Discussion for Applications of Linear Systems
When solving a system of equations, how do you determine which method to use?

Answer:

[tex]A = 2(width*length) + 2(length*height)+2(width*height)\\=> 2(wl+lh+wh)\\where \ w = width,\  l=length \ and \ h = height[/tex]

l = 8, w = 5, h = 6

A = 2(5*8 + 8*6 + 5*6)

A = 236

[tex]Volume = width * length * height\\V = wlh[/tex]

V = 5*8*6

V = 240

Ratio of A to V = A / V

236/240 = 236:240

Step-by-step explanation:

The surface area of any object is the total area of each face of that object. Now, we have a rectangular prism with dimensions given. Considering a face of the prism, there is an opposite face and this is why the area of each face was multiplied by 2. That is how the formula for the surface area of an object is derived

For the volume, the formula is multiplying the height, width and length

When the area and volume is computed, we now take the ratio of the answers to arrive at 236/240 which in ratio form is 236:240

The student's homework question involves the calculation of a room's floor area for carpeting costs and then determining the wall area for papering costs, utilizing given ratios and costs per square meter.

The question involves calculating the area of a room's floor and using that to determine the cost of carpeting, followed by calculating the cost of papering the walls. To solve for this, we use the given ratio of the length being 3 times the breadth (let's say the breadth is 'b', then the length would be '3b'). When the carpeting cost of Rs. 60 per square meter totals Rs. 4500, we can find the area of the floor by dividing the total cost by the per square meter cost, which gives us an area of 75 square meters (i.e., 4500 / 60 = 75 m2).

Assuming the room to be a rectangle, the length and breadth can be determined from the area (since length times breadth = area), and given the relationship between length and breadth, we can solve for 'b' and then find '3b'. Once we have the dimensions of the floor, we can calculate the perimeter (2 times (length + breadth)) and use the given height to find the total area of the walls. The cost of papering is then determined by multiplying the wall area by the cost per square meter (Rs. 6 in this case).

Answer:

Given inequality:

[tex]y>3x-4[/tex]

To graph it.

Solution:

In order to graph the given inequality, we will first graph the equation of the line by replacing the inequality sign with equal to sign.

Thus, we will graph the line [tex]y=3x-4[/tex]

On comparing the line equation with standard equation [tex]y=mx+b[/tex] where [tex](0,b)[/tex] represents the y-intercept point of the line, we can conclude the y-intercept point for the given line is [tex](0,-4)[/tex]

We can find another point by plugging in [tex]x=1[/tex] in the equation.

So, we have:

[tex]y=3(1)-4[/tex]

[tex]y=3-4[/tex]

[tex]y=-1[/tex]

So, the other point is [tex](1,-1)[/tex]

Potting (0,-4) and (1,-1) on the graph.

We can joint the two points and extend it infinity in order to graph the equation of line.

Now, since the inequality sign is greater than,

1) so we make sure that the line of the equation is a broken line as the line is not included in the solution for the inequality.

2) The part of the graph lying above the line would be shaded as the solution for the inequality.

Answer:

The two-liter bottle of soda has a unit price of $0.028/ounce.  

The case of twelve 12 ounces has a unit price of $0.021/ounce.

The case of twelve 12 ounce can is the better bargain.

Step-by-step explanation:

A two-liter bottle of soda i.e. 67.6 ounces cost $1.89.

So, the unit price (price/ounces) of this type of soda is [tex]\frac{1.89}{67.6} = 0.028[/tex] dollars per ounce.

Again, a case of twelve 12 ounce cans of the same soda costs $2.99.

So, the unit price (price/ounces) of this type of soda is [tex]\frac{2.99}{12 \times 12} = 0.021[/tex] dollars per ounce.

Therefore, the case of twelve 12 ounce can is the better bargain. (Answer)

Answer:

The value of x can be 5 or 2

Step-by-step explanation:

In given triangle ABC

Interior Angle A = [tex]x^{2} +3x+20[/tex]

Interior Angle B = [tex]x^{2} +14x-5[/tex]

Exterior Angle I = [tex]3x^{2} +10x+25[/tex]

We know that

Exterior Angle of triangle is summation of corresponding opposite angles

Angle I = Angle A + Angle B

[tex]3x^{2} +10x+25= (x^{2} +3x+20)+(x^{2} +14x-5)[/tex]

[tex]3x^{2} +10x+25= 2x^{2} +17x+15[/tex]

[tex]1x^{2}-7x+10=0[/tex]

[tex]x^{2}-2x-5x+10=0[/tex]

[tex]x(x-2)-5(x-2)=0[/tex]

[tex](x-5)(x-2)=0[/tex]

PLEASE MARK BRAINLIEST!

Answer:

2!!

Step-by-step explanation:

2 ÷ 2 = 1

I hope this helps! If this answer is not correct, please don't hesitate to comment on it, as that way I can fix my answer!

- ibringsExxiback

Answer: X = 2

Step-by-step explanation: In this problem, notice that x is being divided by 2. So to get x by itself, we need to multiply both sides of the equation by 2.

Notice that on the left side, the 2 and 2 cancel each other out so we are just left with x. On the right side, we have 1 x 2 which is 2 so x = 2.

Finally, we can check our answer by plugging a 2 back into the original equation. So we have [tex]\frac{(2)}{2} = 1[/tex] which is a true statement so we know that our answer is correct.

Answer:

Speed of the counselor is represented by 60 and is equal to 60 mph

Step-by-step explanation:

Given:

The expression that represents the distance, in miles, the counselor still needs to drive after 't' hours is given as:

[tex]d=130-60t\\d=-60t+130[/tex]

The above expression is of the form of a standard linear equation:

[tex]y=mx+b[/tex]

Here, 'm' represents the slope or rate of change of 'y' with 'x' and 'b' is the y-intercept or initial value of 'y' when 'x' was 0.

On comparing the standard equation with the given equation, we conclude that:

[tex]m=-60[/tex]

Therefore, the slope here represents the change in distance with time 't'.

We know that, change in distance with time is nothing but the speed.

So, the speed of the counselor is 60 mph.

Therefore, the term 60 represents the speed of the counselor.

Answer:

h(5) = - 22

Step-by-step explanation:

Given that h is a linear function say, h(x) = ax + b ......... (1)

Now, given that h(1) = 10 and h(3) = - 6  

Hence, we can write from equation (1), a(1) + b = 10, ⇒ a + b = 10 .......... (2)

And a(3) + b = - 6, ⇒ 3a + b = - 6 ........ (3)

Now, solving equations (2) and (3) we get (3a - a) = - 6 - 10

⇒ 2a = - 16

⇒ a = - 8

So, from equation (2), we get, b = 10 - a = 18  

Therefore, the linear function is

h(x) = - 8x + 18

Hence, h(5) = - 8(5) + 18 = - 22 (Answer)

Answer:

shown in steps, either 189/15625 or 63/4600 depends on different situation

Step-by-step explanation:

The answer depends on whether the is chip can be put into bin for next pick,

if it can be put back after first pick.. the possibility will be

total chips: 7 + 9 + 3 + 6 = 25

Red -> green -> yellow 7/25 x 9/25 x 3/25 = 189/15625

If the first time picked chip will not put back for second pick:

7/25 x 9/24 x 3/23 = 63/4600

Answer:

A. [tex]\frac{189}{15625}[/tex]

Step-by-step explanation:

I got it right!!

[tex]T = \frac{-9}{2}x + 103[/tex] is the linear equation to find the temperature T at an elevation x on the mountain, where x is in thousands of feet.

Solution:

The linear equation in slope intercept form is given as:

T = cx + k ------ (i)

Where "t" is the temperature at an elevation x

And x is in thousands of feet

Given that its 76 degrees fahrenheit at the 6000-foot level of a mountain

Given, when c = 6 thousand ft and [tex]T = 76^{\circ}[/tex] fahrenheit

This implies,

From (i)

76 = c(6) + k

76 = 6c + k

⇒ k = 76 - 6c  ----- (ii)

Given that 49 degrees Fahrenheit at the 12000-foot level of the mountain

Given, when c = 12 thousand ft and [tex]T = 49^{\circ}[/tex] fahrenheit

This implies,

From (i)

49 = c(12) + k

49 = 12c + k

Substitute (ii) in above equation

49 = 12c + (76 - 6c)

49 = 12c + 76 - 6c

49 - 76 = 6c

6c = -27

[tex]c = \frac{-9}{2}[/tex]

Substituting the value of c in (ii) we get

[tex]k = 76 - 6( \frac{-9}{2})\\\\k = 76 + 27 = 103[/tex]

Substituting the value of c and k in (i)

[tex]T = \frac{-9}{2}x + 103[/tex]

Where "x" is in thousands of feet

Thus the required linear equation is found

The linear equation that represents the temperature T at an elevation x on the mountain (where x is in thousands of feet) is T = -4.5x + 103.

This is a problem involving linear equations in Mathematics. We are given two data points, (6, 76) and (12, 49) where the first value in each pair is an elevation (in thousands of feet) and the second value is the corresponding temperature in degrees Fahrenheit. We can use these points to derive a linear equation of the form T = mx + c where T is the temperature, m is the slope, x is the elevation, and c is the y-intercept. The slope of the line m can be calculated using the formula (y2-y1)/(x2-x1) = (49-76)/(12-6) = -27/6 = -4.5. Therefore, every thousand feet in elevation leads to a temperature drop of 4.5 degrees Fahrenheit. To calculate the y-intercept c we can substitute x and y values from one of our points into the linear equation. For instance, using point (6, 76) we get 76 = -4.5*6 + c, which gives c = 103. Therefore, the linear equation to find the temperature T at an elevation x (in thousands feet) is T = -4.5x + 103.

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The sun is roughly 93,000,000 miles away from Earth

We can round this to 100,000,000

As a power of 10, this is [tex]10^{8}[/tex]

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Answer:

There are an estimated 1 quadrillion ants living on earth. A quadrillion as a power of 10 would be [tex]10^{15}[/tex], because there are 15 zeros in a quadrillion.

Answer:

There are 30 two point questions and 10 four point questions  on the test.

Step-by-step explanation:

Given:

Your teacher is giving you a test worth 100 points containing 40 questions. There are two point and four point questions on the test.

Now, to find number of each type of question are on the test.

Let the two point questions be [tex]x[/tex].

And let the four point questions be [tex]y[/tex].

So, total questions:

[tex]x+y=40.[/tex].........( 1 )

⇒ [tex]y=40-x.[/tex]

Now, total number of points of the questions on the test:

[tex]2x+4y=100.[/tex]

Substituting the value of [tex]y[/tex]:

⇒ [tex]2x+4(40-x)=100[/tex]

⇒ [tex]2x+160-4x=100[/tex]

⇒ [tex]160-2x=100[/tex]

Adding both sides by [tex]2x[/tex] we get:

⇒ [tex]160=100+2x[/tex]

Subtracting both sides by 100 we get:

⇒ [tex]60=2x[/tex]

Dividing both sides by 2 we get:

⇒ [tex]30=x[/tex]

⇒ [tex]x=30[/tex].

The number of two point questions = 30.

Putting the value of [tex]x[/tex] in the equation ( 1 ) we get:

[tex]30+y=40.[/tex]

Subtracting both sides by 30 we get:

⇒ [tex]y=10.[/tex]

The number of four point questions = 10.

Therefore, there are 30 two point questions and 10 four point questions on the test.

The primary solution for [tex]\( \cos \theta = -0.3 \) is approximately \( \theta \approx 109.47^\circ \) within \( [0^\circ, 360^\circ) \).[/tex]

To find the primary solution of  [tex]\( \cos \theta = -0.3 \), we need to determine the angle \( \theta \) within the interval \( [0^\circ, 360^\circ) \) that satisfies this equation.[/tex]

1. **Using Inverse Cosine Function**: Since[tex]\( \cos \theta = -0.3 \), we need to find the angle whose cosine is -0.3. We use the inverse cosine function \( \arccos \) to find \( \theta \):[/tex]

[tex]\[ \theta = \arccos(-0.3) \][/tex]

2. **Evaluate the Inverse Cosine**: We evaluate  [tex]\( \arccos(-0.3) \)[/tex] using a calculator or a trigonometric table. This gives us the angle in radians.

3. **Convert Radians to Degrees (if necessary)**: If the solution is given in radians, we convert it to degrees since the interval given is in degrees. Recall that [tex]\( 180^\circ = \pi \) radians. To convert radians to degrees, multiply by \( \frac{180}{\pi} \).[/tex]

4. **Check Interval**: Finally, we ensure that the angle [tex]\( \theta \)[/tex] is within the interval [tex]\( [0^\circ, 360^\circ) \). If the angle is greater than or equal to \( 360^\circ \), we subtract multiples of \( 360^\circ \)[/tex]until we get an angle within the desired interval.

5. **Final Result**: The angle [tex]\( \theta \)[/tex] found in step 4 is the primary solution of the given equation.

In summary, the detailed stepwise solution involves using the inverse cosine function to find the angle whose cosine is -0.3, evaluating this angle, converting it to degrees if necessary, and ensuring it falls within the desired interval.

The set of equations that represent this situation is:

[tex]y = \frac{1}{6}x + 3[/tex]

[tex]y = \frac{1}{2}x + 3[/tex]

Solution:

Let "x" represent the smaller number

Let "y" represent the larger number

Given that,

Six times a larger number is equal to the sum of a smaller number and 18

Here "times" represents multiplication

Six times a larger number = sum of a smaller number and 18

6 x larger number = smaller number + 18

6y = x + 18

Thus,

[tex]y = \frac{1}{6}(x + 18)\\\\y = \frac{1}{6}x + 3[/tex]

Also given that difference of twice the larger number and the  smaller number is 6

twice the larger number - smaller number = 6

2y - x = 6

Thus,

2y = x + 6

[tex]y = \frac{1}{2}(x + 6)\\\\y = \frac{1}{2}x + 3[/tex]

Thus the set of equations that represent this situation is:

[tex]y = \frac{1}{6}x + 3[/tex]

[tex]y = \frac{1}{2}x + 3[/tex]

Answer:

the correct answer is D

Step-by-step explanation:

Answer:

The right option is 6 units to the left of 3.

Step-by-step explanation:

In the number line, +x means x units moved to the right and -x means x units moved to the left.

Now, in our case, we have to locate the number 3 - 6 on the horizontal number line.

Now, we can start with position 3 on the number line then move 6 units to the left.

Therefore, the right option is 6 units to the left of 3. (Answer)

Answer:

Last option is correct.

6 units to the left of 3​

Step-by-step explanation:

given:

The given number is.

[tex]=3-6[/tex]

[tex]=-3[/tex]

So, the number is -3.

The number -3 located on horizontal line as shown in the attached file.

Therefore, the number -3 located on the horizontal line is 6 units to the left of 3​.

Answer:

Option 3) (0.38,-2.85) and (2.62,3.85)

Step-by-step explanation:

The correct question is

The graphs of y=x^2-3 and y=3x-4 intersect at approximately...

we have

[tex]y=x^2-3[/tex] ----> equation A

[tex]y=3x-4[/tex] ----> equation B

Solve the system by graphing

Remember that the solution of the system is the intersection point both graphs

Equate equation A and equation B

[tex]x^2-3=3x-4[/tex]

[tex]x^2-3x-3+4=0[/tex]

[tex]x^2-3x+1=0[/tex]

The formula to solve a quadratic equation of the form

[tex]ax^{2} +bx+c=0[/tex]

is equal to

[tex]x=\frac{-b\pm\sqrt{b^{2}-4ac}} {2a}[/tex]

in this problem we have

[tex]x^2-3x+1=0[/tex]

so

[tex]a=1\\b=-3\\c=1[/tex]

substitute in the formula

[tex]x=\frac{-(-3)\pm\sqrt{-3^{2}-4(1)(1)}} {2(1)}[/tex]

[tex]x=\frac{3\pm\sqrt{5}} {2}[/tex]

so

[tex]x_1=\frac{3+\sqrt{5}} {2}=2.62[/tex]

[tex]x_2=\frac{3-\sqrt{5}} {2}=0.38[/tex]

Find the values of y (substitute the value of x in equation A or equation B)

For x=2.62 ----> [tex]y=(2.62)^2-3=3.85[/tex]

For x=0.38 ----> [tex]y=(0.38)^2-3=-2.85[/tex]

therefore

The intersection points are approximately  (0.38,-2.85) and (2.62,3.85)

Answer:

A. M = -15n + 300

Step-by-step explanation:

Given:

Money Borrowed =300

Money to be paid each week = $15

Total number of weeks = 'n'

We need to find the equation which shows Money owes 'M' after 'n' weeks by maria.

Money Owes after n week will be equal to Total Money Borrowed minus Money to be paid each week multiplied by number of weeks.

framing in equation form we get;

[tex]M = 300-15n[/tex]

Hence The Equation representing the Money owes by Maria after n week is  [tex]M = 300-15n[/tex]

Answer:

A M= -15n+300

Step-by-step explanation:

Answer:

See attachment

Step-by-step explanation:

You can obtain any two points on the graph of [tex]y=-73x+2[/tex] and use it to draw its graph.

When x=0, [tex]y=-73*0+2=2[/tex]

So you plot (0,2)

When x=1, [tex]y=-73*1+2=-71[/tex]

You again plot (1,-71).

With a straight edge you can now draw a straight line through the two points.

See attachment

Answer:

5

Step-by-step explanation:

The LCM of 1 and 5 is 5, as 1 is a factor of all numbers and thus does not change the other number when finding the LCM.

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both of the given numbers. In the case of 1 and 5, the LCM can be found by simply taking the larger number since 1 is a factor of all integers which therefore, gives the LCM of 1 and 5 as 5. There isn't really a need for a complex calculation because any number multiplied by 1 remains unchanged.

Answer:

Step 1: Distribute each term of the first polynomial to every term of the second polynomial. Remember that when you multiply two terms together you must multiply the coefficient (numbers) and add the exponents.

Step 2: Combine like terms (if you can).

Answer: From least to greatest, we get: 0.6010, 6.010, 6.100. Ordering these decimals from least to greatest we get: 0.601, 6.01, 6.1. Sometimes it is helpful to place a number in a circle to the right of each decimal you are trying to order.

Step-by-step explanation:

Answer:

Teresa is wrong. 20 times does not always mean 20% because it depends on the total times she rolled the die. Since she rolled the die 50 times, the fraction she got five was 20/50, which is 40%.

Answer: 200 miles

Step-by-step explanation: 1/5 is equal to 10miles so you 5/5 is 50 miles or 1 inch on the map.

Final answer:

The distance from the hospital to the airport on the map is 18 inches.

Explanation:

To find the distance from the hospital to the airport, we can use the scale of the map. Since the school and hospital are 3/5 inch apart on the map, and the school is 30 miles away from the hospital, we can set up the proportion:

3/5 inch = 30 miles

x inch = distance from hospital to airport

Cross-multiplying, we get 5x = 3 * 30

x = 18

Therefore, the distance from the hospital to the airport is 18 inches on the map.

Answer:

Step-by-step explanation:

This is a parallelogram.

The formula of an area of a parallelogram:

[tex]A=bh[/tex]

b - base

h - height

From the graph we have b = 21m and h = 33m.

Substitute:

[tex]A=(21)(33)=693\ m^3[/tex]

Answer:

$1.58

Step-by-step explanation:

To find the mean price, add up all the prices and divide by the number of items. The mean price of your purchased items is $1.59.

To find the mean price of the purchased items, you need to add up all the prices and then divide by the number of items. In this case, you purchased 6 items. So, you add $1.98 + $0.59 + $2.99 + $0.99 + $2.49 + $0.49 = $9.53. Then, you divide $9.53 by 6 to get the mean price which is $1.59.

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Answer:

It results -14 in either way

Step-by-step explanation:

Velocity As A Rate Of Change

The velocity of an object can be computed as the rate of change of its displacement (or position taken as a vector) over time. If we compute it as a derivative, it's called instantaneous velocity, and if computed as the slope of the function (difference quotient) at a certain point it's the average velocity

The position of the object as a function of time is

[tex]\displaystyle s(t)=6-14t[/tex]

Computing the derivative

[tex]\displaystyle s'(t)=-14[/tex]

We can see it's a constant value. If we use the slope or rate of change:

[tex]\displaystyle v=\frac{s_2-s_1}{t_2-t_1}[/tex]

Now let's fix two values for time  

[tex]\displaystyle t_1=5\ sec,\ t_2=8\ sec[/tex]

and compute the corresponding positions, by using the given function

[tex]\displaystyle s_1=6-14(5)=-64[/tex]

[tex]\displaystyle s_2=6-14(8)=-106[/tex]

Now we compute the average velocity

[tex]\displaystyle v=\frac{-106-(-64)}{8-5}[/tex]

[tex]\displaystyle v=\frac{-106+64}{3}=\frac{-42}{3}[/tex]

[tex]\displaystyle v=-14[/tex]

We get the very same result in both ways to compute v. It happens because the position is related with time as a linear function, it's called a constant velocity motion.

David collected 13 newspaper

Solution:

Given that,

Michelle collected 7 more than twice the number of newspaper David collected.

Michelle collected 33 newspaper

To find: Number of newspaper collected by David

From given information,

Number of newspaper collected by Michelle = 7 more than twice the number of newspaper David collected

Number of newspaper collected by Michelle = 7 + 2(Number of newspaper collected by David)

33 = 7 + 2(Number of newspaper collected by David)

2(Number of newspaper collected by David) = 33 - 7

2(Number of newspaper collected by David) = 26

Number of newspaper collected by David = [tex]26 \div 2 = 13[/tex]

Therefore David collected 13 newspaper

Answer:

Step-by-step explanation:

ax= 27........I would normally solve this by division....but it says multiply....so multiply both sides by 1/a

example :

3x = 27.....multiply both sides by 1/3

1/3(3/1)x = 27 * 1/3

x = 27/3 which reduces to 9

u have to multiply by the reciprocal of a to get rid of a

Answer:

infinity

Step-by-step explanation: it keeps going

We need to find the slope and the y-intercept to get an equation.

slope=change in y/change in x... or "(y2-y1)/(x2-x1)"

slope=(2-(-14))/(3-(-1))=16/4=4

Equation of a line:y=mx+b

Plug in a point and solve for b, the y intercept.

2=4(3)+b

b=-10

Answer: y=4x-10