rank(A) = 1
Given that A = $\left[\begin{array}{cc}a & b \\ a & b\end{array}\right]$, where $a \neq 0$.Let's find the rank of A:Rank(A) = number of leading 1's in the row echelon form of A.= $\left[\begin{array}{cc}a & b \\ a & b\end{array}\right]$=> $\left[\begin{array}{cc}a & b \\ 0 & 0\end{array}\right]$There are two leading 1's in row 1. Hence, rank(A) = 1.Now, let's find the basis of the null space of A:Null space of A is the set of all solutions to the equation $Ax=0$.So, $Ax$ = $\left[\begin{array}{cc}a & b \\ a & b\end{array}\right]$ $\left[\begin{array}{c}x \\ y\end{array}\right]$= $\left[\begin{array}{c}ax+by \\ ax+by\end{array}\right]$= $\left[\begin{array}{c}(a+b)x \\ (a+b)y\end{array}\right]$= $\left[\begin{array}{c}0 \\ 0\end{array}\right]$=> (a + b) x = 0 and (a + b) y = 0=> $x = \frac{-b}{a}y$=> $\left[\begin{array}{c}x \\ y\end{array}\right]$= $\left[\begin{array}{c}\frac{-b}{a}y \\ y\end{array}\right]$= $y \left[\begin{array}{c}\frac{-b}{a} \\ 1\end{array}\right]$=> Null space of A = $\{y \left[\begin{array}{c}\frac{-b}{a} \\ 1\end{array}\right] | y \in R \}$Let's take a = 1, b = 2. So, the null space of A = $\{y \left[\begin{array}{c}-2 \\ 1\end{array}\right] | y \in R \}$Therefore, the basis of the null space of A = $\left[\begin{array}{c}-2 \\ 1\end{array}\right]$Now, let's find the basis of the row space of A:Row space of A is the span of the rows of A.=> Row space of A = span $\left\{\left[\begin{array}{cc}a & b\end{array}\right]\right\}$Since rank(A) = 1, there is only 1 non-zero row. Therefore, a basis of the row space of A is $\left\{\left[\begin{array}{cc}a & b\end{array}\right]\right\}$.
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Perform the addition or subtraction and simplify.
x
x2 + x − 2
−
9
x2 − 5x + 4
The final simplified expression turns out to be "-8x^2 + 6x - 6"
To perform the addition or subtraction and simplify the expression, we need to combine like terms and simplify the expression. First, we need to subtract the second expression from the first one:
x^2 + x − 2 - (9x^2 − 5x + 4)
Next, we need to distribute the negative sign to each term in the second expression:
x^2 + x − 2 - 9x^2 + 5x - 4
Now we can combine like terms:
-8x^2 + 6x - 6
This is the simplified expression. Therefore, the answer is:
-8x^2 + 6x - 6
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Use the pair of functions to find f(g(x)) and g(f(x)). Simplify your answers.
f(x) = x²+1, g(x) = √x+5
f(g(x)) =
g(f(x)) =
The solutions of the given functions are:
f(g(x)) = x+10√x+26 and g(f(x)) = √(x²+6)
f(x) = x²+1, g(x) = √x+5 are given.
To find f(g(x)), we need to substitute g(x) into the function f(x):
f(g(x)) = f(√x+5) = (√x+5)²+1 = x+10√x+26
To find g(f(x)), we need to substitute f(x) into the function g(x):
g(f(x)) = g(x²+1) = √(x²+1+5) = √(x²+6)
Therefore, the simplified answers are:
f(g(x)) = x+10√x+26
g(f(x)) = √(x²+6)
Note: It is important to include the parentheses when substituting one function into another to ensure the correct order of operations.
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During a rugby contest 54342 people what's the first game or 49990 people what's the second game if 156330 people watched all three games how many what's the third game
The answer would be 51,998
find the variation constant and an equation of variation where y
varies inversely as x and y = 11 when x = 2
the variation constant is...
The variation constant is 22 and the equation of variation where y varies inversely as x is y = 22/x.
When two variables, y and x, are inversely proportional, it means that as one variable increases, the other variable decreases proportionally, and vice versa. Mathematically, this can be expressed as:
y = k/x
where k is the variation constant. To find k, we can use the given information that "y = 11 when x = 2". Substituting these values into the inverse variation equation above, we get:
11 = k/2
Multiplying both sides by 2, we get:
k = 22
So, the variation constant is 22.
Now, we can write the equation of variation as:
y = 22/x
This means that as x increases, y decreases proportionally. For example, if x doubles to 4, y would be halved to 5.5, keeping the product (xy) constant.
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336 fewer the the product of u and 258 is 168
Answer:
335.6511628
Step-by-step explanation:
Ux258-336=168
U-336=0.6511628
U=336.6511628
PLEASE HELP I NEED HELP PLEASE
Answer:
Step-by-step explanation:
You take 528- divided by 2000- and then you get your answer.
The diagonals of a rhombus measure 8cm and 6cm .what is the length of a side of the rhombud
All sides of the given rhombus measure 5 cm respectively.
What is a rhombus?A rhombus is a quadrilateral with four equal-length sides in planar Euclidean geometry.
The term "equilateral quadrilateral" refers to a quadrilateral whose sides all have equal lengths.
Quadrilaterals having four congruent sides are called squares.
Squares are rhombuses by definition since they are quadrilaterals with four congruent sides.
So, consider the following values: AC = 8 cm, BD = 6 cm, OD = OB = 3 cm, and OA = OC = 4 cm.
Now,
OB² + OC² = BC²
3²+ 4² = BC²
9 + 16 = BC²
25 = BC²
5 = BC
A rhombus has an equal number of sides.
Therefore, all sides of the given rhombus measure 5 cm respectively.
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Five different stores sell a bag of flour for one of the following prices: $1.45, $1.55, $1.90, $1.95, $1.45. What are the mean, median, and mode of the data?
The mean, median and mode of the prices that the bags of flour are sold at are:
Mean - $ 1.66Median - $ 1. 55Mode - None How to find the mean, mode and median ?To find the mean, we add up all the prices and divide by the total number of prices:
Mean:
= (1.45 + 1.55 + 1.90 + 1.95 + 1.45) / 5
= 1.66
To find the median, we need to arrange the prices in order from lowest to highest:
1.45, 1.45, 1.55, 1.90, 1.95
There are five prices, so the median is the middle value, which is 1.55.
To find the mode, we need to look for the price that appears most frequently in the data set. In this case, there are two prices that appear twice, 1.45 and 1.55. So there is no unique mode for this data set.
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isosceles trapezoid with base lengths 2ft and 5ft and leg lengths 2.5ft
The area of the isosceles trapezoid with base lengths 2ft and 5ft and leg lengths 2.5ft will be 8.75ft
Isosceles trapezoid, also known as isosceles trapezium, is defined as a convex quadrilateral, which mainly consists of a line of symmetry that bisects one pair of the opposite side, it is also know that the base angles are of equal measure. It has parallel bases with the legs also being of the equal measure, on the other hand the opposite angles of the isosceles trapezoid are supplementary, which makes it a cyclic quadrilateral.
Area of an Isosceles Trapezoid = [(a+b)h]/2 square units,
when we put these values in the formula, we get:
Area = [(2+5)2.5]/2
Area = (7 × 2.5)/2
Area = 17.5/2
Area = 8.75ft²
therefore, we know that the area of the isosceles trapezoid with base lengths 2ft and 5ft and leg lengths 2.5ft will be 8.75ft
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In 1 hour 32 cars pass through a particular intersection. At the same rate, how long would it take for 96 cars to pass through the intersection?
It wοuId take 3 hοurs fοr 96 cars tο pass thrοugh the intersectiοn at the same rate as 32 cars in 1 hοur.
We can use the cοncept οf prοpοrtiοnaIity tο sοIve this prοbIem. The number οf cars passing thrοugh the intersectiοn is directIy prοpοrtiοnaI tο the amοunt οf time taken. This means that if we dοubIe the number οf cars passing thrοugh, we wiII aIsο dοubIe the amοunt οf time taken.
Let x be the amοunt οf time it takes fοr 96 cars tο pass thrοugh the intersectiοn. Then, we can set up the fοIIοwing prοpοrtiοn:
32 cars / 1 hοur = 96 cars / x hοurs
Tο sοIve fοr x, we can crοss-muItipIy and simpIify:
32x = 96
x = 3
Therefοre, it wοuId take 3 hοurs fοr 96 cars tο pass thrοugh the intersectiοn at the same rate as 32 cars in 1 hοur.
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A $900 item is on sale for 20% off.
What is the final price with 7% sales tax?
Money off:
Amt, of tax:
Money paid:
Final price:
AW
IN. I need help I’m stuck
A $900 item on the sale of 20% discount the
Money off: $180
Amt. of tax: $50.40
Money paid: $770.40
Final price: $770.40
what is discount ?
A discount is a reduction in the price of an item or service, often offered as a promotion or incentive to encourage customers to make a purchase. The discount is usually expressed as a percentage of the original price and represents the amount of money that the customer can save on the purchase.
For example:
Percentage discount: a certain percentage of the original price is deducted from the selling price. For example, a 20% discount on a $100 item would reduce the price to $80.
According to the question:
To calculate the final price of the item with sales tax, we can follow these steps:
Calculate the amount of money off by multiplying the original price by the discount percentage:
Money off = 0.20 x $900 = $180.
Calculate the price after the discount by subtracting the money off from the original price:
Price after discount = $900 - $180 = $720.
Calculate the amount of sales tax by multiplying the price after discount by the tax rate:
Amount of tax = 0.07 x $720 = $50.40.
Calculate the money paid by adding the price after discount and the amount of tax:
Money paid = $720 + $50.40 = $770.40.
Therefore, the final price with 7% sales tax is $770.40.
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can anyone please help me with this question -2u^5(-6u^3)
The simplified expression is [tex]12u^8[/tex].
What is simplification?
Simplifying an expression is just another way to say solving a math problem. When you simplify an expression, you're basically trying to write it in the simplest way possible. At the end, there shouldn't be any more adding, subtracting, multiplying, or dividing left to do. For example, take this expression: 4 + 6 + 5.
Here the given expression is ,
=> [tex]-2u^5(-6u^3)[/tex]
Now simplifying the expression,
=> [tex]-2\times -6 \times u^5 \times u^3[/tex]
=> 12[tex]u^{5+3}[/tex]
=> [tex]12u^8[/tex]
Hence the simplified expression is [tex]12u^8[/tex].
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Aaron invested $7,500 in an account paying an interest rate of 1.5% compounded continuously. Assuming no deposits or withdrawals are made, how much money, to the nearest hundred dollars, would be in the account after 8 years?
Answer:
The formula for calculating the amount of money in an account with continuous compounding is:
A = Pe^(rt)
Where:
A = the amount of money in the account after t years
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
t = the time (in years)
Substituting the given values into the formula, we get:
A = 7500e^(0.0158)
A ≈ $9,071.21
Therefore, to the nearest hundred dollars, there would be $9,100 in the account after 8 years.
Step-by-step explanation:
A puppy weighed 28 ounces on Monday. It gained 7 ounces in one week. What is the percent increase of the puppy's weight rounded to the nearest percent?
The percent increase of the puppy's weight is 25%
How to calculate the percentage increase of the puppy's weight?A puppy weighed 28 ounces on Monday
It gained 7 ounces in one week
Total weight is = 28 + 7
= 35
The percent increase can be calculated as follows
= 35/28
= 1.25
= 1.25-1
= 0.25 × 100
= 25%
Hence the percent increase of the puppy's weight is 25%
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Knowledge Check Divide. (x+6)/(3x+9)-:(7)/(9x)+42 Simplify your answer as much as possible.
(3x²+18x)/(7x+21) is the simplified answer of (x+6)/(3x+9)-:(7)/(9x)+42.
To divide the given expressions, we need to first simplify the expressions as much as possible and then use the rule for dividing fractions. The rule for dividing fractions is to multiply the first fraction by the reciprocal of the second fraction.
Simplifying the expressions:
(x+6)/(3x+9) = (x+6)/(3(x+3)) = (x+6)/3(x+3)
(7)/(9x) = 7/9x
Now, using the rule for dividing fractions:
((x+6)/3(x+3)) ÷ (7/9x) = ((x+6)/3(x+3)) * (9x/7)
= (9x(x+6))/(3(x+3)*7)
= (3x(x+6))/(x+3)*7
= (3x²+18x)/(7x+21)
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Please help me with this math question.
Answer:120
Step-by-step explanation:
A quadratic function models the graph of a parabola. The quadratic functions, y=x and y=x² +3, are modeled in the
graphs of the parabolas shown below.
-10-9-8-76
y M
10
-2-1
0
8
40
6
-7
-8
9
-10-
y=x²+3
y=x²
。
Determine which situations best represent the scenario shown in the graph of the quadratic functions, y=x² and y=x²
+3. Select all that apply.
The situations that best represent the scenario shown in the graph of the quadratic functions, y = x² and y = x² + 3 include the following:
B. "From x = -2 to x = 0, the average rate of change for both functions is negative."
C. "For the quadratic function, y = x² + 3, the coordinate (2, 7) is a solution to the equation of the function."
D. "The quadratic function, y = x², has an x-intercept at the origin."
What is the x-intercept?In Mathematics, the x-intercept of the graph of any function simply refers to the point at which the graph of a function crosses or touches the x-coordinate (x-axis) and the y-value or value of "y" is equal to zero (0).
In this context, we can logically deduce that the x-intercepts of the graph of the given equation y = x² is at the origin:
x = (0, 0)
For the the coordinate (2, 7), we would evaluate the quadratic function, y = x² + 3 as follows;
7 = 2² + 3
7 = 4 + 3
7 = 7 (True).
By critically observing the graph of both functions, we can logically deduce that their average rate of change is negative from x = -2 to x = 0.
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find the slope of the line that contains each pair of points (2,6) and (0,1)
5/2 is the slope of line .
What is slope, exactly?
Typically, a line's slope provides information about the steepness and direction of the line. Finding the difference between the coordinates of the locations will allow you to quickly compute the slope of a straight line connecting two points, (x1,y1) and (x2,y2).
The letter "m" is frequently used to signify slope. A line's steepness can be determined by its slope. In mathematics, slope is calculated as "rise over run" (change in y divided by change in x).
points (2,6) and (0,1)
Slope = 1 - 6/0 - 2
= -5/-2
= 5/2
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what even number is not a composite number 6,4,2,8
Answer: 2
Step-by-step explanation:
Answer and Explanation: The only even number that is not a composite number is 2. All even numbers are defined as numbers that are divisible by, or have a divisor of, 2.
Answer:
The answer to your question is, 2
Step-by-step explanation:
The reason why that is our answer is because, 2 can either be multiplied and divided but 2 NUMBERS.
Such as the following:
2 / 1 = 2.
1 x 2 = 2.
What is a composite number?
A composite number are numbers that have more than two factors like the number 2:
2 / 1 = 2.
1 x 2 = 2.
Thus the answer to your problem is, 2
Triangle ABC is enlarged with a scale factor of -2 and the origin of the centre to give triangle ABC . WORK OUT THE COORDINATES OF A and B
The coordinates of the triangle A'B'C' are A'(-2, -6), B'(-14, -2) and C'(-2, -2).
What is a dilation?Dilation is the process of resizing or transforming an object. It is a transformation that makes the objects smaller or larger with the help of the given scale factor. The new figure obtained after dilation is called the image and the original image is called the pre-image.
here, we have,
From the given graph, triangle ABC have A(1, 3), B(7, 1) and (1, 1).
Triangle ABC is enlarged with a scale factor of -2 and the origin of the Centre to give triangle A'B'C'.
Now, A(1, 3) → -2(1, 3)→A'(-2, -6)
B(7, 1) → -2(7, 1) → B'(-14, -2)
C(1, 1) → -2(1, 1) → C'(-2, -2)
Therefore, the coordinates of the triangle A'B'C' are A'(-2, -6), B'(-14, -2) and C'(-2, -2).
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Houa is ordering a taxi from an online taxi service. The taxi charges $4 just for the pickup and then an additional $2 per mile driven. How much would a taxi ride cost if Houa is riding for 2 miles? How much would a taxi ride cost that is mm miles long?
Answer:
A) $8
B) 4+2x
Step-by-step explanation:
If it costs $2 for every mile driven, and Huoa rides for 2 miles, then you multiply 2×2=4. Then you add another $4 for the $4 the driver charges for just getting in, so 4+4=8
Another way to do that is to use the equation that's the answer for part B. As you don't know the number of miles, you substitute that unknown number for x.
4+2x
The x represents the number of miles
The 2 represents the amount charged for each mile
The 4 represents the flat fee for getting into the car
If you plug 2 (for the 2 miles driven) into the equation you get:
4+2(2)
4+4
8
What are the x-intercepts of the quadratic function?
f(x)=x^2+6x−27
enter your answer in the boxes
blank and blank
Step-by-step explanation:
To find the x-intercepts of the quadratic function f(x) = x^2 + 6x - 27, we need to set f(x) equal to zero and solve for x.
So we have:
f(x) = 0
x^2 + 6x - 27 = 0
We can solve for x using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Where a = 1, b = 6, and c = -27.
Substituting these values into the quadratic formula, we get:
x = (-6 ± sqrt(6^2 - 4(1)(-27))) / 2(1)
x = (-6 ± sqrt(180)) / 2
x = (-6 ± 6sqrt(5)) / 2
x = -3 ± 3sqrt(5)
So the x-intercepts of the quadratic function are approximately -10.16 and 4.16.
Hey everyone and anyone of anybody could please help me with this math worksheet i would greatly appreciate it
The side x is 30 miles
The unknown side is 346 yards
What is the sine rule?The sine rule, also known as the law of sines, is a formula that relates the sides and angles of a non-right triangle (a triangle that does not have a 90-degree angle). The sine rule states that the ratio of the length of a side of a triangle to the sine of the opposite angle is the same for all three sides of the triangle.
1) Using the sine rule;
x/sin 65 = 32/sin75
xsin75 = 32sin65
x = 32sin65/sin75
x = 30 miles
2) The angle = π/3 = 180/3 = 60 degrees
Using the cosine rule;
c^2 = (200)^2 + (400)^2 - (2 * 200 * 400) cos 60
c^2 = 200,000 - 80,000
c = 346 yards
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What is the answer for -3i²
Answer:
The answer is 3
Angle N = 40 degrees, side NP = 8, angle Q = 40 degrees, and side QS = 8. What additional information would you need to prove that ΔNOP ≅ ΔQRS by ASA?
a
Angle O is congruent to angle R.
b
Angle P is congruent to angle S.
c
Side NO is congruent to side QR.
d
Side OP is congruent to side RS.
Answer:
Option d: Side OP is congruent to side RS.
To prove that ΔNOP ≅ ΔQRS by ASA, we need to show that:
1. ∠N ≅ ∠Q (given)
2. Side NP ≅ Side QS (given)
3. Side OP ≅ Side RS (additional information needed)
Hence, option d is the correct answer.
Select the correct equation for the following sentence: Twenty-four is the same as 31.4 times a number plus negative 8.4. 31.4n + 8.4 = 24 –8.4n + 31.4 = 24 24 = 31.4n + (–8.4) 24 – 31.4 = –8.4n
The correct equation for the sentence is: 24 = 31.4n - 8.4.
Twenty-four is equal to 31.4 times a number plus a negative 8.4 multiplied by the proper equation, which is:
24 = 31.4n - 8.4
This formula can be rewritten as follows:
31.4n = 24 + 8.4
And after being distilled:
31.4n = 32.4
Finally, by multiplying both sides by 31.4, we can find the value of n:
n = 32.4/31.4
Thus, the following is the proper equation for the phrase:
24 = 31.4n - 8.4.
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Can somebody PLEASE help me ASAP? I really need help. It’s due today!! Please show work!! I will give brainliest.
Answer:
1. c
2. b
3. a
Step-by-step explanation:
Surface Area of a cylinder = 2πrh + 2πr²
1. 2(3.14)(5)(11) + 2(3.14)(5)² = 502.4
2. 2(3.14)(6)(10) + 2(3.14)(6)² = 602.88
3. 2(3.14)(2)(12) + 2(3.14)(2)² = 175.84
Martin buys 2(5)/(6) pounds of peanuts and 4(3)/(4) pounds of almonds. How many pounds of nuts does Martin buy?
Martin buys a total of 2(5)/(6) + 4(3)/(4) = 10/6 + 19/4 = 40/24 + 57/24 = 97/24 pounds of nuts.
To find the total amount of nuts Martin buys, we need to add the amount of peanuts and the amount of almonds he buys.
First, we need to convert the mixed numbers to improper fractions. To do this, we multiply the whole number by the denominator and add the numerator. For 2(5)/(6), we multiply 2 by 6 and add 5 to get 10/6. For 4(3)/(4), we multiply 4 by 4 and add 3 to get 19/4.
Next, we need to find a common denominator in order to add the two fractions. The least common denominator for 6 and 4 is 24, so we multiply the numerator and denominator of 10/6 by 4 to get 40/24 and the numerator and denominator of 19/4 by 6 to get 57/24.
Finally, we add the two fractions by adding the numerators and keeping the same denominator: 40/24 + 57/24 = 97/24.
Therefore, Martin buys a total of 97/24 pounds of nuts.
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Q1. Let \( A=\left[\begin{array}{lll}a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & a\end{array}\right] \), where \( a \in \mathbb{F} \). Prove that \[ A^{n}=\left[\begin{array}{ccc} a^{n} & n a^{n-1} & \frac{n(n-
To prove that \[ A^{n}=\left[\begin{array}{ccc} a^{n} & n a^{n-1} & \frac{n(n-1)}{2}a^{n-2} \end{array}\right], we will use mathematical induction.
Base case: For \( n=1 \), \( A^{1}=A \). This is true since
\[ A=\left[\begin{array}{ccc} a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & a \end{array}\right] \]
which matches the left side of our equation.
Inductive step: Assume that the statement is true for some \( n=k \), i.e.
\[ A^{k}=\left[\begin{array}{ccc} a^{k} & ka^{k-1} & \frac{k(k-1)}{2}a^{k-2} \end{array}\right] \]
We will now show that it is true for \( n=k+1 \).
\[ A^{k+1}=A^{k}\cdot A \]
We substitute the induction hypothesis into this equation and get:
\[ A^{k+1}=\left[\begin{array}{ccc} a^{k} & ka^{k-1} & \frac{k(k-1)}{2}a^{k-2} \end{array}\right]\cdot \left[\begin{array}{ccc} a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & a \end{array}\right] \]
Calculating the matrix product yields:
\[ A^{k+1}=\left[\begin{array}{ccc} a^{k+1} & (k+1)a^{k} & \frac{(k+1)k}{2}a^{k-1} \end{array}\right] \]
which matches the left side of our equation, completing the proof.
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that the denominators are different. Simplify the result, if possible. (2)/(y-5)+(7)/(y+7)
The simplified expression is (3y-7)/(y^2-35)/3.
To simplify the given expression, we need to find the least common denominator (LCD) of the two fractions. The LCD of (y-5) and (y+7) is (y-5)(y+7).
Next, we need to multiply each fraction by the LCD in order to get the same denominator for both fractions.
(2)/(y-5) * (y+7)/(y+7) + (7)/(y+7) * (y-5)/(y-5)
= (2y+14)/(y^2-35) + (7y-35)/(y^2-35)
Now, we can combine the two fractions since they have the same denominator.
(2y+14+7y-35)/(y^2-35)
= (9y-21)/(y^2-35)
Finally, we can simplify the fraction by factoring out a 3 from the numerator.
= (3)(3y-7)/(y^2-35)
= (3y-7)/(y^2-35)/3
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