Answer:
y = (2/3)x - 6Step-by-step explanation:
Slope-intercept form is:
y = mx + bConvert the given equation as following:
2x - 3y = 18 Isolate the term with y3y = 2x - 18 Divide all terms by 3y = (2/3)x - 6Simplify completely.a.4x212 xwhen x +0.b. (2t)(3t)(t)c. (3x² - 4x +8)+(x² +6x-11)d. (3x² + 4x – 8) - (x² + 6x +11)
The expression in 4a) is given below
[tex]\frac{4x^2}{12x}[/tex]Collecting similar terms using the division rule of indices, we will have
[tex]\frac{a^m}{a^n}=a^{m-n}[/tex]The above expression therefore becomes
[tex]\begin{gathered} \frac{4x^2}{12x} \\ =\frac{4x^2}{12x^1} \\ =\frac{1}{3}\times x^{2-1} \\ =\frac{1}{3}\times x \\ =\frac{x}{3} \end{gathered}[/tex]Hence,
The final answer = x/3
A student rolled 2 dice. What is the probability that the first die landed
on a number less than 3 and the second die landed on a number
greater than 3?
I really need help please!
Answer:
n < -15/4
Step-by-step explanation:
You want to use the discriminant to find the values of n for which the quadratic 3z² -9z = (n -3) has only complex solutions.
DiscriminantThe discriminant of quadratic equation ax²+bx+c = 0 is ...
d = b² -4ac
The given quadratic can be put in this form by subtracting (n-3):
3z² -9z -(n -3) = 0
This gives us ...
a = 3b = -9c = -(n -3)and the discriminant is ...
d = (-9)² -4(3)(-(n-3)) = 81 +12(n -3)
d = 12n +45
Complex solutionsThe equation will have only complex solutions when the discriminant is negative:
d < 0
12n +45 < 0 . . . . . use the value of the discriminant
n +45/12 < 0 . . . . . divide by 12
n < -15/4 . . . . . . . subtract 15/4
There will be two complex solutions when n < -15/4.
Find all the values of x where the tangent line is horizontal.3f(x) = x³ - 4x² - 7x + 12X=(Use a comma to separate answers as needed. Type an exact answer, using radicals
Given the function:
[tex]h(x)=x^3-4x^2-7x+12[/tex]Find the first derivative:
[tex]h^{\prime}(x)=3x^2-8x-7[/tex]The first derivative gives us the slope of the tangent line to the graph of the function. When the tangent line is horizontal, the slope is 0, thus:
[tex]3x^2-8x-7=0[/tex]This is a quadratic equation with coefficients a = 3, b = -8, c = -7.
To calculate the solutions to the equation, we use the quadratic solver formula:
[tex]$$x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}$$ [/tex]Substituting:
[tex]x=\frac{-(-8)\pm\sqrt{(-8)^2-4(3)(-7)}}{2(3)}[/tex]Operate:
[tex]\begin{gathered} x=\frac{8\pm\sqrt{64+84}}{6} \\ \\ x=\frac{8\pm\sqrt{148}}{6} \end{gathered}[/tex]Since:
[tex]148=2^2\cdot37[/tex]We have:
[tex]\begin{gathered} x=\frac{8\pm2\sqrt{37}}{6} \\ \\ \text{ Simplifying by 2:} \\ \\ x=\frac{4\pm\sqrt{37}}{3} \end{gathered}[/tex]There are two solutions:
[tex]\begin{gathered} x_1=\frac{4+\sqrt{37}}{3} \\ \\ x_2=\frac{4-\sqrt{37}}{3} \end{gathered}[/tex]Solve for a.5a== ✓ [?]2aPythagorean Theorem: a2 + b2 = c2=
ANSWER
a = √21
EXPLANATION
This is a right triangle, so we have to apply the Pythagorean Theorem to find the value of a.
We know the length of the hypotenuse which is 5, and the length of one of the legs, which is 2. The Pythagorean Theorem for this problem is,
[tex]a^2+2^2=5^2[/tex]Subtract 2² from both sides,
[tex]\begin{gathered} a^2+2^2-2^2=5^2-2^2 \\ a^2=25-4 \end{gathered}[/tex]And take the square root to both sides,
[tex]\begin{gathered} \sqrt[]{a^2}=\sqrt[]{25-4} \\ a=\sqrt[]{21} \end{gathered}[/tex]Hence, the value of a is √21.
The wholesale price for a chair is 194$ . A certain furniture store marks up the wholesale price by 35%. Find the price of the chair in the furniture store.
The price of the chair will be 261.9 $ .
One percent (symbolized 1%) is a hundredth part; thus, 100 percent represents the entirety and 200 percent specifies twice the given quantity. For example, 1 percent of 1,000 chickens equals 1/100 of 1,000, or 10 chickens; 20 percent of the quantity is 20/100 1,000, or 200.
If we say, 5%, then it is equal to 5/100 = 0.05.
To solve percent problems, you can use the equation, Percent · Base = Amount, and solve for the unknown numbers. Or, you can set up the proportion, Percent = , where the percent is a ratio of a number to 100. You can then use cross multiplication to solve the proportion.
Based on given conditions formulate
x = 194 ×(35%+1)
x= 194 ×1.35
x = 261.9 $ .
Thus The price of the chair will be 261.9 $ .
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What is the probability of drawing a red card from a pack of cards and rolling an even number on a standard six-sided die?
Select one:
1/12
1/2
1/4
1/8
Answer:
1/2 because half the cards are red and half the numbers are even
Which of the following expressions is equal to -x2 -36
OA. (-x+6)(x-6i)
OB. (x+6)(x-6i)
OC. (-x-6)(x-6i)
OD. (-x-6)(x+6i)
The expression equivalent to -x² - 36 is the one in option C.
(-x - 6i)*(x - 6i)
Which of the following expressions is equal to -x² - 36?We can rewrite the given expression as:
-x² - 36 = -x² - 6²
And remember that the product of a complex number z = (a + bi) and its conjugate (a - bi) is:
(a + bi)*(a - bi) = a² + b²
Then in this case we can rewrite:
-x² - 6² = -(x² + 6²) = - (x + 6i)*(x - 6i)
= (-x - 6i)*(x - 6i)
The correct option is C.
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URGENT!! ILL GIVE
BRAINLIEST!!!!! AND 100
POINTS!!!!!
If angle a measures 42 degrees, then what other angles would be congruent to angle a and also measure 42 degrees?
If angle "a" measures 42° the the other angles that will be congruent to angle "a" and also measure 42° will be angle d, angle e and angle h .
In the question ,
a figure is given ,
From the figure we can see that 2 parallel lines are cut by a transversal .
So ,
angle a = angle d .......because vertically opposite angles .
angle a = angle e ...because corresponding angles are equal in measure
also
angle e = angle h .... because vertically opposite angles .
Therefore , If angle "a" measures 42° the the other angles that will be congruent to angle "a" and also measure 42° will be angle d, angle e and angle h , the correct option is (a) .
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For people over 50 years old, the level of glucose in the blood (following a 12 hour fast) is approximately normally distributed with mean 85 mg/dl and standard deviation 25 mg/dl ("Diagnostic Tests with Nursing Applications", S. Loeb). A test result of less than 40 mg/dl is an indication of severe excess insulin, and medication is usually prescribed.
What is the probability that a randomly-selected person will find an indication of severe excess insulin?
Suppose that a doctor uses the average of two tests taken a week apart (assume the readings are independent). What is the probabiltiy that the person will find an indication of severe excess insulin?
Repeat for 3 tests taken a week apart:
Repeat for 5 tests taken a week apart:
Using the normal distribution and the central limit theorem, it is found that:
There is a 0.0359 = 3.59% probability that a randomly-selected person will find an indication of severe excess insulin.Considering the mean of two tests, there is a 0.0054 = 0.54% probability that the person will find an indication of severe excess insulin.Three tests: 0.0009 = 0.09%.Five tests: 0% probability.Normal Probability DistributionThe z-score of a measure X of a normally distributed variable that has mean represented by [tex]\mu[/tex] and standard deviation represented by [tex]\sigma[/tex] is given by the following rule:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure X is above or below the mean, depending if the z-score score is positive or negative.From the z-score table, the p-value associated with the z-score is found, which represents the percentile of the measure X in the distribution of interest.By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].The mean and the standard deviation of the glucose levels are given, respectively, by:
[tex]\mu = 85, \sigma = 25[/tex]
The probability of a reading of less than 40 mg/dl(severe excess insulin) is the p-value of Z when X = 40, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (40 - 85)/25
Z = -1.8.
Z = -1.8 has a p-value of 0.0359.
For the mean of two tests, the standard error is:
s = 25/sqrt(2) = 17.68.
Hence, by the Central Limit Theorem:
[tex]Z = \frac{X - \mu}{s}[/tex]
Z = (40 - 85)/17.68
Z = -2.55.
Z = -2.55 has a p-value of 0.0054.
For 3 tests, we have that:
s = 25/sqrt(3) = 14.43.
Z = (40 - 85)/14.43
Z = -3.12.
Z = -3.12 has a p-value of 0.0009.
For 5 tests, we have that:
s = 25/sqrt(5) = 11.18.
Z = (40 - 85)/11.18
Z = -4.03
Z = -4.03 has a p-value of 0.
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Solve-2x-16=2x-20.
Ox=1
O no solutions
○ * = −1
all real numbers
What is the area of the triangle 9 2 12
The given diagram is a traingle with base 12 units, height 4 units, and one side 9 units.
Since base (b) and height (h) are known, we can use the following formula for the area (A) of the triangle,
[tex]A=\frac{1}{2}bh[/tex]Substitute the values and simplify the expression,
[tex]\begin{gathered} A=\frac{1}{2}\times12\times4 \\ A=6\times4 \\ A=24 \end{gathered}[/tex]Thus, the area of the given triangle is 24 square units.
Andre was trying to write 7^4/7^-3 with a single exponent and write 7^4/7^-3= 7^4-3=7^1 Exploit to Andre what his mistake was and what the answer should be...PLEASE THE ANSWER IS URGENT!!
Here, we want to get what Andre's mistake was and correct it
To answer this, we are supposed to use the division law of indices
We have this as;
[tex]\frac{a^x}{a^y\text{ }}=a^{x-y}[/tex]Now, in the case of this question, x is 4 and y is -3
So, we have the expression as;
[tex]7^{4-(-3)}=7^{4+3}=7^7[/tex]His mistake is thus adding the exponents instead of subtracting
20 >= 4/5 w
Solve the inequality. Grab the solution
Solve the inequality. Grab the solution
-8<-1/4m
In inequality 20 >= 4/5 w, w is 25 or any real no. lower than< 25 and in inequality -8<-1/4m, m = any real no. greater than> 24 is are the solution.
What is inequality?An inequality compares two values and indicates whether one is lower, higher, or simply not equal to the other.
A B declares that a B is not equal.
When a and b are equal, an is less than b.
If a > b, then an is bigger than b.
(those two are called strict inequality)
The phrase "a b" denotes that an is less than or equal to b.
The phrase "a > b" denotes that an is greater than or equal to b.
We have give the inequality to solve
20 = 4/5w
w = 20 × 5/4
= 25 or any real no. lower than< 25
Let -8 = -1/4m
m = -8 × -4
m = 24
so m = any real no. greater than> 24
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what is the area of a circular pool with a diameter of 36 ft?
Answer:
1,017.36ft^2
Explanation:
Area of the circular pool = \pi r^2
r is the radius of the pool
Given
r = d/2
r = 36/2
r = 18ft
Area of the circular pool = 3.14(18)^2
Area of the circular pool = 3.14 * 324
Area of the circular pool = 1,017.36ft^2
Solve the system [tex]\left \{ {{5x1 + 5x2 = 5} \atop {2x1 + 3x2 = 4}} \right.[/tex]
The solution for the given system of equations is x[1] = -1 and x[2] = 2.
What is system of equations?A system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
Given are the following equations as -
5 x[1] + 5 x[2] = 5
2 x[1] + 3 x[2] = 4
Assume that -
x[1] = a
x[2] = b
Then, we can write the equations as -
5a + 5b = 5
2a + 3b = 4
Now -
5a + 5b = 5
5(a + b) = 5
a + b = 1
a = 1 - b
So, we can write -
2a + 3b = 4
as
2(1 - b) + 3b =4
2 - 2b + 3b = 4
b = 4 - 2
b = 2 = x[2]
Then
a = 1 - 2
a = -1 = x[1]
Therefore, the solution for the given system of equations is x[1] = -1 and x[2] = 2.
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7Lines a and bare parallel cut by transversal line t solve for the value of x25x + 4a3x + 14
These angles measure the same they are interior alternate angles.
3x + 14 = 5x + 4
Solve for x
3x - 5x = 4 - 14
Simplify like terms
-2x = -10
x = -10/-2
Result
x = 5
If the Manufacturing Overhead account is closed proportionally to Work in Process, Finished Goods, and Cost of Goods Sold, the related entry will include a ________.
If the Manufacturing Overhead account is closed proportionally to Work in Process, Finished Goods, and Cost of Goods Sold, the related entry will include a Credit to cost of goods sold for $12000
How to solve for the costThe cost of a good is the total amount that was used in the purchase of a particular good form the market.
we have the manufacturing overhead to be = 30000 dollars
the work in progress = 30000 x 25 %
= 30000 x 0.25
= 7500
The finished goods = 30000 x 35 %
= 30000 x 0.35
= 10500
The cost of good sold = 30000 - 10500 - 7500
= 12000
Hence the cost of good sold is 12000
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complete questionManufacturing overhead applied $ 150,000
Actual amount of manufacturing overhead costs 120,000
Amount of overhead applied during the year that is in:
Work in Process $ 37,500 25 %
Finished Goods 52,500 35 %
Cost of Goods Sold 60,000 40 %
Total overhead applied $ 150,000 100 %
If the Manufacturing Overhead account is closed proportionally to Work in Process, Finished Goods, and Cost of Goods Sold, the related entry will include a ________.
debit to Cost of Goods Sold for $12,000
credit to Cost of Goods Sold for $12,000
credit to Cost of Goods Sold for $30,000
debit to Work in Process for $7,500
What needs to occur for a geometric series to converge?
Given a Geometric Series:
[tex]\sum_{n\mathop{=}1}^{\infty}a\cdot r^{n-1}[/tex]Where "r" is the ratio.
By definition:
[tex]undefined[/tex]Write the equation of a Circle with the given information.End points of a diameter : (11, 2) and (-7,-4)
The form of the equation of the circle is
[tex](x-h)^2+(y-k)^2=r^2[/tex]Where (h, k) are the coordinates of the center
r is the radius
Since the endpoints of the diameter are (11, 2) and (-7, -4), then
The center of the circle is the midpoint of the diameter
[tex]\begin{gathered} M=(\frac{11+(-7)}{2},\frac{2+(-4)}{2}) \\ M=(\frac{4}{2},\frac{-2}{2}) \\ M=(2,-1) \end{gathered}[/tex]The center of the circle is (2, -1), then
h = 2 and k = -1
Now we need to find the length of the radius, then
We will use the rule of the distance between the center (2, -1) and one of the endpoints of the diameter we will take (11, 2)
[tex]\begin{gathered} r=\sqrt[]{(11-2)^2+(2--1)^2} \\ r=\sqrt[]{9^2+3}^2 \\ r=\sqrt[]{81+9} \\ r=\sqrt[]{90} \\ r^2=90 \end{gathered}[/tex]Now substitute them in the rule above
[tex]undefined[/tex]A
piece of ribbon
was cut into
three parts in the ratio of 1:3'5
If the shortest was 11cm how long was the ribbon
Answer: Total Length of ribbon is 99 cm
Step-by-step explanation:
Here ribbon was cut into three parts in the ratio 1:3:5
let x be the common multiple of the above ratio
therefore, the lengths of the three parts of the ribbon is 1x,3x,5x
now, given is that the shortest part i.e 1x is equals to 11cm
i.e 1x=11
x=[tex]\frac{11}{1}[/tex]=11cm
now lengths of the ribbon will be
1x=11cm, 3x=3*11=33cm, 5x=5*11=55cm
now total length of piece of ribbon = 1x+3x+5x=9x=9*11=99cm
15 Which of the digits from 2 to 9 is 5544
divisible by?
Answer:
All of em, except 5
Step-by-step explanation:
5544 / 2 = 2772
5544 / 3 = 1848
5544 / 4 = 1386
5544 / 6 = 924
5544 / 7 = 792
5544 / 8 = 693
5544 / 9 = 616
Two cars start moving from the same point. One travels south at 24 mi/h and the other travels west at 18 mi/h. At what rate (in mi/h) is the distance between the cars increasing four hours later?
mi/h
The rate at which the distance between the two cars increased four hours later is 30 mi/h.
How to determine the rate?First of all, we would determine the distances travelled by each of the cars. The distance travelled by the first car after four (4) hours is given by:
Distance, x = speed/time
Distance, x = 24/4
Distance, x = 6 miles.
For the second car, we have:
Distance, y = speed/time
Distance, y = 18/4
Distance, y = 4.5 miles.
After four (4) hours, the total distance travelled by the two (2) cars is given by this mathematical expression (Pythagorean theorem):
z² = x² + y²
Substituting the parameters into the mathematical expression, we have;
z² = 6² + 4.5²
z² = 36 + 20.25
z² = 56.25
z = 7.5 miles.
Next, we would differentiate both sides of the mathematical expression (Pythagorean theorem) with respect to time, we have:
2z(dz/dt) = 2x(dx/dt) + 2y(dy/dt)
Therefore, the rate of change of speed (dz/dt) between the two (2) cars is given by:
dz/dt = [x(dx/dt) + y(dy/dt)]/z
dz/dt = [6(24) + 4.5(18)]/7.5
dz/dt = [144 + 81]/7.5
dz/dt = 225/7.5
dz/dt = 30 mi/h.
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Answer:
30 mi / hr
Step-by-step explanation:
First find out how far the cars are apart after 4 hours
24 * 4 = 96 mi = y
18 * 4 = 72 mi = x
Now use the pythagorean theorem
s^2 = ( x^2 + y^2 ) shows s = 120 miles apart at 4 hours
Now s^2 = x^2 + y^2 Differentiate with respect to time ( d / dt )
2 s ds/dt = 2x dx/ dt + 2y dy / dt
ds/dt = (x dx/dt + y dy/dt)/s
= (72(18) + 96(24)) / 120
ds/dt = 30 mi/hr
help pleaseeeeeeeeeeeeeeee
Answer:
(b) f(2) = 28
(c) f(-2) = -20
Step-by-step explanation:
f(2) = -2^3 + 7 x 2^2 - 2 x 2 + 12 f(-2) = -2^3 + 7 x -2^2 - 2 x -2 + 12
1. Calculate Exponents
-2^3 = -8 -2^3 = -8
2^2 = 4 -2^2 = -4
-8 + 7 x 4 - 2 x 2 + 12 -8 + 7 x -4 - 2 x -2 + 12
2. Multiply (left to right)
7 x 4 = 28 7 x -4 = -28
2 x 2 = 4 2 x -2 = -4
-8 + 28 - 4 + 12 -8 - 28 - (-4) + 12
3. Add (left to right)
-8 + 28 = 20 -8 + -28 = -36
20 - 4 = 16 -36 - (-4)= -32
16 + 12 = 28 -32 + 12 = -20
Use the rules of significant figures to answer the following question:43.5694 * 22.07A. 961.58B. 961C. 961.577D. 961.7
we have that
43.5694 * 22.07=961.576658
therefore
the answer is
961.577 -----> 6 figures
(remember that 43.5694 has 6 figures)
option C
A parents' evening was planned to start at
15h45. There were 20 consecutive
appointments of 10 minutes each and a
break of 15 minutes during the evening. At
what time was the parents evening due to
finish?
C O 19h15
O 19h20
O 19h00
O 20h00
O 19h30
The time on which parents evening was due to finish was 19 hour 20 minutes.
What is time and its unit?
Time is the ongoing pattern of existence and things that happen in what seems to be an irreversible order from the past, through the present, and into the future.
It is a component quantity of various measurements used to order events, compare the length of events or the time gaps between them, and quantify rates of change of quantities in objective reality or in conscious experience. Along with the three spatial dimensions, time is frequently considered a fourth dimension.
The International System of Units is built upon the seven base units of measurement stipulated by the Système International d'Unités (SI), from which all other SI units are derived. The primary unit of time is the second. The second can be shortened using either the letter S or the letter sec.
20 consecutive appointments of 10 mins = 20 × 10 mins
= 200 min
= 3 hours 20 mins
A break of 15 mins = 3 hours 20 mins + 15 min
= 3 hours 35 mins
The time that the parents evening due to finish = 15h 45 min +3h 35 mins
= 19h 20min
Thus, the time on which parents evening was due to finish was 19h 20min.
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A local video game store sells used games and new games. A new game costs$64, including tax. A used game costs $43, including tax. Luis bought 3 more used games than new games. Luis spent $343. How many used games did Luis purchase?
Given:
new game cost - $ 64
used game cost - $ 43
Luis spent $ 343
Required:
Number of used games Luis purchased
Solution
Let: x be the number of new games Luis bought
x + 3 be the number of used games Luis bought
Total cost = $ 343
Total Cost = (No. of new games bought)(Cost of new games) + (No. of used games bought)(Cost of used games)
$ 343 = ( x ) ( $ 64 ) + ( x + 3 ) ( $ 43 )
343 = 64x + 43 ( x + 3 )
343 = 64x + 43x + 129
343 - 129 = 107x
214 = 107x
2 = x
x = 2
x be the number of new games Luis bought 2
x + 3 be the number of used games Luis bought 2 + 3 = 5
Answer:
Luis purchased 5 used games
To check:
Substitute x into the equation,
343 = x ( 64 ) + (x + 3 ) (43)
343 = 2 ( 64 ) + (2 + 3)(43)
343 = 2 (64) + (5) (43)
343 = 128 + 215
343 = 343
The computed value of x satisfies the equation, Our answer is correct.
The sum of the squares of three consecutive odd numbers is 83. Find the numbers.
In order to represent three consecutive odd numbers, we can use the expressions "x", "x+2" and "x+4".
If we add the square of each number, the result is 83, so we can write the following inequality:
[tex]\begin{gathered} x^2+(x+2)^2+(x+4)^2=83\\ \\ x^2+x^2+4x+4+x^2+8x+16=83\\ \\ 3x^2+12x+20=83\\ \\ 3x^2+12-63=0\\ \\ x^2+4x-21=0 \end{gathered}[/tex]Let's solve this quadratic equation using the quadratic formula, with a = 1, b = 4 and c = -21:
[tex]\begin{gathered} x=\frac{-b\pm\sqrt{b^2-4a}c}{2a}\\ \\ x=\frac{-4\pm\sqrt{16+84}}{2}\\ \\ x=\frac{-4\pm10}{2}\\ \\ x_1=\frac{-4+10}{2}=\frac{6}{2}=3\\ \\ x_2=\frac{-4-10}{2}=\frac{-14}{2}=-7 \end{gathered}[/tex]If we assume the numbers are positive, the numbers are 3, 5 and 7.
(The other result, with negative numbers, would be -7, -5 and -3).
Hi, can you help me answer this question please, thank you!
Given:
The test claims that night students' mean GPA is significantly different from the mean GPA of day students.
Null hypothesis: the population parameter is equal to a hypothesized value.
Alternative hypothesis: it is the claim about the population that is contradictory to the null hypothesis.
For the given situation,
[tex]\begin{gathered} \mu_N_{}=\text{ Night students} \\ \mu_D=Day\text{ students} \end{gathered}[/tex]Null and alternative hypothesis is,
[tex]\begin{gathered} H_0\colon\mu_N=\mu_D \\ H_1\colon\mu_N_{}\ne\mu_D \end{gathered}[/tex]Answer: option f)
What point in the feasible region maximizes the objective function?
x>0
Y≥0
Constraints
-x+3≥y
{ y ≤ ½ x + 1
objective function: C = 5x - 4y
Answer:
(3, 0)
Maximum Value of Objective Function = 15
Step-by-step explanation:
This is a problem related to Linear Programming(LP)
In linear programming, the objective is to maximize or minimize an objective function subject to a set of constraints.
For example, you may wish to maximize your profits from a mix of production of two or more products subject to resource constraints.
Or, you may wish to minimize cost of production of those products subject to resource constraints..
The given LP problem can be stated in standard form as
Max 5x - 4y
s.t.
-x + 3 ≥ y
y ≤ 0.5x + 1
x ≥ 0, y ≥ 0
The last two constraints always apply to LP problems which means the decision variables x and y cannot be negative
It is standard to express these constraints with the decision variables on the LHS and the constant on the RHS
Rewriting the above LP problem using standard notation,
Let's rewrite the constraints using the standard form:
- x + 3 ≥ y
→ -x - y ≥ -3
→ x + y ≤ 3 [1]
y ≤ 0.5x + 1
→ -0.5x + y ≤ 1 [2]
The LP problem becomes
Max 5x - 4y
s. t.
x + y ≤ 3 [1]
-0.5x + y ≤ 1 [2]
x ≥ 0 [3]
y ≥0 [4]
With an LP problem of more than 2 variables, we can use a process known as the Simplex Method to solve the problem
In the case of 2 variables, it is possible to solve analytically or graphically. The graphical process is more understandable so I will use the graphical method to arrive at the solution
The feasible region is the region that satisfies all four constraints shown.
The graph with the four constraint line equations is attached. The feasible region is the dark shaded area ABCD
The feasible region has 4 corner points(A, B,C, D) whose coordinates can be computed by converting each of the inequalities to equalities and solving for each pair of equations.
It can be proved mathematically that the maximum of the objective function occurs at one of the corner points.
Looking at [1] and [2] we get the equalities
x + y = 3 [3]
-0.5x + y = 1 [4]
Solving this pair of equations gives x = 4/3 and y = 5/3 or (4/3, 5/3)
Solving y = 0 and x + y = 3 gives point x = 3, y =0 (3,0)
The other points are solved similarly, I will leave it up to you to solve them
The four corner points are
A(0,0)
B(0,1)
C(4/3, 5/3)
D(3,0)
The objective function is 5x - 4y
To find the values of x and y that maximize the objective function,
plug in each of the x, y values of the corner points
Ignoring A(0,0)
we get the values of the objective function at the corner points as
For B(0,1) => 5(0) - 4(1) = -4
For C(4/3, 5/3) => 5(4/3) - 4(5/3) = 20/3 - 20/3 = 0
For D(3, 0) => 5(3) - 4(0) = 15
So the values of x and y which maximize the objective function are x = 3 and y = 0 or point D(3,0)