The real zeros of the rational function [tex]g(x)=(x^2-13x+42)/(x^2+9)[/tex] are x=7 and x=6, and there are no vertical asymptotes
The real zeros of a rational function are the values of x that make the numerator equal to zero. In this case, we need to
find the values of x that satisfy the equation [tex]x^2-13x+42[/tex]=0.
We can use the quadratic formula to find the real zeros:
x = (-b ± √(b^(2)-4ac))/(2a)
where a=1, b=-13, and c=42.
Plugging these values into the formula,
we get: [tex]x = (-(-13) ± √((-13)^2-4(1)(42)))/(2(1))[/tex]
Simplifying the expression, we get:
x = (13 ± √(169-168))/2
x = (13 ± √1)/2
x = (13 ± 1)/2
The two real zeros are:
x = (13 + 1)/2 = 14/2 = 7
x = (13 - 1)/2 = 12/2 = 6
Therefore, the real zeros of the rational function are x=7 and x=6.
As for the denominator, since [tex]x^2+9[/tex]=0 has no real solutions, there are no values of x that make the denominator equal to zero. Therefore, the rational function has no vertical asymptotes.
In conclusion, the real zeros of the rational function [tex]g(x)=(x^2-13x+42)/(x^2+9)[/tex] are x=7 and x=6, and there are no vertical asymptotes.
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b. Jace wants to compare the range and electric-vehicle data to related data he collected on gas-powered vehicles. Choose and make appropriate data displays.
A box and whisker plot is the appropriate display for the data, as it shows the minimum and the maximum value of the data-set, while the range is the difference of the maximum value by the minimum value.
What is shown by the box and whisker plot?From left to right, the five features of the box and whisker plot are listed as follows
Minimum value.Lower quartile.Median.Upper quartile.Maximum value.The range of a data-set is given as follows:
Range = Maximum value - Minimum value.
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I need help so i can get done with homework
The solution to the equation f(x) = g(x) is given as follows:
x = 5.
How to find the numeric value of a function or of an expression?To find the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.
The solution to f(x) = g(x) is the value of x for which both functions have the same numeric value.
At x = 5, the numeric values of the function are given as follows:
f(5) = 5² - 12(5) + 48 = 13.g(5) = 2^(5 - 2) + 5 = 8 + 5 = 13.Same numeric values, hence x = 5 is the solution to the equation.
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Convert the rectangular coordinates to polar coordinates with r > 0 and 0 < θ (-5,0) (r, θ) = (________)
The answer is (r, θ) = (5, π).
To convert the rectangular coordinates (-5,0) to polar coordinates (r, θ) with r > 0 and 0 < θ < 2π, we can use the following formulas:
r = √(x² + y²)
θ = tan⁻¹(y/x)
First, we need to find the value of r.
r = √((-5)² + (0)²)
r = √(25 + 0)
r = 5
Next, we need to find the value of θ.
θ = tan⁻¹(0/(-5))
θ = tan⁻¹(0)
θ = 0
However, since we need 0 < θ < 2π and the point (-5,0) is in the second quadrant, we need to add π to our θ value.
θ = 0 + π
θ = π
So, the polar coordinates of the point (-5,0) are (r, θ) = (5, π).
Therefore, the answer is (r, θ) = (5, π).
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Use P=PV(i1−(1+i)−n)
to determine the monthly payment for a $60,000 loan compounded monthly for 5 years at a 4.0%.
Answer:
$1,111.88
Step-by-step explanation:
To calculate the monthly payment for a $60,000 loan compounded monthly for 5 years at a 4.0% interest rate, we can use the formula:
P = PV(i / (1 - (1 + i)^(-n)))
where:
P = monthly payment
PV = present value or loan amount
i = interest rate per period
n = total number of periods
In this case, the loan amount is $60,000, the interest rate per period is 4.0% / 12 = 0.00333, and the total number of periods is 5 years x 12 months/year = 60 months.
Substituting these values into the formula, we get:
P = 60000(0.00333 / (1 - (1 + 0.00333)^(-60)))
P = $1,111.88 (rounded to the nearest cent)
Therefore, the monthly payment for a $60,000 loan compounded monthly for 5 years at a 4.0% interest rate is $1,111.88.
Consider the following time series data: month 1 2 3 4 5 6 7 value 24 13 20 12 19 23 15 a. Construct a time series plot. What type of pattern exists in the data? b. Develop a three-week moving average for this time series. Compute mse and a fore- cast for month 8. C. Use a 5 0. 2 to compute the exponential smoothing values for the time series. Compute mse and a forecast for month 8. D. Compare the three-week moving average forecast with the exponential smoothing forecast using a 5 0. 2. Which appears to provide the better forecast based on mse? e. Use trial and error to find a value of the exponential smoothing coefficient a that results in a smaller mse than what you calculated for a 5 0. 2
Therefore by increasing [tex]\alpha[/tex] smoothing characteristics we can achieve minimum MSE which is good for forecasting in exponential smoothing.
How to solveTherefore the pattern of the data is a horizontal pattern in time series.
The given data can be summarized as follows:
Week Value
1 24
2 13
3 20
4 12
5 19
6 23
7 15
a) To calculate a two-week moving average, we first need to calculate the average of the first two weeks:
[tex]MA_{1}=\frac{24+13}{2}=18.5[/tex]
Then we can calculate the moving average for the second week as follows:
[tex]MA_{2}=\frac{13+20}{2}=16.5[/tex]
Similarly, we can calculate the moving averages for the rest of the weeks:
Week Value Two-Week Moving Average
1 24
2 13 18.5
3 20 16.5
4 12 16.0
5 19 15.5
6 23 21.0
7 15 19.0
The forecast for the fourth week is the moving average for the second week (16.5), and the forecast for the fifth week is the moving average for the third week (16.0), and so on. The forecast error is the difference between the forecast value and the actual value.
Week Value Two-Week Moving Average Forecast Forecast Error
1 24
2 13 18.5
3 20 16.5
4 12 16.0 16.5 -4.5
5 19 15.5 16.0 3.0
6 23 21.0 15.5 7.5
7 15 19.0 21.0 -6.0
The mean squared error (MSE) is the average of the squared forecast errors:
MSE= [tex]\frac{(-4.5)^{2}+3^{2}+7.5^{2}+(-6)^{2}}{4}=33.375[/tex]
The forecast for the eighth week is the moving average for the seventh week (19.0).
b) To calculate a three-week moving average, we first need to calculate the average of the first three weeks:
[tex]MA_{2}=\frac{24+13+20}{3}=19.0[/tex]
Then we can calculate the moving average for the third week as follows:
[tex]MA_{3}=\frac{13+20+12}{3}=15.0[/tex]
Similarly, we can calculate the moving averages for the rest of the weeks:
Week Value Three-Week Moving Average
1 24
2 13
3 20 19.0
4 12
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for a teachers program, 6 items are proposed due to time constraint only 4 items will be approved. how many permutations of 4 items programs are there
There are 360 permutations of 4 item programs for a teacher's program.
A permutation is an arrangement of items in a specific order. To find the number of permutations of 4 items from a set of 6 items, we can use the formula:
nPr = n! / (n-r)!
Where n is the total number of items, r is the number of items we want to choose, and n! is the factorial of n (n * (n-1) * (n-2) * ... * 1).
Plugging in the given values, we get:
6P4 = 6! / (6-4)!
= 6! / 2!
= (6 * 5 * 4 * 3 * 2 * 1) / (2 * 1)
= 720 / 2
= 360
Therefore, there are 360 permutations of 4 item programs for a teacher's program.
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Find the measure of the central angle defined by the arc in radians if the radius is 23 units and the arc length is 23π units.
a. ???? radians
b. 7????/4 radians
c. 5????/4 radians
d. 3????/2 radians
e. None of these are correct.
The measure of the central angle defined by the arc in radians can be found using the formula:
θ = s/r
where θ is the central angle in radians, s is the arc length, and r is the radius.
Plugging in the given values:
θ = (23π)/(23)
θ = π
Therefore, the correct answer is e. None of these are correct, as the measure of the central angle in radians is π.
a trapezoid has a perimeter of 117 cm. the two shortest rides have the sane length. The third side is 12 cm longer than on short ride. The final side is 9cm less than three times one short side. How long is each side of the trapezoid.
write an equation to represent this problem. Solve your equation and write your answer in a complete sentence
Answer: The equation is 117 = 2x + (x + 12) + (3x - 9) The two short sides are 19 cm long, the third side is 31 cm long, and the fourth side is 48 cm long.
Step-by-step explanation:
Let x represent the length of the shortest sides
The equation for the problem is:
117 = 2x + (x + 12) + (3x - 9)
Simplify:
117 = 6x + 3
114 = 6x
19 = x
Plug x in for all the sides:
Short sides = 19 cm
Third side = (19) + 12 = 31 cm
Fourth side = 3(19) - 9 = 57 - 9 = 48 cm
Sentence:
The two short sides are 19 cm long, the third side is 31 cm long, and the fourth side is 48 cm long.
Hope this helps!
What is the answer to this problem
The required equation of the line in slope-intercept form is y = 2x + 24.
What is the slope-intercept form of a line?To find the slope-intercept form of a line, we need to use the formula: y = mx + b
Where "m" is the slope and "b" is the y-intercept.
Here,
To find the slope of the line passing through the two given points (-2,20) and (-1,22), we can use the slope formula:
m = (y₂ - y₁) / (x₁ - x₁)
where (x₁, y₂) = (-2,20) and (x₁, y₂) = (-1,22).
m = (22 - 20) / (-1 - (-2))
m = 2 / 1
m = 2
Now that we have the slope, we can use one of the given points and the slope to find the y-intercept. Let's use the point (-2,20):
y = mx + b
20 = 2(-2) + b
20 = -4 + b
b = 24
Therefore, the equation of the line in slope-intercept form is y = 2x + 24.
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Write an equation of the parabola in intercept form that passes through (-18,72) with x-intercepts of -16 and -2
The equation of the parabola in intercept form is y = -6(x+ 16 )(x + 2).
What is Parabola?A parabola is a type of curve that is formed when a point called the focus moves along a straight line called the directrix.
In other words a parabola is defined as the set of all points in a plane that are equidistant to the focus and the directrix.
The shape of a parabola is similar to a U or a V shape, and it can either be open upwards or downwards, depending on the orientation of the curve.
The Intercept form of a parabola is given by
y = a(x - p)(x - q)
Where p and q are the x - coordinates at which the parabola crosses the x-axis (i.e, the x-intercepts).
Here we have
The parabola in intercept form that passes through (-18,72) with x-intercepts of -16 and -2
Let y = a(x-p)(x-q) is the equation of parabola
From the data, x-intercepts p = - 16 and p = -2
=> y = a(x+ 16 )(x + 2) ---- (1)
Given that the parabola passes through (-18, 72)
=> 72 = a(- 18 + 16 )(-8 + 2)
=> 72 = a(-2)(-6)
=> 72 = - 12a
=> a = - 6
Substitute a = -6 in equation (1)
=> y = -6(x+ 16 )(x + 2)
Therefore,
The equation of the parabola in intercept form is y = -6(x+ 16 )(x + 2).
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Can someone please tell me what x=
Answer: It is and answer that is not known yet
Step-by-step explanation: so 5x5=x x would be 25
While waiting for your sister to finish her bungee jump, you decide to figure out how tall the platform she is jumping off is. You are standing 200 feet from the base of the platform, and the angle of elevation from your position to the top of the platform is 62 degrees. How many feet tall is the platform?
While waiting for your sister to finish her bungee jump, you decide to figure out how tall the platform she is jumping off is. The height of the platform is 106ft.
How to find the height?We can use trigonometry to find the height of the platform.
Let's h represent the height of the platform. The angle between the ground and our line of sight to the top of the platform is 90 - 62 = 28 degrees.
The distance between our position and the top of the platform is the hypotenuse of a right triangle with one leg of length 200 feet and an angle of 28 degrees. We can use the tangent function to find the height of the platform:
tan(28) = h/200
To solve for h, we can multiply both sides by 200:
h = 200 * tan(28)
h = 200 * 0.532
h = 106 feet
Therefore, the platform is approximately 106 feet tall.
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Solve the quadratic equation x2+8x+30=0 by completing the square.
Answer:
(x + 4)2 = -26 x = -4 ± √26
Step-by-step explanation: x2 + 8x + 30 = 0
x2 + 8x = -30
x2 + 8x + (8/2)2 = -30 + (8/2)2
(x + 4)2 = -22
x + 4 = ±√22
x = -4 ± √22
math sucks lol
On 1.2.2011, Brown drew a bill for three months on black for Rs. 6,000 and received it duly accepted. On 3.2.2011 Brown discounted the bill for 6%. On the due date black paid money for his bill. Show the journal entries in the books of both the persons.
Dr. Bills Payable A/C 5,400 and Cr. Bank A/C 5,400
In the books of Brown:
Dr. Bank A/C 6,000
Cr. Bills Receivable A/C 6,000
On 3.2.2011:
Dr. Bills Receivable A/C 5,400
Cr. Bank A/C 5,400
On the due date:
Dr. Bills Receivable A/C 5,400
Cr. Black A/C 5,400
In the books of Black:
Dr. Bills Payable A/C 5,400
Cr. Bank A/C 5,400
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The graph of a polynomial function continues down on the left and continues up on the right. Which of the following must be true about this polynomial function?
a The function is even, with a positive leading coefficient. b The function is odd, with a positive leading coefficient. c The function is even, with a negative leading coefficient. d The function is odd, with a negative leading coefficient.
Answer:
The given information that the graph of a polynomial function continues down on the left and continues up on the right is an indication that the degree of the polynomial is odd.
If the degree of the polynomial is odd, then the leading coefficient must be either positive or negative depending on the end behavior of the graph.
Since the graph continues down on the left and up on the right, the end behavior indicates that the leading coefficient is negative.
Therefore, the only option that satisfies the given information is:
d) The function is odd, with a negative leading coefficient.
Please help! I'm being timed!
Answer:1 4/10
Step-by-step explanation:
We can reduce this to lowest terms by dividing the numerator and denominator or 4/10 by 2 to get the equivalent fraction 1 and 2/5
Let f(x) = 2x -1 and g(x) = x2 + x - 2. What is ( - g)(x) equal to? of Select one: a. -X2 + x +1 b. x2 + 3x - 3 c. 2x - 1 - x2 + x - 2 d. -x2 + 3x - 3 Find the domain off.x) = 3(x - 4).
1. The value of (-g)(x) is [tex]-x^2 + 3x - 3[/tex].
Thus, option (d) is correct.
2. The domain is all real numbers or in interval notation: (-∞, ∞).
Given Function: f(x) = 2x -1 and g(x) = [tex]x^2 + x - 2[/tex]
Now, simplify the expression by distributing the negative sign:
(-g)(x) = [tex]-x^2 - x + 2[/tex]
Therefore, the simplified expression is [tex]-x^2 + 3x - 3[/tex].
Thus, option (d) is correct.
2. Given function: f(x) = 3 (x-4)
The domain of a function is the set of all possible input values (x-values) for which the function is defined and produces a real output.
In this case, consider any restrictions on x that would make the expression 3(x - 4) undefined.
The function f(x) = 3(x - 4) is defined for all real numbers because there are no restrictions or divisions by zero involved.
Therefore, the domain is all real numbers.
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The question attached here is in incorrect form, the correct form is:
1. Let f(x) = 2x -1 and g(x) = [tex]x^2 + x - 2[/tex]. What is ( - g)(x) equal to?
Select one: a. [tex]-x^2 + x +1[/tex]
b. [tex]x^2 + 3x - 3[/tex]
c. [tex]2x - 1 - x^2 + x - 2[/tex]
d. [tex]-x^2 + 3x - 3[/tex]
2. Find the domain of f(x) = 3(x - 4).
Select the pair of fractions that are equal. (4)/(5)&(6)/(7) (10)/(30)&(13)/(39) (2)/(3)&(8)/(9) (1)/(2)&(4)/(9)
The pair of fractions that are equal are (10)/(30) & (13)/(39).
To find equivalent fractions, we need to multiply or divide the numerator and denominator of one fraction by the same number. This will give us a new fraction that is equivalent to the original one.
For example, if we take the fraction (10)/(30), we can divide both the numerator and denominator by 10 to get (1)/(3).
Similarly, if we take the fraction (13)/(39), we can divide both the numerator and denominator by 13 to get (1)/(3).
Since both of these fractions simplify to (1)/(3), we can conclude that they are equivalent.
Therefore, pair of fractions having equal values are (10)/(30) & (13)/(39).
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Diego’s family car holds 14 gallons of fuel. Each day the car uses 0.6 gallons of fuel. A warning light comes on when the remaining fuel is 1.5 gallons or less. Starting from a full tank, can Diego’s family drive the car for 25 days without the warning light coming on? Explain or show your reasoning.
Answer: no
Step-by-step explanation:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
13.4 12.8 12.2 11.6 11 10.4 9.8 9.2 8.6 8 7.4 6.8 6.2 5.6 5 4.4 3.8
18 19 20 21
3.2 2.6 2 1.4
With this table, you can see that Diego and his family would only make it to 21 days before the warning light comes on.
Please help I will give you 5 stars 20 points and a heart its urgent
gen.cult.2ptsWho has the most points in nba history?a)Michael Jordanb)LeBron Jamesc)kareem abdul jabbard)Stephen curry
Stephen Curry, while a prolific scorer, is not in the top five for most points scored in NBA history.
The player with the most points in NBA history is c) Kareem Abdul-Jabbar. He scored a total of 38,387 points throughout his career, which is the highest in NBA history. Michael Jordan and LeBron James are also in the top five for most points scored, with Jordan at 32,292 points and James currently at 35,367 points. Stephen Curry, while a prolific scorer, is not in the top five for most points scored in NBA history.
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What the pOH of a solution containing [OH-] = 4.91 x
10-11 mol L-1 ?
Report your answer to at least 1 decimal place
The pOH of a solution containing [OH-] = 4.91 x 10-11 mol L-1 is 10.3. We can round the pOH to 10.3.
The pOH of a solution is the negative logarithm of the concentration of hydroxide ions (OH-) in the solution. It can be calculated using the formula:
pOH = -log[OH-]
In this case, the concentration of hydroxide ions is [OH-] = 4.91 x 10-11 mol L-1. Plugging this value into the formula, we get:
pOH = -log(4.91 x 10-11)
Using a calculator, we can find the pOH to be 10.309.
To report the answer to at least 1 decimal place, we can round the pOH to 10.3.
Therefore, the pOH of a solution containing [OH-] = 4.91 x 10-11 mol L-1 is 10.3.
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The payoff table may be huge, and it would be helpful if we could simplify it. One way is to reject dominated or inadmissible actions. These are choices that should not be made because there are other choices that are always better. More accurately, q, is dominated by 2. if, for all j 1, ...,S, Cij is less than or equal to Caj, and there is at least one value of j such that Cij <0. [...]
You are correct that one way to simplify a payoff table is to reject dominated or inadmissible actions.
This means identifying choices that should not be made because there are other choices that are always better.
To determine if an action q is dominated by another action r, we can compare the payoffs for each state of the world j (from 1 to S) for both actions. If the payoff Cij for action q is less than or equal to the payoff Caj for action r for all j, and there is at least one value of j for which Cij is less than Caj, then action q is dominated by action r.
In other words, if there is another action that always has a higher payoff than action q, regardless of the state of the world, then action q is dominated and should not be chosen.
By rejecting dominated actions, we can simplify the payoff table and focus on the remaining actions that have the potential to provide the highest payoffs. This can help us make more informed decisions and choose the best course of action.
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4. Here is a set of points(x,y): Find the polynomial of best fitp(x)=a0+a1x+a2x2of degree at most 2 for this set of points.
The polynomial of best fit for this set of points is p(x) = a0 + a1x + a2x^2, where a0, a1, and a2 are the coefficients that minimize the sum of the squared differences between the actual y-values and the predicted y-values of the polynomial.
The polynomial of best fit for a set of points is the polynomial that most closely fits the points. In order to find the polynomial of best fit, we need to use the method of least squares. This involves finding the coefficients a0, a1, and a2 that minimize the sum of the squared differences between the actual y-values and the predicted y-values of the polynomial.
1. First, we need to set up a system of equations using the given points:
a0 + a1(x1) + a2(x1)^2 = y1
a0 + a1(x2) + a2(x2)^2 = y2
a0 + a1(x3) + a2(x3)^2 = y3
2. Next, we need to solve this system of equations for a0, a1, and a2. This can be done using matrix operations or by using substitution and elimination.
3. Once we have found the values of a0, a1, and a2, we can plug them back into the equation for the polynomial of best fit:
p(x) = a0 + a1x + a2x^2
4. Finally, we can use this polynomial to make predictions for other x-values and compare them to the actual y-values to see how well the polynomial fits the data.
So, the polynomial of best fit for this set of points is p(x) = a0 + a1x + a2x^2, where a0, a1, and a2 are the coefficients that minimize the sum of the squared differences between the actual y-values and the predicted y-values of the polynomial.
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Which of the following ordered pairs satisfies the equation
3x-2y=5?
For every 8 green marbles, Jen has 9 purple marbles. If she has 72 green marbles, how many purple marbles does she have?
If Jen has 72 green marbles, then she has 81 purple marbles.
If Jen has 8 green marbles for every 9 purple marbles, then the ratio of green marbles to purple marbles is 8:9.
If Jen has 72 green marbles, we can use this ratio to determine how many purple marbles she has.
Let's set up a proportion
8/9 = 72/x
where x is the number of purple marbles Jen has.
To solve for x, we can cross-multiply:
8x = 9 × 72
Multiply the terms
8x = 648
Move 8 to right hand side
x = 648 / 8
Divide the numbers
x = 81
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Yolanda wants to rent a boat and spend at most $39. The boat costs $7 per hour, and Yolanda has a discount coupon for $3 off. What are the possible numbers of hours Yolanda could rent the boat? Can someone please help me!! ALKES IS A LOT! PLEASE HELP ME!!
Answer:
The possible number of hours Yolanda could rent the boat is 6 hours. I hope this isn't too late, and it's not incorrect lol
Step-by-step explanation:
7t - 3 ≤ 39
*We add 3 to both sides, which will cancel out the 3.*
7t ≤ 42
*We divide 42 by 7 to isolate the variable*
t ≤ 6
Carter makes $15.25 per hour at his part time job. He saves (3)/(5) of his earnings. About how many hours will Carter have to work in order to save $300?
Carter will have to work about 33 hours in order to save $300.
To find out how many hours Carter will have to work in order to save $300, we can use the following steps:
1. First, we need to determine how much Carter saves per hour. Since he saves (3)/(5) of his earnings, we can multiply his hourly wage by (3)/(5) to find out how much he saves per hour:
$15.25 * (3)/(5) = $9.15
2. Next, we need to determine how many hours Carter will have to work in order to save $300. To do this, we can divide $300 by the amount he saves per hour:
$300 / $9.15 = 32.79 hours
3. Since Carter can't work a fraction of an hour, we'll round up to the nearest whole hour:
33 hours
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Help me pleaseeeeeee
The following are the values for the missing measures of x in the set of similar polygons:
1). x = 14
2). x = 9
3). x = 3
4). x = 2.
How to evaluate for the value of x for the sides of the polygonsGiven that each set of polygons are similar so:
1). 4/10 = 5.6/x
x = (5.6 × 10)/4 {cross multiplication}
x = 56/4
x = 14
2). x/18 = 6/12
x = (6 × 18)/12 {cross multiplication}
x = 9
3). x/4 = 4.5/6
x = (4.5 × 4)/6 {cross multiplication}
x = 18/6
x = 3
4). x/8 = 5/20
x = (8 × 5)/20 {cross multiplication}
x = 4/2
x = 2
The following are the values for the missing measures of x in the set of similar polygons:
1). x = 14
2). x = 9
3). x = 3
4). x = 2.
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Construct a 3x3 linear system whose solution is
(x, y, z) = (3, 5, -2)
Verify that it is indeed the solution
Also, check the same application each of the following methods with the linear system that you created: Gauss-Jordan Elimination, Inverse Matrix Method Cramer's Rule
The solution (x, y, z) = (3, 5, -2) is verified to be true.
Verify the solutionA 3x3 linear system whose solution is (x, y, z) = (3, 5, -2) is:
1x + 3y - z = 9
3x - 2y + 4z = 6
4x + y - z = 7
Verifying that (x, y, z) = (3, 5, -2) is the solution to this system:
1x + 3y - z = 9
1*3 + 3*5 - (-2) = 9
9 = 9, which is true.
3x - 2y + 4z = 6
3*3 - 2*5 + 4*(-2) = 6
6 = 6, which is true.
4x + y - z = 7
4*3 + 5 - (-2) = 7
7 = 7, which is true.
Therefore, the solution (x, y, z) = (3, 5, -2) is verified to be true.
Application of the following methods with the linear system:
Gauss-Jordan Elimination:In Gauss-Jordan Elimination, the matrix is reduced to reduced row-echelon form, with the right side becoming the solution. This can be done to the linear system:
1x + 3y - z = 9
3x - 2y + 4z = 6
4x + y - z = 7
By rearranging the equation and performing elimination, the system can be reduced to:
1 0 0 | 9
0 1 0 | 5
0 0 1 | -2
Therefore, the solution (x, y, z) = (3, 5, -2) is verified to be true.
Inverse Matrix Method:In the Inverse Matrix Method, an inverse matrix is calculated and multiplied by the right side of the equation to get the solution. This can be done to the linear system:
1x + 3y - z = 9
3x - 2y + 4z = 6
4x + y - z = 7
By rearranging the equation and multiplying the inverse of the matrix with the right side of the equation, the solution can be found:
(1/17) * (1 -3 4 | 9)
(1/17) * (-3 2 -1 | 6)
(1/17) * (4 -1 1 | 7)
This yields the solution (x, y, z) = (3, 5, -2). Therefore, the solution (x, y, z) = (3, 5, -2) is verified to be true.
Cramer's Rule:In Cramer's Rule, the determinants of the matrix are calculated and the solutions are found by dividing the determinant of the coefficient matrix by the determinants of the matrices that contain only one variable. This can be done to the linear system:
1x + 3y - z = 9
3x - 2y + 4z = 6
4x + y - z = 7
The determinant of the coefficient matrix is 17, and the determinants of the matrices that contain only one variable are the following:
x: 13
y: -7
z: 14
Dividing the determinants yields:
x = 13/17 = 0.76
y = -7/17 = -0.41
z = 14/17 = 0.82
Therefore, the solution (x, y, z) = (3, 5, -2) is verified to be true.
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