Only (A) and (B) are true.
Explanation:Variable distributions:
X: Since we are sampling a specific number (20) of individuals with a known probability of color blindness (8%), X follows a binomial distribution with n = 20 and p = 0.08.
Y: Similarly, Y follows a binomial distribution with n = 40 and p = 0.01.
Z: Z is not a simple binomial because it combines two independent binomial variables (X and Y) with different parameters. Therefore, Z's distribution is not directly binomial.
Probability of 2 color-blind males:
Using the binomial probability formula for X, the probability of exactly 2 color-blind males (out of 20) is:
P(X = 2) = 20C2 * 0.08^2 * (1 - 0.08)^18 ≈ 0.2711
Therefore, only statements (A) and (B) are true:
(A) True: X is binomial with n = 20 and p = 0.08.
(B) True: Y is binomial with n = 40 and p = 0.01.
Statements (C), (D), and the answer choices for the probability of 2 color-blind males are incorrect.
Find the quotient. -71/-12
Answer:
5.92
Step-by-step explanation:
-71/-12=5.92
Answer:
[tex]\frac{71}{12}[/tex]
Step-by-step explanation:
Reduce the expression, if possible, by cancelling the common factors.
Exact Form :
[tex]\frac{71}{12}[/tex]
Decimal Form :
5.916
Mixed Number Form :
5[tex]\frac{11}{12}[/tex]
Hope this helps,
Davinia.
If the cost to the store is $20 and the selling price is $32 what is the markup?
Answer:
60%
Step-by-step explanation:
It went up 12 dollars, and you need to find how much 12 is of 20 as a percent. You do 12/20 times 5/5 and get 60/100, so 60% markup
Answer:
Step-by-step explanation:
The Answer Well -12$
20 - 32 = -12
A circle has a diameter of 26 units what is the area of the circle to the nearest hundredth of a square unit
Answer:
The area of the circle is [tex]530.93\ units^{2}[/tex]
Step-by-step explanation:
we know that
The area of a circle is equal to
[tex]A=\pi r^{2}[/tex]
we have
[tex]r=26/2=13\ units[/tex] ----> the radius is half the diameter
assume
[tex]\pi=3.1416[/tex]
substitute
[tex]A=(3.1416)(13)^{2}=530.93\ units^{2}[/tex]
Select the graph of the solution. Click until the correct graph appears.
|x| = 7
Answer:
Picture 1 is the answer.
Step-by-step explanation:
The expression states that the absolute value of a number x , is equal to the number 7. Absolute Values have an 2 inputs for every output (except for 0), the negative and positive inputs both output the same positive number.
Example: abs(-5) = abs(5) = 5
The Absolute value of -5 and 5 both output 5. Therefore, there are two possible x values for the answer to be 7 and those values are -7 and 7. Since these are the only possible values they would be represented on a number line as closed dots.
The only picture with closed dots on both -7 and 7 is picture 1. So that is the answer.
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
Answer: graph one
Step-by-step explanation:
The independent variable of a data set is y while the dependent variable is z. Which of these is a response variable
Answer:
Dependant Variable
Step-by-step explanation:
Answer:
Step-by-step explanation:
The response variable is the dependent variable.
Parallelogram PQRS is shown on the coordinate grid. Parallelogram PQRS is dilated with the origin as the center of dilation using the rule (x, y) → (0.75x, 0.75y) to create parallelogram P'Q'R'S'.
Which statement is true?
A) Parallelogram P'Q'R'S' is larger than parallelogram PQRS, because the scale factor is greater than 1.
B) Parallelogram P'Q'R'S' is smaller than parallelogram PQRS, because the scale factor is less than 1.
C) Parallelogram P'Q'R'S' is smaller than parallelogram PQRS, because the scale factor is greater than 1.
D) Parallelogram P'Q'R'S' is larger than parallelogram PQRS, because the scale factor is less than 1.
The transformed parallelogram P'Q'R'S' is smaller than the original parallelogram PQRS, due to a scale factor of less than 1 (0.75) being applied in this dilation.
Explanation:The given transformation (x, y) → (0.75x, 0.75y), the scale factor is 0.75, indicating a reduction in size. This means that the transformed parallelogram P'Q'R'S' is smaller than the original PQRS. The scale factor of 0.75 signifies that both the x and y coordinates of every point in the original parallelogram are multiplied by 0.75, leading to a proportional reduction in dimensions. Therefore, option B) stating that Parallelogram P'Q'R'S' is smaller than parallelogram PQRS due to the scale factor being less than 1, accurately describes the transformation, illustrating the diminished size of the transformed figure in comparison to the original one.
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The correct answer is B: Parallelogram P'Q'R'S' is smaller than parallelogram PQRS because the dilation scale factor is 0.75, which is less than 1.
The question asks about the effect of a given dilation on a geometric figure, specifically a parallelogram named PQRS, which is transformed into parallelogram P'Q'R'S'.
This transformation is performed using the dilation rule (x, y) → (0.75x, 0.75y), which indicates that every point on the original parallelogram is multiplied by the scale factor of 0.75 when creating the new parallelogram.
Since the scale factor is less than 1, each coordinate is reduced to 75% of its original value, resulting in a figure that is proportionally smaller than the original.
Therefore, the correct statement about the relationship between the original parallelogram PQRS and the dilated parallelogram P'Q'R'S' is Option B: Parallelogram P'Q'R'S' is smaller than parallelogram PQRS, because the scale factor is less than 1.
A building in a downtown business area casts a shadow that measures 88 meters along the ground. The straight-line distance from the top of the building to the end of the shadow it creates a 32 angel with the ground. What is the approximate height of the building? Round your answer to the nearest meter
Answer:
=55m
Ahh. Correct me if I'm wrong.
A sprinkler that sprays water in a circular area can be adjusted to spray up to 30 feet. Turning the radius-reduction screw on the top of the nozzle let's people reduce the radius by up to 25percent. To the nearest tenth, what is the maximum area of lawn that can be watered by the sprinkler if the radius reduction is used at full capacity?
Answer:
Area_lawn = 393.75 π ft^2
Step-by-step explanation:
Maximum radius : 30 feet
Minimum radius: 30 feet - 0.25*(30feet) = 22.5 feet
(25 percent reduction)
To find the area of lawn that can be watered, we just need to calculate the area for the maximum radius and the minimum radius, and then subtract them.
Since the sprinklers have a circular area:
Area = π*radius^2
Max area = π*(30 ft)^2 = 900π ft^2
Min area = π*(22.5 ft)^2 = 506.25π ft^2
Maximum area of lawn that can be watered by the sprinkler:
Area_lawn = Max area - Min area = 900π ft^2 -506.25π ft^2
Area_lawn = 393.75 π ft^2
Answer:
1,590.4 sq ft
Step-by-step explanation:
what is a formula for the nth term of a given sequence
-12,-16,-20...
an = -12(-4)^n
a^n = -12 - 4 (n+1)
an = -8 - 4n
an = -12(-4)^n-1
Answer:
[tex]a_n=-12-4(n-1)[/tex]
Step-by-step explanation:
The given sequence is
-12,-16,-20...
The first term of this sequence is [tex]a_1=-12[/tex].
The common difference is
[tex]d=-16--12[/tex]
[tex]d=-16+12=-4[/tex]
The nth term of this arithmetic sequence is;
[tex]a_n=a_1+d(n-1)[/tex]
We substitute the values for the first term and the common difference to obtain;
[tex]a_n=-12-4(n-1)[/tex]
Answer:
[tex]a_{n} = -12-4(n-1)[/tex]
Step-by-step explanation:
We have given a arithmetic sequence.
-12,-16,-20,...
We have to find formula for a given sequence.
The general formula for nth term of sequence is :
[tex]a_{n} = a_{1}+d(n-1)[/tex]
In given sequence,
[tex]a_{1} = -12[/tex]
d is the common difference between consecutive terms.
d = -16-(-12) = -16+12
d = -4
Putting given values in formula, we have
[tex]a_{n} = -12-4(n-1)[/tex] which is the answer.
Harry got 42 out of 49 correct in his test. What fraction of marks did he get wrong
Answer:
42 correct; if all were answered, that means 7 were wrong. 7/49 reduces to 1/7; one seventh were wrong.
What Is The Surface Area Of this triangular prism? The Base Of Each Triangle 42m And the height of the triangular base is 20m. (Not A Multiple Choice Question)
Answer:
The surface area is [tex]2,440\ m^{2}[/tex]
Step-by-step explanation:
see the attached figure to better understand the problem
we know that
The surface area of a triangular prism is equal to
[tex]SA=2B+PH[/tex]
where
B is the area of the triangular base
P is the perimeter of the triangular face
H is the height of the prism
Find the area of the base B
[tex]B=\frac{1}{2}(42)(20)=420\ m^{2}[/tex]
Find the perimeter of the base P
[tex]P=(42+29+29)=100\ m[/tex]
we have
[tex]H=16\ m[/tex]
substitute the values
The surface area is
[tex]SA=2(420)+100(16)=2,440\ m^{2}[/tex]
Which of the following is the best estimate of the area of the irregular shape? HELP PLEASE!!
ANSWER: 15.5 units^2
(Please show steps)
Graph the logarithmic function y=log(x-2)
Answer:
So the answer is (3, 0) on a graph.
Step-by-step explanation:
What I use is a graphing calculator. It helps tremendously. It is called desmos.com. Hope this helps.
Answer:
The given function is
[tex]y=log(x-2)[/tex]
To graph any function, we just have to five arbitrary values to x-variable, and see the y-values that gives to formed coordinate pairs to graph them then.
In this case, the table that shows these values is gonna be
X Y
2.5 -0.3
3 0
3.5 0.2
4 0.3
4.5 0.4
Notice that we started from [tex]x=2.5[/tex]. The reason is beacuse the given logarithmic function is not defined for values equal or lower than 2, when taht happens, the function becoms undetermined. This means the domain of the given function has to be restricted to [tex]D= (x>2)[/tex], otherwise the function won't be defined.
Next, we evalute the function for each case.
[tex]x=2.5[/tex]
[tex]y=log(x-2)=log(2.5-2)=log(0.5) \approx -0.3[/tex]
[tex]x=3\\y=log(x-2)=log(3-2)=log(1)=0[/tex]
[tex]x=3.5\\y=log(x-2)=log(3.5-2)=log(1.5) \approx 0.2[/tex]
[tex]x=4\\y=log(x-2)=log(4-2)=log(2) \approx 0.3[/tex]
[tex]x=4.5\\y=log(x-2)=log(4.5-2)=log(2.5) \approx 0.4[/tex]
When you have the table completed, you proceed to graph each coordinate.
In this case, the graph of the logarithmic function would be as the image attached. That's all the process to graph the function.
(In the graph, you can observe how fast the function drops when is near to x=2, that's the undetermination behaviour we talk about lines above).
HELP ASAP PLEASE!!!
In the equation
y=f(x)+k, the k value
A. Shifts the graph down k units
B. Shifts the graph up k unit
C. Shifts the graph to the left k units
D. Shifts the graph to the left k unit
im pretty sure it’s D
The k value in the equation y = f(x) + k causes a vertical shift of the graph. If k is positive, the graph shifts up; if k is negative, it shifts down. Thus, the correct answer is B.
In the function transformation y = f(x) + k, the term k represents a vertical shift of the graph. Specifically:
If k is positive, the graph of the function f(x) shifts up by k units.If k is negative, the graph of the function f(x) shifts down by the absolute value of k units.Thus, the correct answer to the question is:
B. Shifts the graph up k units.
To summarize, the value k in the equation y = f(x) + k causes a vertical shift upwards if it's positive and downwards if it's negative.
Please answer this question only if you know the answer!
Answer:
b. 6 cm
Step-by-step explanation:
Q is the midpoint of PR, so QR = PR/2 = (12 cm)/2 = 6 cm
Please Help Me. How do you form an augmented matrix from the system of linear equations and solve.
This is Algebra 2. I need help solving questions 6.3 & 6.4. I've read the instructions and I am not understanding what their asking me.
Answer:
6.3 (x, y) = (3, 1)
6.4 (x, y) = (-4, 2)
Step-by-step explanation:
An augmented matrix in this context is an array of the coefficients in the equation. For example, the equation x -2y = 3 can be represented by the row of numbers 1 -2 3. When writing the matrix by hand, we usually just separate the elements in a row using blank space. Some calculators require a comma or other character between matrix elements. The entire array is enclosed in square brackets.
An augmented matrix may have a vertical line separating the square matrix of coefficients from the rightmost column of constants.
Many graphing calculators have functions available for entering and solving linear equations in this form.
__
6.3 The augmented matrix is ...
[tex]\left[\begin{array}{cc|c}2&3&9\\1&-4&-1\end{array}\right][/tex]
This array can be used to solve the system of equations using "row operations." The idea is that any row can be multiplied by any number, and any row can be added to any other row (multiplied by some number or not). So, the same techniques used to solve a system by elimination can be used here.
We can, for example, subtract twice the second row from the first. Then we have ...
[tex]\left[\begin{array}{cc|c}2-2(1)&3-2(-4)&9-2(-1)\\1&-4&-1\end{array}\right]\\\\\left[\begin{array}{cc|c}0&11&11\\1&-4&-1\end{array}\right] \quad\text{simplified}\\\\\left[\begin{array}{cc|c}0&1&1\\1&-4&-1\end{array}\right] \quad\text{first row divided by 11}\\\\\left[\begin{array}{cc|c}0&1&1\\1+4(0)&-4+4(1)&-1+4(1)\end{array}\right] \quad\text{4 times row 1 added to row 2}\\\\\left[\begin{array}{cc|c}0&1&1\\1&0&3\end{array}\right] \quad\text{simplified}[/tex]
When we turn this last matrix back into equations, we get ...
0x +y = 1
1x + 0y = 3
That is, the solution is (x, y) = (3, 1).
__
6.4 The augmented matrix is ...
[tex]\left[\begin{array}{cc|c}4&5&-6\\3&2&-8\end{array}\right][/tex]
Let's start the solution of this one by dividing the first row by 4.
[tex]\left[\begin{array}{ccc}1&\frac{5}{4}&-\frac{3}{2}\\3&2&-8\end{array}\right] \\\\\left[\begin{array}{ccc}1&\frac{5}{4}&-\frac{3}{2}\\3-3(1)&2-3(\frac{5}{4})&-8-3(-\frac{3}{2})\end{array}\right] \quad\text{row 2 minus 3 times row 1}\\\\\left[\begin{array}{ccc}1&\frac{5}{4}&-\frac{3}{2}\\0&-\frac{7}{4}&-\frac{7}{2}\end{array}\right] \quad\text{simplify}\\\\\left[\begin{array}{ccc}1&\frac{5}{4}&-\frac{3}{2}\\0&1&2\end{array}\right] \quad\text{multiply row 2 by -4/7}[/tex]
We can eliminate the y-term in the first row by subtracting 5/4 times the second row. This will put the matrix in the form that shows us the solution directly.
[tex]\left[\begin{array}{ccc}1-\frac{5}{4}(0)&\frac{5}{4}-\frac{5}{4}(1)&-\frac{3}{2}-\frac{5}{4}(2)\\0&1&2\end{array}\right] \quad\text{row 1 minus 5/4 times row 2}\\\\\left[\begin{array}{ccc}1&0&-4\\0&1&2\end{array}\right][/tex]
This tells us ...
x = -4
y = 2
So the solution is (x, y) = (-4, 2).
An augmented matrix organizes the coefficients of a system of linear equations. After setting up the matrix with the coefficients from the equations, methods like Gaussian elimination or Gauss-Jordan elimination can be used to solve the matrix and consequently, solve the system.
Explanation:An augmented matrix is a compact way of keeping track of the coefficients in a system of linear equations. Here's how to form it and solve the system:
Identify the coefficients of the variables in the system of linear equations. Form a matrix based on these coefficients. For example, for the system of equations 2x + 3y = 7 and x - 4y = 11, the augmented matrix would be (2,3,7) on the first row and (1,-4,11) on the second row. Once the augmented matrix is formed, you can then use Gaussian elimination or Gauss-Jordan elimination methods to solve it.
Gaussian elimination involves using elementary row operations to manipulate the matrix to a form where the solution is easier to find, often called reduced row echelon form. For the Gauss-Jordan method, the main idea is to get zeros below (and above when possible) a leading 1, and make sure that leading 1 is the only nonzero entry in its column.
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Which statement regarding these four states is true?
The state with the lowest population has the greatest population density.
The state with the second lowest population has the lowest population density.
The state with the lowest population has the lowest population density.
The state with the second greatest population has the lowest population density.
Answer:
The state with the second lowest population has the lowest population density.
Step-by-step explanation:
The state with the second lowest population has the lowest population density.
State B has the second lowest population, and it has the lowest population distance considering it has a substantial amount of area compared to other states, yet 1, 333, 089 people, in comparison it therefore as the lowest population density.
The statement "The state with the second-lowest population has the lowest population density." is true.
How to find population density?The population density can be found by dividing the total population of an area by the area.
The correct statement can be determined below:We must calculate the population densities of all the states to determine which state has the highest density.
This can be done as shown below:
State A:
Population density = Population/Area
= 1,055,173/2,677
= 394.162
State B:
Population density = Population/Area
= 1,333,089/36,418
= 36.605
State C:
Population density = Population/Area
= 3,596,677/5,543
= 648.87
State D:
Population density = Population/Area
= 6,745,408/10,555
= 639.072
We can see that the state with the second-lowest population has the lowest population density.
Therefore, we have found that the statement "The state with the second-lowest population has the lowest population density." is true. The correct answer is option B.
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A model car is 12 inches long. Using a scale of 3 inches : 4 feet, how long is the actual car?
Answer:
16 feet
Step-by-step explanation:
Since the scale is 3 in : 4 ft, the model car is 16 ft.
To find the actual length of a model car given a scale of 3 inches : 4 feet with the model car being 12 inches long, set up a proportion to calculate the actual car length, which is 16 feet.
A scale of 3 inches : 4 feet means that for every 3 inches on the model car, the actual car is 4 feet long. Given that the model car is 12 inches long, you can set up a proportion:
3 inches = 4 feet12 inches = x feetSolving for x, you get the actual length of the car to be 16 feet.
A square is 8 inches long each side a rectangle is 6 inches wide and 10 inches long wich shape has the greater perimeter? Explain
After calculating the perimeters, both the square and the rectangle have the same perimeter of 32 inches when the square has each side measuring 8 inches and the rectangle has dimensions of 6 inches by 10 inches.
Explanation:To determine which shape has the greater perimeter, calculate the perimeter of both the square and the rectangle. The formula for the perimeter of a square is 4 × side length, and the formula for the perimeter of a rectangle is 2 × (length + width).
For the square with each side measuring 8 inches, its perimeter is 4 × 8 = 32 inches.
For the rectangle with a width of 6 inches and a length of 10 inches, its perimeter is 2 × (10 + 6) = 2 × 16 = 32 inches.
Therefore, both the square and the rectangle have the same perimeter of 32 inches.
Madison enjoys the game of golf. He knows that he will one-putt a green 15% of the time, two-putt 20% of the time, three-putt 35% of the time, and four-putt 30% of the time. Find the expected value for the number of putts Madison will need on any given green. Make sure to write down the entire equation that you used to solve this problem.
Answer:
The formula to calculate standard deviation from probability is \sqrt(n*p*(1-p)). n is the sample size, and 200 in this case (number of putts for practice). p is 80% or 0.8, the probability that he can make it. So the standard deviation is \sqrt(200*0.8*(1-0.8)=\sqrt(200*0.8*0.2)=\sqrt(16)=4.
Please help me out :)
Answer: x=12
Explanation:
In a parallelogram, adjacent angles equals 180 (consecutive angles). Therefore, (132-x)+(6x-12)=180
Simplify:
-x+6x+132-12=180
5x+120=180
5x=60
x=12
What is the area of this triangle? Enter your answer as a decimal. Round only your final answer to the nearest tenth.
___in^2
Answer:
56.1 in^2
Step-by-step explanation:
When given sides "a" and "b" and the angle α between them, the applicable formula for the area of the triangle is ...
A = (1/2)ab·sin(α)
Substituting the given values, we find the area to be ...
A = (1/2)(14.2 in)(8 in)sin(99°) ≈ 56.1 in^2
Answer:
56.1
Step-by-step explanation:
Diagram
Let's look at the diagram for a moment.
The 8 in line has been extended.
The dashed line is the height associated with the 8 inch line
The dashed line and the 8 inch line's extension meets at right angles.
Discussion
Normally you would take the 8 inch line and the height and divide their product by 2. You don't have the height. So you have to somehow get an expression for it.
The 81o angle is the supplement of the 99o angle. If you take the sine of both of them the sines are equal.
h = sin(81) * 14.2 because
sin(81) = opposite / hypotenuse.
opposite = hypotenuse * sin(81)
opposite = height = hypotenuse* sin(81)
height = 14.2 * sin(81)
Now we can find the area
Area
Area = height * base
base = 8
height = 14.2 * sin(81)
Area = 1/2 * 8 * 14.2 * sin(81)
Area = 56.1
Which expression is equivalent to
Answer:
Its the First choice.
Step-by-step explanation:
(3 m-2 n)^-3 / 6m n^-2
= (1/27) m^6 n^-3 / 6 m n^-2
= m^5 n ^-3 / 27*6 n^-2
= m^5 n^2 / 162 n^3
= m^4 / 162 n (answer).
The correct answer is option (A) m5 /162n
Step-by-step explanation:Let us solve step by step
See the attached figure for solution
(-3,5) is located in
It is located in the 2nd quadrant.
(x,y) - 1st quadrant
(-x,y) - 2nd quadrant
(-x,-y) - 3rd quadrant
(x,-y) - 4th quadrant
Answer:
Second quadrant
Step-by-step explanation:
That coordenates are located in the second quadrant of the cartesian plane
Best regards
Which is 2logx-6log(x-9) written as a single logarithm
Answer: option c.
Step-by-step explanation:
To solve the given exercise and write the expression as a single logarithm, you must keep on mind the following properties:
[tex]m*log(a)=log(a)^m[/tex]
[tex]log(a)-log(b)=log(\frac{a}{b})[/tex]
Therefore, by applying the properties shown above, you can rewrite the expression given, as following:
[tex]2logx-6log(x-9)=logx^2-log(x-9)^6=log(\frac{x^2}{(x-9)^6})[/tex]
Then as you can see, the answer is the option c.
If Paul is 5 years old and mei is 2 years old, how old will mei be when Paul is 15
5$ to get in and 0.80 to ride but he can only spend 25$ how many rides can he get on
Determine whether the point (2, 0) is a solution to the system of equations. Explain your reasoning in complete sentences. graph of a line 3 times x plus 2 and the absolute value of x minus 1 plus one. The graphs intersect at the point 0 comma 2
Answer:
(2,0) is not a solution of the system. The point does not belong to any of the graphs.
Step-by-step explanation:
To easily solve this question, we can graph both equations in a graphing calculator, and verify the intersection point, which is equal to the solution of the system of equations.
Plotting the graphs
y = 3x + 2
g = |x-1| +1
We obtain the intersection point
(0,2)
Solution of the system of equations
The area of a triangle is 3.6 cm2. If the triangle has a base of 6 cm, what is the height?
A. 0.6 cm
B. 1.2 cm
C. 12 cm
D. 3 cm
The height of a triangle with an area of 3.6 cm² and a base of 6 cm is calculated using the area formula, resulting in a height of 1.2 cm.
Explanation:To find the height of a triangle when the area and base are known, we use the formula for the area of a triangle, which is area = 1/2 × base × height. The given area is 3.6 cm2, and the base is 6 cm. Rearranging the formula to solve for the height, we get height = (2 × area) / base.
Plugging the values in, we get height = (2 × 3.6 cm2) / 6 cm = 7.2 cm2 / 6 cm = 1.2 cm.
Thus, the height of the triangle is 1.2 cm, which corresponds to option B.
The height of a triangle with an area of 3.6 cm² and a base of 6 cm is found by rearranging the formula for the area of a triangle. The calculated height comes out to be 1.2 cm.
To find the height of a triangle given the area and base, we can use the formula for the area of a triangle which is Area = (1/2) × base × height. In this case, the area is 3.6 cm² and the base is 6 cm.
Rearranging the area formula to find the height, we get height = (2 × Area) / base. Plugging in the values, we have:
height = (2 × 3.6 cm²) / 6 cm = 7.2 cm² / 6 cm = 1.2 cm.
Therefore, the correct answer to the question is B. 1.2 cm.
How many solutions does the system have? {y=−2x+2y=x2−3x Enter your answer in the box.
The system has two solutions: [tex]\( (2, -2) \) and \( (-1, 4) \),[/tex] found by solving their simultaneous equations.
let's solve the system step by step:
Given the system of equations:
[tex]1. \( y = -2x + 2 \)\\2. \( y = x^2 - 3x \)[/tex]
We want to find the values of [tex]\( x \) and \( y \)[/tex] that satisfy both equations simultaneously.
First, we set the equations equal to each other since they both equal [tex]\( y \):[/tex]
[tex]\[ -2x + 2 = x^2 - 3x \][/tex]
Now, let's rearrange this equation to get a quadratic equation:
[tex]\[ x^2 - 3x + 2x - 2 = 0 \]\[ x^2 - x - 2 = 0 \][/tex]
Now, we can use the quadratic formula to solve for [tex]\( x \):[/tex]
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \][/tex]
where [tex]\( a = 1 \), \( b = -1 \), and \( c = -2 \).[/tex]
[tex]\[ x = \frac{{-(-1) \pm \sqrt{{(-1)^2 - 4(1)(-2)}}}}{{2(1)}} \]\[ x = \frac{{1 \pm \sqrt{{1 + 8}}}}{2} \]\[ x = \frac{{1 \pm \sqrt{9}}}{2} \]\[ x = \frac{{1 \pm 3}}{2} \][/tex]
So, we have two potential values for [tex]\( x \): \( x = 2 \) and \( x = -1 \).[/tex]
Now, let's find the corresponding [tex]\( y \)[/tex] values for each [tex]\( x \)[/tex] value using either of the original equations:
For [tex]\( x = 2 \):[/tex]
[tex]\[ y = -2(2) + 2 = -4 + 2 = -2 \][/tex]
For [tex]\( x = -1 \):[/tex]
[tex]\[ y = -2(-1) + 2 = 2 + 2 = 4 \][/tex]
So, we have two solutions for the system:[tex]\( (2, -2) \) and \( (-1, 4) \).[/tex] Therefore, the system has 2 solutions.