Which choice is equivalent to the expression
4 (1/2x - 3y)

Answer:

The solution of this system is the yellow region which is the area of overlap. In other words, the solution of the system is the region where both inequalities are true. The y coordinates of all points in the yellow region are both greater than x + 1 as well as less than x.

Since the distance between P and S is greater than 10 km, the captain is incorrect. The ship is still more than 10 km away from P.

In mathematics, a function is a rule that maps a set of inputs (domain) to a set of outputs (codomain) in such a way that each input is associated with exactly one output. A function can be represented using various notations, including equations, graphs, and tables. A function can be thought of as a machine that takes an input and produces an output. The input is usually represented by the variable x, and the output is represented by the variable y. The relationship between the input and output is defined by the function rule. Functions are used in many areas of mathematics, science, and engineering to describe various phenomena, such as motion, growth, and decay. They are also used to model and analyze data in statistics and economics, and to design and control systems in engineering and computer science.

Here,

a) To solve this problem, we can use the Law of Cosines to find the distance and the Law of Sines to find the bearing.

First, let's label the points as follows:

P: the starting point

Q: the point reached after sailing 46 km on a bearing of 104

R: the final destination reached after sailing 32 km on a bearing of 310

To find the distance QR, we can use the Law of Cosines:

QR² = PQ² + PR² - 2(PQ)(PR)cos(QPR)

where PQ is the distance sailed on the first leg, PR is the distance sailed on the second leg, and angle QPR is the angle between the two legs.

We can calculate PQ and PR using basic trigonometry:

PQ = 46cos(14)

PR = 32cos(50)

Substituting these values into the Law of Cosines, we get:

QR² = (46cos(14))² + (32cos(50))² - 2(46cos(14))(32cos(50))cos(206)

Simplifying this expression using a calculator, we get:

QR ≈ 67.7 km

To find the bearing of QR, we can use the Law of Sines:

sin(QRP) / QR = sin(QPR) / PR

where angle QRP is the bearing we want to find.

Substituting the known values, we get:

sin(QRP) / 67.7 = sin(310 - 50) / (32sin(14))

Solving for sin(QRP) and taking the inverse sine, we get:

sin(QRP) ≈ 0.493

QRP ≈ 30.1°

Therefore, the ship is approximately 67.7 km away from P on a bearing of 30.1°.

b) If the ship travels west until it is due north of P, it will reach a point S that forms a right triangle with P and Q. Let's label the angles as follows:

angle QPS: the angle between PQ and PS

angle PQS: the angle between PS and QS

angle QSP: the right angle at S

We know that angle QPS is 90° (since the ship travels due north) and that PQ = 46 km. To find PS, we can use basic trigonometry:

PS = PQtan(QPS)

PS = 46tan(90-104)

PS ≈ 26.8 km

To check if the captain is correct, we need to find the distance between P and S. We can use the Pythagorean theorem:

PS² + SP² = PQ²

SP² = PQ² - PS²

SP² = (46)² - (26.8)²

SP ≈ 38.5 km

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Step-by-step explanation:

D option is the correct one.

(a)  cos(y) = adjacent/hypotenuse = 3x/√(64 + 9x^2) b) The angle whose secant is 67 is approximately 89.149 degrees. C) The exact value of cos^-1(cos(-5phi/6)) is phi/3.

For x > 0, if y = arccot(3x/8), find cos(y).Solution: First, we draw the reference triangle with y = arccot(3x/8). Since y = arccot(3x/8), we have:
tan(y) = 8/3x. This means that the opposite side is 8 and the adjacent side is 3x.

Using the Pythagorean Theorem, we can find the hypotenuse: h = √(8^2 + (3x)^2) = √(64 + 9x^2). Now, we can find cos(y) using the definition of cosine: cos(y) = adjacent/hypotenuse = 3x/√(64 + 9x^2)

To find sec^-1 (67) using a TI calculator, we can use the inverse cosine function: sec^-1 (67) = cos^-1 (1/67) On the TI calculator, we can enter: cos^-1 (1/67). And the calculator will give us an approximate value of 89.149 degrees.

The angle whose secant is 67 is approximately 89.149 degrees.
To find the exact value of cos^-1(cos(-5phi/6)).

The reference triangle for 5phi/6 is a 30-60-90 triangle, with the hypotenuse equal to 2, the opposite side equal to √3, and the adjacent side equal to 1.

Therefore, we have: cos(5phi/6) = adjacent/hypotenuse = 1/2. Now, we can find the exact value of cos^-1(cos(-5phi/6)): cos^-1(cos(-5phi/6)) = cos^-1(1/2) = phi/3. Therefore, the exact value of cos^-1(cos(-5phi/6)) is phi/3.

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The product of ((2)/(3)x+(4)/(3)) and (2x+(5)/(6)) as a trinomial in simplest form is (4/3)x^2 + (13/9)x + (10/9).

To express the product of ((2)/(3)x+(4)/(3)) and (2x+(5)/(6)) as a trinomial in simplest form, we need to multiply the two binomials using the distributive property.

First, we will multiply the first term of the first binomial by each term of the second binomial:
(2/3)x * 2x = (4/3)x^2
(2/3)x * (5/6) = (10/18)x = (5/9)x

Next, we will multiply the second term of the first binomial by each term of the second binomial:
(4/3) * 2x = (8/3)x
(4/3) * (5/6) = (20/18) = (10/9)

Now we will combine like terms:
(4/3)x^2 + (5/9)x + (8/3)x + (10/9) = (4/3)x^2 + (13/9)x + (10/9)

Therefore, the product of ((2)/(3)x+(4)/(3)) and (2x+(5)/(6)) as a trinomial in simplest form is (4/3)x^2 + (13/9)x + (10/9).

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x = 23°

4) To find the value of Angle B, we can use the alternate interior angles theorem, which states that if two parallel lines are cut by a transversal, the alternate interior angles are equal. In this case, we can say that ∟AB=∟CD, since ABCD is a parallelogram. Therefore, we can set 6x-15 = 4x-11 and solve for x to get x = 4. Plugging this back in, we can find that ∟AB = 6(4)-15 = 15°.

5) To find the value of Angle R, we can use the same method as above. We can set ∟QR=∟PS, since PQRS is a parallelogram. Therefore, we can set 8x-12 = 3x+5 and solve for x to get x = 7. Plugging this back in, we can find that ∟QR = 8(7)-12 = 56°.

7) To find the measure of x, we can use the fact that the quadrilateral is a rhombus. A rhombus is a quadrilateral with all four sides equal in length, so we can say that ∟AB = ∟CD. Therefore, we can set 23=x and solve for x to get x = 23°.

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Answer:

To multiply 640 by 36, we use the standard multiplication algorithm:

  640

x  36

------

 3840   (6 times 640)

+25600   (3 times 640, shifted one digit to the left)

-------

23040

Therefore, 640 times 36 equals 23,040.

The order of the number from least to greatest will be -12, |3|, |-4|, 10. Then the correct option is D.

It is the order of the numbers in which a smaller number comes first and then followed by the next number and then the last number will be the biggest one.

The numbers are given below.

|-4|, 10, -12, |3|

Determine the absolute value of the numbers, then we have

|-4| = 4

10 = 10

-12 = -12

|3| = 3

The order of the number from least to greatest will be given as,

-12, 3, 4, 10

-12, |3|, |-4|, 10

Thus, the correct option is D.

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Answer:

7.55

Step-by-step explanation:

3.50 + x = 11.05, x = 11.05 - 3.50, x = 7.55

Answer:

Step-by-step explanation:

Let [tex]x[/tex] be the cost of a juice bottle. Then

[tex]3.50+5x=11.05[/tex]

Solve this:

[tex]5x=7.55[/tex]   (subtracted 3.50 from both sides)

[tex]x=7.55/5=1.51[/tex]  (divided both sides by 5)

So juice bottles cost $1.51

The general solutions for this equation are [tex]\( x = \frac{\pi}{18} + \frac{2n\pi}{3} \)[/tex] and [tex]\( x = \frac{5\pi}{18} + \frac{2n\pi}{3} \)[/tex].

The general solutions of the given trigonometric equations can be found by applying the basic trigonometric identities and solving for the unknown variable.

a. [tex]\( 4 \sin ^{2}(x)-4 \sin (x)+1=0 \)[/tex]

This equation can be solved by using the quadratic formula. Let \( u = \sin (x) \), then the equation becomes [tex]\( 4u^2 - 4u + 1 = 0 \)[/tex]. Using the quadratic formula, we get:

[tex]\( u = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(4)(1)}}{2(4)} \)\\\( u = \frac{4 \pm \sqrt{16 - 16}}{8} \)\\\( u = \frac{4}{8} = \frac{1}{2} \)[/tex]

Now, we can substitute back \( u = \sin (x) \) and solve for x:

[tex]\( \sin (x) = \frac{1}{2} \)\\\( x = \arcsin (\frac{1}{2}) \)\\\( x = \frac{\pi}{6} + 2n\pi \) or \( x = \frac{5\pi}{6} + 2n\pi \)[/tex], where n is an integer.

The general solutions for this equation are [tex]\( x = \frac{\pi}{6} + 2n\pi \)[/tex] and[tex]\( x = \frac{5\pi}{6} + 2n\pi \)[/tex].

b. [tex]\( \tan (x) \sin (x)+\sin (x)=0 \)[/tex]

This equation can be solved by factoring out \( \sin (x) \):

[tex]\( \sin (x)(\tan (x) + 1) = 0 \)[/tex]

This equation will be true if either[tex]\( \sin (x) = 0 \) or \( \tan (x) + 1 = 0 \)[/tex].

For \( \sin (x) = 0 \), the general solutions are[tex]\( x = n\pi \)[/tex] where n is an integer.

For \( \tan (x) + 1 = 0 \), the general solutions are [tex]\( x = \frac{3\pi}{4} + n\pi \)[/tex], where n is an integer.

The general solutions for this equation are [tex]\( x = n\pi \) and \( x = \frac{3\pi}{4} + n\pi \).[/tex]

c. [tex]\( 3 \csc ^{2}(\theta)=4 \)[/tex]

This equation can be solved by isolating [tex]\( \csc ^{2}(\theta) \)[/tex] and taking the square root of both sides:

[tex]\( \csc ^{2}(\theta) = \frac{4}{3} \)[/tex]

[tex]\( \csc (\theta) = \pm \sqrt{\frac{4}{3}} \)[/tex]

Now, we can use the identity[tex]\( \csc (\theta) = \frac{1}{\sin (\theta)} \)[/tex]to solve for \( \theta \):

[tex]\( \frac{1}{\sin (\theta)} = \pm \sqrt{\frac{4}{3}} \)\( \sin (\theta) = \pm \sqrt{\frac{3}{4}} \)[/tex]

The general solutions for this equation are[tex]\( \theta = \arcsin (\pm \sqrt{\frac{3}{4}}) + 2n\pi \)[/tex], where n is an integer.

d. [tex]\( 2 \sin (3 x)-1=0 \)[/tex]

This equation can be solved by isolating[tex]\( \sin (3x) \)[/tex] and taking the inverse sine of both sides:

[tex]\( \sin (3x) = \frac{1}{2} \)\( 3x = \arcsin (\frac{1}{2}) \)\\\( 3x = \frac{\pi}{6} + 2n\pi \) or \( 3x = \frac{5\pi}{6} + 2n\pi \)[/tex], where n is an integer.

The general solutions for this equation are [tex]\( x = \frac{\pi}{18} + \frac{2n\pi}{3} \)[/tex] and [tex]\( x = \frac{5\pi}{18} + \frac{2n\pi}{3} \)[/tex].

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a. For A2 = 0, it is proved that  (I - A)-1 = I + A.

b.  For A3 = 0, (I - A)-1 = I + A + A2.

a. If A2 = 0, then  (I - A)-1 = I + A

Proof:

(I - A)-1 = (I + A)-1

(I + A)-1 = (I + A)(I - A)-1

(I + A)(I - A)-1 = I + A

Therefore, (I - A)-1 = I + A

b. If A3 = 0 then  (I - A)-1 = I + A + A2

Proof:

(I - A)-1 = (I + A + A2)-1

(I + A + A2)-1 = (I + A + A2)(I - A)-1

(I + A + A2)(I - A)-1 = I + A + A2

Therefore, (I - A)-1 = I + A + A2

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The probability of rolling a triangle is 50%

From the question, we have the following parameters that can be used in our computation:

Sides = 12

triangles = 6

Circles = 3

Squares = 3

The die has a total of 6 + 3 + 3 = 12 sides.

Since there are 6 triangles on the die, the probability of rolling a triangle is:

P(triangle) = number of favorable outcomes / total number of outcomes

Substitute the known values in the above equation, so, we have the following representation

P(triangle) = 6 / 12

Evaluate

P(triangle) = 50%

Hence, the probability of rolling a triangle on the die is 1/2 or 0.5.

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let's recall that a year has 12 months, thus 15 months is really 15/12 of a year.

[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$9800\\ r=rate\to 5\%\to \frac{5}{100}\dotfill &0.05\\ t=years\to \frac{15}{12}\dotfill &\frac{5}{4} \end{cases} \\\\\\ I = (9800)(0.05)(\frac{5}{4}) \implies I = 612.5[/tex]

In a circle whose equation is [tex]x^2 + y^2 - 4x + 6y -12 = 0[/tex]. The correct answer is option C.  9.59.

Equation of the circle in standard form by completing the square for both x and y terms:
[tex](x^2 - 4x) + (y^2 + 6y) = 12(x - 2)^2 + (y + 3)^2 = 12 + 4 + 9(x - 2)^2 + (y + 3)^2 = 25[/tex]
then  Center: (2, -3) and Radius: 5

Equation of the line passing through the point (8,6) and the center of the circle (2,-3): Slope =

[tex](6 - (-3))/(8 - 2) = 9/6 = 3/2y - 6 = (3/2)(x - 8)y = (3/2)x - 6[/tex]
Substitute the equation of the line into the equation of the circle and solve for x:
[tex](x - 2)^2 + ((3/2)x - 6 - 3)^2 = 25(x - 2)^2 + ((3/2)x - 9)^2 = 25 (5/4)x^2 - (19/2)x + 80 = 0[/tex]

Use the quadratic formula to find the x-intercepts of the line and circle:
[tex]x = (-(-19/2) ± √((-19/2)^2 - 4(5/4)(80)))/(2(5/4))x = (19/2 ± √(361/4 - 400))/(5/2)x = (19/2 ± √(-39/4))/(5/2)x = (19 ± √(-39))/(5)[/tex]

Use the distance formula to find the distance between the point (8,6) and the x-intercepts:
[tex]Distance = √((x - 8)^2 + (y - 6)^2)Distance = √(((19 ± √(-39))/5 - 8)^2 + ((3/2)(19 ± √(-39))/5 - 6)^2)Distance = √((-19/5)^2 + (-39/5)^2)Distance = √(961/25 + 1521/25)Distance = √(2482/25)Distance = 9.59[/tex]

Therefore, the tangent distance from the point (8,6) to the circle is 9.59. The correct answer is option C

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The mapping diagram represents a relation between the set of x-values {negative 8, negative 5, negative 1, 1, 12} and the set of y-values {negative 4, negative 2}. The arrows between the circles show how the x-values are related to the y-values.

A mapping diagram is a visual way to represent a relation between two sets of values, where the arrows show how the values in one set are related to the values in the other set. The input values are typically shown in one circle or column, while the output values are shown in another circle or column.

Calculus is a branch of mathematics that deals with the study of rates of change and the accumulation of small changes to determine their effects on a larger scale. It is a major part of modern mathematics and is used extensively in science, engineering, and economics.

In the given question,

The mapping diagram represents a relation between the set of x-values {negative 8, negative 5, negative 1, 1, 12} and the set of y-values {negative 4, negative 2}. The arrows between the circles show how the x-values are related to the y-values.

Specifically, the arrow from negative 8 to negative 4 means that the input value of x = negative 8 is related to the output value of y = negative 4. Similarly, the arrow from negative 5 to negative 2 means that the input value of x = negative 5 is related to the output value of y = negative 2.

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The width of a rectangle is 2 ft less than 4 times the length, so the model for the width W in terms of the length L is W = 4L - 2.

To solve this problem, we need to create a model for the width W in terms of the length L. According to the problem, the width of the rectangle is 2 ft less than 4 times the length. This can be written as:
W = 4L - 2

This equation represents the relationship between the width and the length of the rectangle. It shows that the width is equal to 4 times the length, minus 2. Therefore, the correct answer is a. W=4L - 2.

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Answer:

q - 8

Step-by-step explanation:

q - 8

Answer:

q=0

Step-by-step explanation:

The minimum possible parameter of the rectangle is 34 units.

A rectangle is a quadrilateral having four sides and the sum of the angles is 180 in the rectangle the opposite two sides are equal and parallel and the two sides are at 90-degree angles.

Let's assume the length of the rectangle is L and the width is W. The area of the rectangle is given as 60 sq. units.

Area of rectangle = Length × Width = L × W = 60

We are looking for the minimum perimeter of the rectangle. Perimeter of rectangle = 2(L + W)

To find the minimum perimeter, we need to find the minimum values of L and W that satisfy the condition that the area is 60.

The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

If we choose L = 1 and W = 60, then the area is 1 × 60 = 60.

If we choose L = 2 and W = 30, then the area is 2 × 30 = 60.

If we choose L = 3 and W = 20, then the area is 3 × 20 = 60.

If we choose L = 4 and W = 15, then the area is 4 × 15 = 60.

If we choose L = 5 and W = 12, then the area is 5 × 12 = 60.

If we choose L = 6 and W = 10, then the area is 6 × 10 = 60.

The minimum perimeter occurs when L and W are the closest in value, which is achieved when L = 5 and W = 12. Thus, the minimum perimeter of the rectangle is:

Perimeter = 2(L + W) = 2(5 + 12) = 34

Therefore, the minimum possible parameter of the rectangle is 34 units.

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The general solution to a 2nd order derivative for f(x) with real coefficients can be obtained by finding the other root of the auxiliary equation and then using those roots to write the general solution.

Since one of the roots of the auxiliary equation is 3 + 7i, the other root must be the conjugate of this root, which is 3 - 7i. This is because the coefficients of the auxiliary equation are real, so the roots must come in conjugate pairs.

Now that we have both roots, we can write the general solution to the 2nd order derivative as:

f(x) = e^(3x)(C1*cos(7x) + C2*sin(7x))

where C1 and C2 are arbitrary constants.

This is the general solution to the 2nd order derivative for f(x) with real coefficients when one of the roots of the auxiliary equation is 3 + 7i.

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There are 24 five-pound bags of potatoes in the crate.

The crate contains a total of 6 bags of potatoes weighing 10 pounds each, which is a total of 60 pounds.

So, the remaining weight of the crate is 180 pounds - 60 pounds = 120 pounds.

Let's assume that there are x five-pound bags of potatoes in the crate.

Therefore, the total weight of these x bags of potatoes would be 5x pounds.

We know that the weight of the entire crate is 180 pounds,

60 pounds (weight of 6 bags of 10-pound potatoes) + 5x pounds (weight of x 5-pound bags of potatoes) = 180 pounds (weight of the entire crate)

60 + 5x = 180

5x = 180 - 60

5x = 120

x = 24

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Shiro purchased 11 pounds of beef and 4 pounds of chicken.

Shiro purchased a total of 18 pounds of meat, spending $96.

We can designate the beef as x pounds, and the chicken as y pounds. To find the solution, we set up two equations:

6x + 4.5y = 96 x + y = 18

To solve this system of equations, we can subtract 6x from both sides of the first equation, giving us 4.5y = 96 - 6x.

We can substitute this expression into the second equation, giving us x + (96 - 6x) = 18.

We can then simplify this equation, giving us 7x = 78, and thus x = 11.

Now we can substitute x = 11 into the original equation, 6x + 4.5y = 96, to find y = 4.

Therefore, Shiro purchased 11 pounds of beef and 4 pounds of chicken.

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On a given day where the price of the stock is 2900AED, the stock broker should sell the stock.

The stock broker should sell the stock if the price is in the top 4% range. To determine if the price of 2900AED is in the top 4% range, we need to calculate the z-score and compare it to the z-score for the top 4% range.

The z-score formula is:
z = (x - μ) / σ

Where:
x = the value we are interested in (2900AED)
μ = the mean (2540AED)
σ = the standard deviation (150AED)

Plugging in the values, we get:
z = (2900 - 2540) / 150
z = 360 / 150
z = 2.4

The z-score for the top 4% range is 1.75. Since the z-score of 2.4 is greater than 1.75, the price of 2900AED is in the top 4% range. Hence, he should sell the stocks.

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The determinant of the inverse of an invertible matrix is the reciprocal of the determinant of the original matrix.

Determinants: The determinant of an invertible matrix A is a scalar value that is used to indicate the invertibility of the matrix. If the determinant of A is non-zero, then the matrix is invertible. The determinant of the inverse of an invertible matrix is the reciprocal of the determinant of the original matrix. In other words, det(A−1)=det(A)1​.

To prove this, we can use the property of determinants that det(AB)=det(A)det(B) for any two square matrices A and B. Since A is an invertible matrix, we know that AA−1=I, where I is the identity matrix. Taking the determinant of both sides of this equation gives us:

det(AA−1)=det(I)

Using the property of determinants mentioned above, we can rewrite the left-hand side of the equation as:

det(A)det(A−1)=det(I)

Since the determinant of the identity matrix is 1, we can simplify the equation to:

det(A)det(A−1)=1

Dividing both sides of the equation by det(A) gives us:

det(A−1)=det(A)1​

Therefore, the determinant of the inverse of an invertible matrix is the reciprocal of the determinant of the original matrix.

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The measures ∠2 = 135°, ∠5 = 45°, ∠6 = 135°, ∠4 = 45° and  ∠3 = 135°.

An angle is formed when two straight lines or rays meet at a common endpoint.

The angle ∠1 = 45° is given.

Types of the angles are as follows;

Supplementary angle - Two angles are said to be supplementary angles if their sum is 180 degrees.

Corresponding angle - If two lines are parallel then the third line. The corresponding angles are equal angles.

Vertically opposite angle - When two lines intersect, then their opposite angles are equal.

Alternate angle - If two lines are parallel then the third line will make z -angle and z-angles are equal angles with the parallel lines.

∠1 + ∠2 = 180°    (Supplementary angle)

45° + ∠2 = 180°

∠2 = 135°

∠1 = ∠5 = 45°   (Corresponding angle)

∠2= ∠6 = 135°

∠4 = 45° (Alternate angle)

∠3 = 135°   (Alternate angle)

Hence, the measures ∠2 = 135°, ∠5 = 45°, ∠6 = 135°, ∠4 = 45° and  ∠3 = 135°.

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Answer:x = 3 and y = 5

Step-by-step explanation:

We have two equations:

-3x + 3y = 6 ........(1)

x + 3y = 18 ........(2)

We can use the second equation to solve for y in terms of x:

-4x = -12

x = 4

Now we substitute this expression for y into the second equation:

3 + 3y = 18

3y = 15

y = 5

Therefore, the solution to the system of equations is x = 3 and y = 5.

equation 1: - 3x + 3y = 6

equation 2: x + 3y = 18

start with equation 2:

x + 3y = 18

subtract 3y from both sides:

x = 18 - 3y

plug the new equation into equation 1:

- 3x + 3y = 6

- 3 (18 - 3y) + 3y = 6

multiply:

-54 + 9y + 3y = 6

collect like terms:

-54 + 12y = 6

add 54 to both sides, then divide the whole equation by 12:

12y = 60

y = 5

plug the y value into either equation to solve for x. for example, here is equation 2:

x + 3y = 18

x + 3(5) = 18

x + 15 = 18

x = 3

check answer:

x + 3y = 18

3 + 3(5) = 18

3 + 15 = 18

18 = 18 is true

Answer: 0.875 h or for a fraction 8 3/4

Step-by-step explanation:

June spent 15 minutes practicing her piano, 30 minutes cleaning her room and 7.5 minutes putting away her dirty laundry so a total of 52.5 minutes

Answer:

8 3/4

Step-by-step explanation: IK IM SORRY im sooo confusing!! And it makes me frustrated too...

Answer:

3

Step-by-step explanation:

cuz ik