For (a): 52 + 35/10m, rationalizing the denominator gives the result (25√2 + 6√5)/10.
For (b): 33x + 3y/3xy, rationalizing the denominator gives the result √3(x+y)/xy.
To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression is the same expression with the sign between the terms changed. For example, the conjugate of (a+b) is (a-b) and the conjugate of (a-b) is (a+b). By multiplying by the conjugate, we eliminate any radicals in the denominator.
(a) 5/√2 + 3/√5
= (5/√2)(√2/√2) + (3/√5)(√5/√5)
= (5√2)/2 + (3√5)/5
= (25√2 + 6√5)/10
(b) (3√3x + 3√y)/(3√xy)
= (3√3x + 3√y)(3√xy)/(3√xy)(3√xy)
= (9x√3xy + 9y√3xy)/(9xy)
= (9√3xy(x+y))/(9xy)
= √3(x+y)/xy
Therefore, the rationalized denominators are (a) (25√2 + 6√5)/10 and (b) √3(x+y)/xy.
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Joni says that a rectangular prism has two bases. How many possible pairs of bases does a rectangular prism really have? Explain
A rectangular prism actually has three pairs of bases. Each pair of bases is a set of two opposite and parallel faces on the rectangular prism.
The first pair of bases is the two opposite and parallel rectangles on the top and bottom of the prism. The second pair of bases is the two opposite and parallel rectangles on the front and back of the prism. The third pair of bases is the two opposite and parallel rectangles on the left and right sides of the prism.
Each pair of bases is congruent, meaning they have the same shape and size. This is because a rectangular prism is a type of polyhedron with six rectangular faces, and each pair of opposite faces is parallel and congruent to each other.
In conclusion, a rectangular prism has three pairs of bases, each consisting of two opposite and parallel rectangular faces.
I hope this helps! Let me know if you have any further questions.
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O POLYNOMIALS AND FACTORING Polynomial long division: Problem type 1 Divide. (3x^(2)+20x+30)-:(x+5) Your answer should give the quotient and the remainder.
The quotient of the polynomial long division problem (3x2 + 20x + 30) : (x + 5) is 3x + 25 and the remainder is 0.
To solve this, use polynomial long division. First, divide the leading coefficient of the dividend (3) by the leading coefficient of the divisor (1). The quotient is 3 and this is placed above the dividend.
Then, multiply the quotient (3) by the divisor (x + 5) and subtract the product from the dividend (3x2 + 20x + 30). The difference is 3x + 25.
Finally, multiply the quotient (3x + 25) by the divisor (x + 5) and subtract the product from the difference (3x + 25). Since the difference is 0, the remainder is 0.
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hil spent $21 of his $115 pocket money on eating out for one meal. What percent of Phil's pocket money did he spend on eating out for one meal? Round your answer to the nearest hundredth.
Step-by-step explanation:
for % questions always find 100% and/or 1%. everything else can be calculated based on these.
100% = $115
1% = 100%/100 = 115/100 = $1.15
how many % are $21 ?
as many as times 1.15 (1%) fits into 21 :
21/1.15 = 18.26086957...% ≈ 18.26%
Rachel and David were shopping for holiday gifts when they noticed a Thanksgiving sweater on the discount rack. Rachel really wanted the sweater, even though she wouldn’t be wearing it until Thanksgiving of 2021! .Rachel has a coupon for an additional 25% off the sale price of the sweater. If she pays for the shirt with a $10 bill, what will her change be?
The change is negative, Rachel needs to provide an additional $1.25 to pay for the sweater.
What is the basic arithmetic operations?
The four basic arithmetic operations are Addition, subtraction, multiplication, and division.
To calculate Rachel's change, we first need to determine the sale price of the sweater after the discount and then subtract it from the $10 bill.
Let's say the original price of the sweater was $30, and it was discounted by 50% to $15. Rachel's coupon gives her an additional 25% off the sale price, so she can get the sweater for 75% of the sale price.
To calculate the final price Rachel will pay, we can multiply the sale price by 0.75:
Final price = $15 * 0.75 = $11.25
Rachel pays with a $10 bill, so her change will be:
Change = $10 - $11.25 = -$1.25
Hence, the change is negative, Rachel needs to provide an additional $1.25 to pay for the sweater.
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In the 2020 NFL season, Drew Brees completed 73.5% of his attempted passes, and Patrick Mahomes completed 68.8% of his attempted passes. Based on those statistics, which of the following statements is true?
A. Drew Brees must have attempted more passes
B.Patrick Mahomes must have completed more passes
C.Either quarterback could have completed more passes
D.. Drew Brees must have completed more passes
Statement C is true: Either quarterback could have completed more passes
What are basic attributes of a good conclusion?
Rephrase the thesis statement first, then explain what you mean and back it with information from the prompt to show what you mean. Describe how the evidence supports your claims in further detail.
We cannot determine which quarterback completed more passes based solely on their completion percentages.
For example, if Drew Brees attempted 100 passes and completed 73.5% of them, he would have completed 73.5 passes. If Patrick Mahomes attempted 200 passes and completed 68.8% of them, he would have completed 137.6 passes, which is more than Drew Brees.
On the other hand, if Drew Brees attempted 200 passes and completed 73.5% of them, he would have completed 147 passes, which is more than Patrick Mahomes' 137.6 completed passes in the above example.
Therefore, statement C is true: either quarterback could have completed more passes, depending on the number of passes attempted.
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Which of the following is a proportional relationship?10 apples cost $4 plus surcharge $0.20.Bacteria doubles every day.Volume of a cube and its side.4 books cost $9.
Therefore , the solution of the given problem of proportionality comes out to be connection is not proportional as a result.
What is the process of proportionality?When a relationship consistently has the same number, it is considered to be proportional. For instance, the average amount of apples created by each tree determines how many trees there are in an area and how many apples are harvested. Mathematicians refer to a linear relationship between two numbers or factors as being proportional. The other amount doubles if the original amount does.
Here,
When there is a constant multiple between two variables in a relationship, that relationship is called a proportional relationship.
The expense of the 10 apples, which cost $4 plus a $0.20 surcharge, is not directly related to the quantity because the surcharge is a set price regardless of the quantity of apples.
The quantity being measured is growing by a fixed proportion (in this instance, 100%) every time period, so the bacteria doubling every day is an exponential relationship rather than a proportional one.
Last but not least, because the price per book varies, the $9 cost of 4 volumes is also not proportional. We would anticipate spending $18 if we were to purchase 8 books, which is not twice as much as 4 books. The connection is not proportional as a result.
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Isn't it supposed to be one black triangle and one black square?
The value of the model expression when simplified is (d)
How to determine the expressionFrom the question, we have the following representation that can be used in our computation:
Black triangle = -xWhite triangle = xBlack square = -1White square = 1Using the above as a guide, we have the following:
The given expression is
3 * Black triangle + 2 * White triangle + 2 * Black square + 1 * White square
This gives
-3x + 2x - 2 + 1
Evaluate the like terms
-x - 1
This means
1 Black triangle and 1 Black square
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Find an equation of the tangent line of y=√3x (square root of
3x) passing through (3,3) using Descartes method of slope.
The equation of the tangent line of y = sqrt(3x) passing through (3, 3) is y = (1/2)x + 3/2.
To use Descartes' method of slopes, we need to find the derivative of the function first.
Taking the derivative of y = sqrt(3x) gives:
y' = (1/2) * (3x)^(-1/2) * 3 = (3/2√(3x))
Now we can find the slope of the tangent line at (3, 3) by plugging in x = 3 into the derivative:
y' = (3/2√(3*3)) = 3/6 = 1/2
So the slope of the tangent line is 1/2.
Next, we use the point-slope form of a linear equation to find the equation of the tangent line:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is the point (3, 3).
Plugging in the values, we get:
y - 3 = (1/2)(x - 3)
Simplifying, we get:
y = (1/2)x + 3/2
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Find a function of the form y= A sin(kx) + C or y= A cos(kx) + C whose graph matches the function
shown below:
The equation of the sinusoidal graph will be y = 3 sin[(2/π)x] - 3.
What is trigonometry?Trigonometric functions examine the interaction between the dimensions and angles of a triangular form.
The graph is represented by the sinusoidal equation.
From the graph, the amplitude of the graph is 3 units and the graph is shifted 3 units down.
And the value of constant k is given as,
k = 4/2π
k = 2/π
Thus, the equation of the sinusoidal graph will be y = 3 sin[(2/π)x] - 3.
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e) Solve: f(x) = g(x) for x = [-240°; 240°]
Answer:
218.88
Step-by-step explanation:
In Exercises \( 1-8, W \) is a subset of \( R^{2} \) consisting of vectors of the form \[ \mathbf{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right] \] In each case determine whether \( W \)
Yes
Yes, \( W \) is a subset of \( R^{2} \) since it consists of vectors of the form
\[ \mathbf{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right] \]. Therefore, for Exercises \( 1-8 \), the answer is yes.
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Three friends, Martin, Tyreese, and Braydon, are collecting donations to help in their community's clean-up initiative. Their total contribution goal is represented by the expression 7x2 − 4xy + 8. The friends have already collected the following amounts:
Martin: 5xy + 16
Tyreese: x2
Braydon: 4x2 − 7
Which expression represents the amount of money the friends still need to collect to meet their goal?
2x2 − 9xy − 1
2x2 + xy + 17
5x2 + 5xy + 9
12x2 + xy + 17
2x2 - 9xy - 1 is the expression for how much money the buddies still need to gather to reach their objective.
What exactly is a phrase?A mathematical expression is made up of a statement, at least two integers or variables, and one or more arithmetic operations. This mathematical procedure makes it possible to multiply, divide, add, or subtract quantities.
The total of what they have already raised towards the overall fundraising target.
The total contribution goal is given as: 7x² - 4xy + 8
The amount Martin has already collected is: 5xy + 16
The amount Tyreese has already collected is: x²
The amount Braydon has already collected is: 4x² - 7
Adding the amounts already collected:
= (5xy + 16) + x² + (4x² - 7)
= 5xy + x² + 4x² + 16 - 7
= 5xy + 5x² + 9
By deducting the total of the contributions from the overall contribution target:
= (7x² - 4xy + 8) - (5xy + 5x² + 9)
= 2x² - 9xy - 1
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Answer: A
Step-by-step explanation: i wish other people would just put the letter, but here ya go c:
Eight triangles are drawn within a square to create the shaded region in the figure.
The area of the shaded region is 90 cm².
What is an Area?The space filled by the surface of an object or any flat shape can be thought of as the area in terms of geometry. The quantity of unit squares that cover an object's surface when it is closed is known as the area of the object. Inches, millimeters, square feet, and other square measurements are used to determine the area.
The area of shaded region = area of given square - area of all given triangles
Now, Area of given square = side × side
= 12 × 12
= 144 cm²
Similarly, Area of all triangles
= 4 × (area of small triangle) + 4 × (area of big triangle)
= 4 × ( 1/2 × 3× 3) + 4 × ( 1/2 × 3 × 6)
= 18 + 36
= 54 cm²
∴ The area of shaded region = 144 - 54
= 90 cm²
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What type of triangles has no equal sides and one right angle?
• isosceles triangle
O right scalene triangle
• right isosceles triangle
• right triangle
Answer:
[tex]\large\boxed{\textsf{Option B. Right Scalene Triangle.}}[/tex]
Step-by-step explanation:
[tex]\textsf{For this problem, we are asked which triangle has no equal sides and a right angle.}[/tex]
[tex]\textsf{Let's review the types of triangles, according to the options.}[/tex]
[tex]\large\underline{\textsf{What is an Isosceles Triangle?}}[/tex]
[tex]\textsf{An Isoceles Triangle is a triangle that has \underline{2 congruent sides}.}[/tex]
[tex]\textsf{Option A is incorrect.}[/tex]
[tex]\large\underline{\textsf{What is a Right Scalene Triangle?}}[/tex]
[tex]\textsf{An Isoceles Triangle is a triangle that has \underline{no congruent sides} and has \underline{1 right angle}.}[/tex]
[tex]\boxed{\textsf{Option B is correct.}}[/tex]
[tex]\large\underline{\textsf{What is a Right Isosceles Triangle?}}[/tex]
[tex]\textsf{A Right Isoceles Triangle is a triangle that has \underline{2 congruent sides}. and 1 right angle}[/tex]
[tex]\textsf{Option C is incorrect.}[/tex]
[tex]\large\underline{\textsf{What is a Right Triangle?}}[/tex]
[tex]\textsf{A Right Triangle is a triangle that has \underline{1 right angle}.}[/tex]
[tex]\textsf{We're not sure how many equal sides this triangle has as it can be none, or 2.}[/tex]
[tex]\textsf{Option D is incorrect.}[/tex]
On a certain day, the temperature of the pool was 76°F at 10 A.M. and 91°F at 1 P.M. If the temperature rose at a constant rate from 10 A.M. to 1P.M. on that day, what was the temperature at 11 A.M.? a. 79°F d. 89°F b. 81°F e. 90°F c. 86°F
The correct answer is b. 81°F.
On a certain day, the temperature of the pool rose at a constant rate from 76°F at 10 A.M. to 91°F at 1 P.M. To find the temperature at 11 A.M., we need to find the rate of change in temperature and then use it to calculate the temperature at 11 A.M.
The rate of change in temperature can be found by dividing the difference in temperature by the difference in time:
Rate of change = (91°F - 76°F) / (1 P.M. - 10 A.M.) = 15°F / 3 hours = 5°F per hour
Now, we can use this rate of change to find the temperature at 11 A.M.:
Temperature at 11 A.M. = 76°F + (5°F per hour)(1 hour) = 81°F
Therefore, the correct answer is b. 81°F.
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Find the values of x and y
. Write your answer in
simplest form.
X
45°
X =
y =
y
16
&
Answer:
X: 45° Y:16°
Step-by-step explanation:
by help of teacher
You are given the principal, the annual interest rate, and the compounding period. Determine the value of the account at the end of the specified time period. $6000, 4%, quarterly, 3 years A. $6181.81 B.$6749.18 C.$6760.95 D. $9606.19
The value of the account at the end of the specified time period is option B.$6749.81. It can be calculated using the formula for compound interest, which is:
A = P(1 + r/n)^(nt)
Where A is the final amount, P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.
Plugging in the given values, we get:
A = 6000(1 + 0.04/4)^(4*3)
A = 6000(1.01)^12
A = 6749.18
Therefore, the correct answer is B. $6749.18 using the compound interest formula.
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Find the constant of variation k for the direct variation.
f(x)
0
-1
-2
-3.5
x02
0
4
7
The value of the constant of variation for the direct variation is k = -1/2.
What is constant of variation?Variation depicts the relationship between two variables and how it changes. Often, a ratio is used to illustrate this connection. When we remark that a variation is continuous, we are referring to how consistently the ratio changes. Hence, whenever you read the phrase "constant of variation," just keep in mind that it refers to the stability of the relationship between the variables.
The direct variation is given by the following relations:
y = kx
Now, using the table substitute the value of y = f(x) and x:
y = kx
-2 = k(4)
k = -2/4
k = -1/2
Hence, the value of the constant of variation for the direct variation is k = -1/2.
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Practice and Problem S Subtract using the vertical form. 1. (5g^(2)+6g-10)-(2g^(2)+2g+9)
(5g^(2)+6g-10)-(2g^(2)+2g+9) = 3g^(2) + 4g - 19
To subtract the two polynomials, we will use the vertical form. This means that we will line up the like terms and subtract them.
Step 1: Line up the like terms in the vertical form.
5g^(2) + 6g - 10
- (2g^(2) + 2g + 9)
Step 2: Distribute the negative sign to each term in the second polynomial.
5g^(2) + 6g - 10
- 2g^(2) - 2g - 9
Step 3: Subtract the like terms.
5g^(2) - 2g^(2) = 3g^(2)
6g - 2g = 4g
-10 - 9 = -19
Step 4: Write the final answer.
3g^(2) + 4g - 19
Therefore, the answer is 3g^(2) + 4g - 19.
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At noon, the temperature on a mountaintop and the temperature on the ground had opposite values. The temperature on the mountaintop was 20° lower than temperature on the ground. What was the temperature in both places?
The temperature on mountaintop was -10° and the temperature on the ground was 10°. The solution has been obtained by using the linear equation.
What is a linear equation?
One is the degree of the linear equation. It is obvious that there are no variables in linear equations with exponents greater than 1. The graph's equation results in a straight line.
We are given that the temperature on the mountaintop was 20° lower than temperature on the ground.
Let the temperature on the ground be 'x'.
So, temperature on the mountaintop = x - 20
Also, the temperature on a mountaintop and the temperature on the ground had opposite values.
So,
x = - (x - 20)
On solving this, we get
⇒x = -x + 20
⇒2x = 20
⇒x = 10°
So, x - 20 = 10 - 20 = -10°
Hence, the temperature on mountaintop was -10° and the temperature on the ground was 10°.
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Find the minimum and maximum of P=4x+2y subject to 3x+5y>=20 3x+y<=16 -2x+y<=2 x>=0,y>=0
The minimum of P is 16 and the maximum of P is 24.
Calculating the value of P for these points, we get:
P(2,6) = 4*2 + 2*6 = 20
P(0,8) = 4*0 + 2*8 = 16
P(4,2) = 4*4 + 2*2 = 24
P(3,3) = 4*3 + 2*3 = 18
Therefore, the minimum of P is 16 and the maximum of P is 24.
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Calculate the interest earned when $46 230 is invested for 9 years at 7.8% p.a., with interest compounded twice a year.
Given:
What is the interest earned when $46,230 is invested for 9 years at 7.8% p.a?
To find:
The interest
Solution:
[tex]interest = \frac{p \times r \times t}{100} [/tex]let the interest be x. [tex]x = \frac{46230 \times 9 \times 7.8}{100} [/tex][tex]x = 32456.46[/tex]But the interest is compounded twice a year so, [tex]32456.46 \times 2 \times 9[/tex][tex] = 64912.92 \times 9[/tex][tex] = 584216.28[/tex]therefore, the final answer is 584,216.28Mrs.may mixed 4 gal of red paint with every 6 gal of blue paint in order to make purple paint. which ratio of gallons of red paint to gallons of blue pint will make the same shade of purple paint?
Answer: 20 gallons of red paint and 30 gallons blue paint
Step-by-step explanation:
Use an appropriate infinite series method about x = 0 to find two solutions of the given differential equation. (Enter the first four nonzero terms for each linearly independent solution, if there are fewer than four nonzero terms then enter all terms. In each case, the first term has been provided for you. )
3xy" y+y=0
Y₁ = 1 -
12 = x2/3
The two linearly independent solutions are y₁(x) = 1 - (x²/3) + (x⁴/81) - (x⁶/2187) + … and y₂(x) = x - (x³/9) + (x⁵/405) - (x⁷/21870) + ...
We can use the power series method to find the two solutions of the given differential equation:
Let y = ∑anxn be a power series solution. Then, we have:
y' = ∑nanxn-1
y'' = ∑nan(n-1)xn-2
Substituting these into the differential equation and equating coefficients of like powers of x, we get:
3x∑nan(n-1)xn-1 + ∑anxn = 0
Simplifying, we obtain:
∑[3nan(n-1) + an-1]xn = 0
Since x ≠ 0, the coefficients of like powers of x must be zero. This gives us the recurrence relation:
an = -an-1/(3n(n-1)), n ≥ 1
Using this recurrence relation and the initial condition y(0) = 1, we can find the power series solution:
y₁(x) = 1 - (x²/3) + (x⁴/81) - (x⁶/2187) + ...
To find a second linearly independent solution, we can use the method of reduction of order. Let y₂(x) = v(x)y₁(x). Then, we have:
y₂' = v'y₁ + vy₁'
y₂'' = v''y₁ + 2v'y₁' + vy₁''
Substituting these into the differential equation and simplifying, we get:
v''y1x + (2v'y1' + vy1'')x + 3vy1 = 0
Since y₁(x) is a solution, we have:
y₁'' + (1/3x)y₁ = 0
Multiplying by y1 and integrating, we obtain:
(y₁')² + y₁²/3 = C
where C is a constant of integration. Using the initial condition y₂(0) = 0, we can find the second solution:
y₂(x) = x - (x³/9) + (x⁵/405) - (x⁷/21870) + ...
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In Exercises 11-14|, use a matrix equation to solve the system of linear equations. \( 11\left\{\begin{array}{r}x+2 y=3 \\ 2 x+6 y=5\end{array}\right. \)
x = 4/3 and y = 2/3
To solve this system of linear equations using a matrix equation, we first need to construct a matrix by writing the coefficients of the variables and the constants from the equation in a rectangular array.
The matrix for the given equations will be:
Now, we can solve the system by multiplying the inverse of the matrix of coefficients with the matrix of constants. To find the inverse of the matrix, we can use the following equation:
where |A| is the determinant of the matrix, and adj(A) is the adjoint of the matrix.
We can use this equation to find the inverse of the matrix, which is:
We can then multiply this inverse by the matrix of constants to get the solution for the system:
Therefore, the solution for the system is x = 4/3 and y = 2/3.
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Work the problem using the three-step method. Show your work.
The sum of three consecutive integers is 18. Find the integers.
WRITER
Answer:
5,6 and 7.
Step-by-step explanation:
I am not sure what the three step method is, but here is how I solved it.
Consecutive integers means integers in a row. For example, 5,6 7 are consecutive integers or -4,-3,-2 are consecutive integers.
Integers are all the whole numbers (0,1,2,3...) and their opposites (..., -3, -2, -1) 1/2 or -.23 are not integers.
Let x = the first number
Let x + 1 = the second number
Let x + 2 = the third number
x + (x + 1) + (x + 2) = 18 I do not need the parentheses, but I want you to see that the first number, plus the second number, plus the third number equals 18.
x + x + 1 + x + 2 = 18 Combine like terms
3x + 3 = 18 Subtract 3 from both sides of the equation
3x + 3 - 3 = 18 - 3
3x = 15 Divide both sides by 3
[tex]\frac{3x}{3}[/tex] = [tex]\frac{15}{3}[/tex]
x = 5
The first number is 5. The second number is 6 and the last number is 7.
Check:
5 + 6 + 7 = 18
18 =18 Checks.
Helping in the name of Jesus.
can You Please Help Me! I will mark You Brainliest! <33
Answer:
Step-by-step explanation:
1) ballet club
2) ballet club
What is the solution to this equation?
9 ^x- 1 = 2
O A. 1
O B. 2
O C.
1/2
OD. -1/2
The solution to this equation 9ˣ - 1 = 2 is 1/2.
The correct option is A.
What is an equation?
Two algebraic expressions having same value and symbol '=' in between are called as an equation.
Given:
An equation:
9ˣ - 1 = 2
Simplifying,
9ˣ = 3
3²ˣ = 3
Comparing, we get,
2x = 1
x = 1/2
Therefore, the solution is x = 1/2.
-8y + 25 ≤ -31
Please Answer
Answer:
Hi
Step-by-step explanation:
im pretty sure this the answer
hope this helps
good luck;)
Answer:
y≤7
Step-by-step explanation:
-8y+25≤-31
take away 25 from both sides
-8y≤-56
then divided by -8 on both sides
y≤7
ch help video solution of the system of equations. 5x-y=44 -3x-y=-12 Submit Answer
The solution of the system of equations is x=7 and y=-9.
To find the solution of the system of equations, we need to solve for x and y. We can do this using the elimination method.
Multiply the first equation by 3 and the second equation by 5 to eliminate the x variable.
3(5x-y=44) -> 15x-3y=132
5(-3x-y=-12) -> -15x-5y=-60
Add the two equations together to eliminate the x variable.
15x-3y=132
-15x-5y=-60
-----------
-8y=72
Step 3: Solve for y.
-8y=72
y=-9
Substitute the value of y back into one of the original equations to solve for x.
5x-(-9)=44
5x+9=44
5x=35
x=7
Check the solution by substituting the values of x and y back into the original equations.
5(7)-(-9)=44
35+9=44
44=44
-3(7)-(-9)=-12
-21+9=-12
-12=-12
The solution is correct.
Submit the answer.
The solution of the system of equations is x=7 and y=-9.
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