~~~~~~~~~~~~~~~~ \left. - \int_{17\pi/8}^8 \sin\left(2x-\frac\pi4\right) \, dx\right)\)

The signs of each integral are decided by whether \(\sin\left(2x-\frac\pi4\right)\) lies above or below axis over each interval. These integrals are totally doable, but rather tedious. You should end up with

\(\displaystyle \iint_R dA = 40\sqrt2 - 4 (1 + \cos(16) + \sin(16)) \\\\ ~~~~~~~~ \approx 57.5508\)

Similarly, we compute the slightly more complicated \(x\)-integral to be

\(\displaystyle \iint_R x dA = \int_0^8 \int_{\min(8\sin(2x), 8\cos(2x))}^{\max(8\sin(2x),8\cos(2x))} x \, dy \, dx \\\\ ~~~~~~~~ = 8\sqrt2 \int_0^8 x \left|\sin\left(2x-\frac\pi4\right)\right| \, dx \\\\ ~~~~~~~~ \approx 239.127\)

and the even more complicated \(y\)-integral to be

\(\displaystyle \iint_R y dA = \int_0^8 \int_{\min(8\sin(2x), 8\cos(2x))}^{\max(8\sin(2x),8\cos(2x))} y \, dy \, dx \\\\ ~~~~~~~~ = \frac12 \int_0^8 \left(\max(8\sin(2x),8\cos(2x))^2 - \min(8\sin(2x),8\cos(2x))^2\right) \, dx \\\\ ~~~~~~~~ \approx 11.5886\)

Then the centroid of \(R\) is

\((x,y) = \left(\dfrac{239.127}{57.5508}, \dfrac{11.5886}{57.5508}\right) \approx \boxed{(4.15518, 0.200064)}\)

Answer:

Marcus spent $6.36.

Step-by-step explanation:

First off, the fact that tax is included makes it easier for you, you don't need to worry about any multiplying!

Alright, so, he bought 2 candy bars for $0.79 EACH - that's 2 * $0.79 = $1.58.

Next, a drink for $1.49 and a bag of chips for $3.29. That's $1.49 + $3.29 = $4.78.

Now you add those two numbers up: $1.58 + $4.78 = $6.36.

Therefore, Marcus spent $6.36!

Answer:

48°

Step-by-step explanation:

1) The sum of the interior of any triangle is 180°. a + b + c = 180°. We need to find the value of x first before finding the angle F.

2) Substitute the values into a + b + c = 180, where \(a=6x-4\), \(b =5x-7\), \(c=8x-18\).

Hence,

\(6x-4+5x-7+8x-18=180\\19x-29=180\\19x=180+29\\19x=209\\x=\frac{209}{19} \\x=11\)

3) Find the angle of F by substituting 11 into x.

\(=5(11)-7\\=55-7\\=48\)

Therefore, angle F has a measure of 48°.

Step-by-step explanation:

x1, y1 = (-5,-2)

slope,m = -6/5

general equation, y= MX + C

(y-y1)/(x-x1)=m

(y+2)/(x+5)=-6/5

5y+10=-6x-30

5y=-6x-40

y=-(6x/5) - 8

Answer:

Step-by-step explanation:

This problem is based on compound interest, and the expression for the compound interest is given as

Given Data

A = final amount  =  ?

P = initial principal balance  = $2,500

r = interest rate = 2.1%= 0.021

t = number of time periods elapsed= 14 years

Substituting our data into the compound interest formula we can solve for the final amount

\(A= 2500(1+0.021)^1^4\\A= 2500(1.021)^1^4\\A= 2500*1.3377\\A= 3344.25\\\)

Hence to the nearest hundred we have the account balance has $3,000

Answer:3400

Step-by-step explanation:

Answer:

9 . (4)

10 . 17

Step-by-step explanation:

Answer:

x = -4 + 3 i or x = -4 - 3 i

Step-by-step explanation:

Solve for x:

(x + 4)/(x + 5) = (x - 5)/(2 x)

Hint: | Multiply both sides by a polynomial to clear fractions.

Cross multiply:

2 x (x + 4) = (x - 5) (x + 5)

Hint: | Write the quadratic polynomial on the left hand side in standard form.

Expand out terms of the left hand side:

2 x^2 + 8 x = (x - 5) (x + 5)

Hint: | Write the quadratic polynomial on the right hand side in standard form.

Expand out terms of the right hand side:

2 x^2 + 8 x = x^2 - 25

Hint: | Move everything to the left hand side.

Subtract x^2 - 25 from both sides:

x^2 + 8 x + 25 = 0

Hint: | Using the quadratic formula, solve for x.

x = (-8 ± sqrt(8^2 - 4×25))/2 = (-8 ± sqrt(64 - 100))/2 = (-8 ± sqrt(-36))/2:

x = (-8 + sqrt(-36))/2 or x = (-8 - sqrt(-36))/2

Hint: | Express sqrt(-36) in terms of i.

sqrt(-36) = sqrt(-1) sqrt(36) = i sqrt(36):

x = (-8 + i sqrt(36))/2 or x = (-8 - i sqrt(36))/2

Hint: | Simplify radicals.

sqrt(36) = sqrt(4×9) = sqrt(2^2×3^2) = 2×3 = 6:

x = (-8 + i×6)/2 or x = (-8 - i×6)/2

Hint: | Factor the greatest common divisor (gcd) of -8, 6 i and 2 from -8 + 6 i.

Factor 2 from -8 + 6 i giving -8 + 6 i:

x = 1/2-8 + 6 i or x = (-8 - 6 i)/2

Hint: | Cancel common terms in the numerator and denominator.

(-8 + 6 i)/2 = -4 + 3 i:

x = -4 + 3 i or x = (-8 - 6 i)/2

Hint: | Factor the greatest common divisor (gcd) of -8, -6 i and 2 from -8 - 6 i.

Factor 2 from -8 - 6 i giving -8 - 6 i:

x = -4 + 3 i or x = 1/2-8 - 6 i

Hint: | Cancel common terms in the numerator and denominator.

(-8 - 6 i)/2 = -4 - 3 i:

Answer: x = -4 + 3 i or x = -4 - 3 i

Answer:

\(x=-1/2\)

Step-by-step explanation:

\(4 x - 6 = 10 x - 3\)

\(4 x -10x= - 3+6\)

\(-6x=3\)

\(-6x/6=-3/6\)

\(x=-1/2\)

Answer:

\(x = - \frac{1}{2} \)

Step-by-step explanation:

\(4x - 6 = 10x - 3 \\ - 6 + 3 = 10x - 4x \\ - 3 = 6x \\ \frac{ - 3}{6} = \frac{6x}{6} \\ x = - \frac{1}{2} \)

Answer:

You would realize that this is a right triangle with angles of 45-45-90. This means that the sides are 1-3 root 2-2. Hope this helps!!

Step-by-step explanation:

Answer:

b

Step-by-step explanation:

a) The revenue R as a function of p is given by R(p) = -5p^2 + 100p.

b) This equation has two solutions: p = 0 and p = 20. Since R is nonnegative, the domain of R is the set of all values of p greater than or equal to 20, or {p | p ≥ 20}.

c)  The price that maximizes revenue is $10.

d)  The maximum revenue is $500.

a) The revenue R is given by R = xp. Since x = -5p + 100, we can substitute this into the equation for R to get:

R(p) = p(-5p + 100) = -5p^2 + 100p

Therefore, the revenue R as a function of p is given by R(p) = -5p^2 + 100p.

(b) We are given that the revenue R is nonnegative. To find the domain of R, we need to find the values of p that make R nonnegative. We can do this by finding the zeros of the quadratic equation -5p^2 + 100p = 0:

-5p(p - 20) = 0

This equation has two solutions: p = 0 and p = 20. Since R is nonnegative, the domain of R is the set of all values of p greater than or equal to 20, or {p | p ≥ 20}.

(c) To find the price p that maximizes revenue, we can take the derivative of R(p) with respect to p, set it equal to zero, and solve for p:

R'(p) = -10p + 100 = 0

-10p = -100

p = 10

Therefore, the price that maximizes revenue is $10.

(d) To find the maximum revenue, we can substitute p = 10 into the equation for R(p):

R(10) = -5(10)^2 + 100(10) = $500

Therefore, the maximum revenue is $500.

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