If f varies inversely as x, and y= 2 when x= 2, find y when x= 1

The volume of the solid obtained by rotating the region about the y-axis is 3036π cubic units.

How to find the volume by rotating the region?

To find the volume of the solid obtained by rotating the region enclosed by the graphs of y=27-x, y=3x-9, and x=0 about the y-axis, we need to use the method of cylindrical shells.

Here the region is bounded by the graphs of y=27-x, y=3x-9, and x=0. We can see that the region is a trapezoid with vertices at (0, -9), (6, 9), (18, 9), and (27, 0).

Next, we need to determine the height and radius of each cylindrical shell. The height of each shell is the difference between the y-coordinates of the top and bottom of the shell. The radius of each shell is the x-coordinate of the shell.Let's consider a shell with x-coordinate x. The height of the shell is given by (27 - x) - (3x - 9) = 36 - 4x. The radius of the shell is x. The volume of the shell is given by 2πrhΔx, where Δx is the thickness of the shell.

Therefore, the total volume of the solid is given by:

[tex]V = \int0^6 2πx(36 - 4x)dx + \int6^18 2πx(9)dx + \int18^27 2πx(27 - x - 9)dx[/tex]

Simplifying and evaluating the integral, we get:

V = 2π(216) + 2π(243) + 2π(243/2) = 3036π

Therefore, the volume of the solid obtained by rotating the region about the y-axis is 3036π cubic units.

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Rotating the area about the y-axis yields a solid with a volume of 3036 cubic units.

The cylindrical shell method must be used to determine the solid's volume after rotating the region enclosed by the graphs of

y=27-x, y=3x-9, and x=0 about the y-axis.

In this case, the graphs of y=27-x, y=3x-9, and x=0 define the region's boundaries. The region is a trapezium, as can be seen, with vertices at (0, -9), (6, 9), (18, 9), and (27, 0).

The height and radius of each cylindrical shell must then be determined. Each shell's height is determined by the difference between its top and bottom y-coordinates.

Each shell's x-coordinate corresponds to its radius.Consider a shell with the coordinates x and y. (27 - x) - (3x - 9) = 36 - 4x gives the height of the shell. The shell has a radius of x. 2rhx, where x is the shell's thickness, gives the volume of the shell.

As a result, the solid's entire volume can be calculated using:

When we simplify and assess the integral, we obtain:

V = 2π(216) + 2π(243) + 2π(243/2)

  = 3036π

As a result, the region's volume after being rotated about the y-axis is 3036 cubic units.

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Therefore, y = (1/3) sin ((1/2)x²) is the solution of the initial value problem y′=(2/3)x√(1−9y²); y(0) = 0.

To solve the differential equation y′=(2/3)x√(1−9y²)

The differential equation to be solved is: y′=(2/3)x√(1−9y²).

Here, we need to find y.

For this, we will separate the variables and integrate both sides. Integration gives us:

`∫1/(√(1−9y²))dy=∫(2/3)x dx`

.On integrating the left side, we will use u-substitution.

u = 3y → du = 3 dy

dy = (1/3) du → y = (1/3) u.

Now the equation becomes `∫du/(√(1−u²))=(2/3)∫xdx`.

Now, substituting u = sin t in the left integral, we have: `

∫du/(√(1−u²))

=∫cos(t)dt

=[sin⁻¹(u)]+C`.

So, the left-hand side is `

[sin⁻¹(u)]+C

= [sin⁻¹(3y)] + C`

Now, the right-hand side will be:

∫xdx=(1/2)x²+D`

On combining both sides, we get the solution to the differential equation as: `

[sin⁻¹(3y)]+C=(1/2)x²+D`

On solving for y, we get:

y = (1/3) sin ((1/2)x² + D' ) or  y = (1/3) sin ((1/2)x²)

since we can choose D' = C.

To solve the initial value problem

y′=(2/3)x√(1−9y2); y(0) = 0

To solve the initial value problem

y′=(2/3)x√(1−9y2)

y(0) = 0

we will substitute x = 0, y = 0 in the general solution that we obtained in part .

y = (1/3) sin ((1/2)x²)

y = (1/3) sin ((1/2)0²) = 0.

So the required solution is y = (1/3) sin ((1/2)x²).

Therefore, y = (1/3) sin ((1/2)x²) is the solution of the initial value problem y′=(2/3)x√(1−9y²); y(0) = 0.

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Thus, the surface area with the dimensions of a rectangular prism as 6m by 4 m by 15 m is found as: S = 348 sq. m.

A rectangular prism is a 3 solid that is surrounded by 6 rectangular faces, 2 of which are the bases (the top face and bottom face), and the remaining 4 are lateral faces. It likewise has 12 edges and 8 vertices.

A rectangular prism is sometimes known as a cuboid because of its shape. A shoe box, an ice cream bar, or a matchbox are some instances of rectangular prisms in everyday objects.

Dimensions of rectangular prism :

surface area of a rectangular prism:

S = 2(lw + wh + hl)

S = 2(6*4 + 4*15 + 15*6)

S = 348 sq. m.

Thus, the surface area with the dimensions of a rectangular prism as 6m by 4 m by 15 m is found as: S = 348 sq. m.

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Answer: 50

Step-by-step explanation:

2*6+2*4+2*15=12+8+30=50

A′(2, 1), B′(6, 1), and C′(2, 5) are the image's positions as a result of moving the triangular 6 units to the right.

The order of the numbers is always latitude first, semicolon after which is followed by longitude. For instance, New York City is located approximately at 40°N and 74°W in terms of latitude and longitude. The approximate coordinates for Sydney are 34°S and 150°E. When using a chart or globe, your longitude as well as latitude won't be precisely accurate.

To translate a point (x, y) to the right by 6 units, we add 6 to the x-coordinate.

So, the image of vertex A(-8, 1) would be A'(-8 + 6, 1) = A'(-2, 1).

Similarly, the image of vertex B(0, 1) would be B'(0 + 6, 1) = B'(6, 1).

And, the image of vertex C(-4, 5) would be C'(-4 + 6, 5) = C'(2, 5).

Therefore, the coordinates of the image after translating the triangle 6 units to the right are:

A′(−2, 1), B′(6, 1), C′(2, 5).

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The value of FG( SAY X) = x=131.

An angle is the result of the intersection of two lines.

An "angle" is the length of the "opening" between these two beams.

Angles are commonly measured in degrees and radians, a measurement of circularity or rotation.

In geometry, an angle can be created by joining the extremities of two rays. These rays are intended to represent the angle's sides or limbs.

The two primary components of an angle are the limbs and the vertex.

The joint vertex is the common terminal of the two beams.

According to our question-

35 - 3x + 2x + 14= 180

49-x=180

-x=180-49

x=131

The value of FG( SAY X) = x=131.

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Answer:

223

Step-by-step explanation:

i am in 6th grade

Answer:

A

Step-by-step explanation:

the exterior angle of a triangle is equal to the sum of the 2 opposite interior angles.

99° is an exterior angle of the triangle , then

2x + 43 = 99 ( subtract 43 from both sides )

2x = 56 ( divide both sides by 2 )

x = 28

Using the definition of cosecant and cotangent, the expression can be rewritten as 1/sine (x) + 1/cosine (x).

1/sine (x) + 1/cosine (x) = 1 confirms that the Pythagorean identity.

The cosecant of an angle is defined as 1/sine (x), and the cotangent of an angle is defined as 1/cosine(x).

Using the definition of cosecant and cotangent, the expression can be rewritten as 1/sine (x) + 1/cosine (x).

Using the fundamental Pythagorean identity, which states that

sine²(x) + cosine² (x) = 1, the expression can be further simplified to

sine²(x) + cosine² (x) + 1/sine (x) + 1/cosine (x) = 1.

To verify the identity, we can substitute sine²(x) + cosine² (x) with 1, leaving us with 1 + 1/sine (x) + 1/cosine (x) = 1.

Simplifying further, we get 1/sine (x) + 1/cosine (x) = 1, which is the original expression. This confirms that the Pythagorean identity is true for the given expression.

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Given equation represents a circle with center at (1, 1) and vertex form [tex](x - 1)^2 + (y - 1)^2 = 2.[/tex]

A circle is a closed shape in geometry that consists of all points that are equidistant from a central point called the center. A circle is defined by its radius, which is the distance from the center to any point on the circumference (outer boundary) of the circle. The diameter of a circle is twice the radius, and the circumference is the total distance around the circle.

The given equation [tex]x^2 + y^2 - 2x - 2y = 1[/tex]represents a conic section known as a circle.

To rewrite the given equation in vertex form for a circle, we need to complete the square for both x and y terms separately. The vertex form of a circle is given by:

[tex](x - h)^2 + (y - k)^2 = r^2[/tex]

where (h, k) is the center of the circle and r is the radius.

Let's complete the square for x and y terms:

[tex]x^2 - 2x + y^2 - 2y = 1[/tex]

Adding and subtracting the necessary terms to complete the square:

[tex]x^2 - 2x + 1 + y^2 - 2y + 1 = 1 + 1[/tex]

Rewriting the equation in vertex form:

[tex](x - 1)^2 + (y - 1)^2 = 2[/tex]

So, the conic section represented by the given equation is a circle with its center at (1, 1) and a radius of √2, and the vertex form of the circle is [tex](x - 1)^2 + (y - 1)^2 = 2.[/tex]

Therefore, Given equation represents a circle with center at (1, 1) and vertex form [tex](x - 1)^2 + (y - 1)^2 = 2.[/tex]

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Answer:

y = (-1/4)x - 1

Step-by-step explanation:

We can write an equation for this line in slope-intercept form:

[tex]y=mx+b[/tex],

where [tex]m[/tex] is the slope of the line and [tex]b[/tex] is the y-coordinate of the y-intercept.

First, to solve for the slope:

slope = rise / run

slope = -1 / 4

slope = -1/4

Next, to find the y-intercept, we can see that the line crosses the y-axis (vertical axis) when y = -1.

Finally, we can form an equation using these two pieces of information.

y = (-1/4)x - 1

The volume of the prism is 0.343 cubic inches.

What is Volume ?

Volume is a measure of the amount of space occupied by a three-dimensional object. It is the quantity of space that a solid object occupies in three dimensions. Volume is often expressed in cubic units, such as cubic meters, cubic centimeters, or cubic inches , depending on the system of measurement used.

To find the volume of a rectangular prism, we multiply its length, width, and height.

In this case, the length, width, and height of the prism are all 0.7 inch. Therefore, the volume of the prism is:

Volume = (length) x (width) x (height)

= (0.7 x (0.7) x (0.7)

= 0.343 cubic inches

Therefore, the volume of the prism is 0.343 cubic inches.

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The center and radius of a circle given by the equation x² - 10x + y² + 6y - 30 = 0  is (5,-3) and√64 = 8 units.

When finding the center and radius of a circle given by the equation x² - 10x + y² + 6y - 30 = 0, one can use the following steps:

The first step is to rearrange the equation into the standard form, (x - a)² + (y - b)² = r². This is done by completing the square for both the x and y terms in the equation.

x² - 10x + y² + 6y - 30 = 0x² - 10x + 25 + y² + 6y + 9 - 30 = 25 + 9(x - 5)² + (y + 3)² = 64 Therefore, the center of the circle is (5,-3), and the radius is √64 = 8 units.

Explanation:  Given the equation x² - 10x + y² + 6y - 30 = 0, we want to find the center and radius of the circle. The standard form of the equation of a circle with center (a,b) and radius r is (x - a)² + (y - b)² = r². We will begin by completing the square for the x terms and the y terms separately: For the x terms: x² - 10x= x² - 10x + 25 - 25= (x - 5)² - 25 For the y terms: y² + 6y= y² + 6y + 9 - 9= (y + 3)² - 9 Now we can substitute these expressions back into the original equation and simplify: x² - 10x + y² + 6y - 30 = 0(x - 5)² - 25 + (y + 3)² - 9 - 30 = 0(x - 5)² + (y + 3)² = 64 The equation is now in standard form,

which means that the center of the circle is (5,-3) and the radius is √64 = 8 units.

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Answer:

6/7x

Step-by-step explanation:

g/7 x 6 = 6/7 x

Answer: g / 7 x 6

Explanation:

Answer:

This rectangle is shifted 1 unit to the left and then 4 units down. So the algebraic rule is (x - 1, y - 4).

All tests of hypothesis are based on the assumption that the null hypothesis is true. C. the null hypothesis is true is the correct option.

In hypothesis testing, two hypotheses are compared:

The null hypothesis (H0) and the alternative hypothesis (Ha).The null hypothesis is the statement that is assumed to be true, while the alternative hypothesis is the statement that the researcher is trying to prove.

For example, if the researcher wants to test whether a new drug is effective in treating a certain disease, the null hypothesis would be that the drug has no effect, while the alternative hypothesis would be that the drug is effective.

Therefore, all tests of hypothesis are based on the assumption that the null hypothesis is true option c is correct..

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Answer:

To estimate the number of pentagons in the tub, we can use the proportion of pentagons in the sample of geometric shapes that Mrs. Chen selected and apply it to the total number of geometric shapes in the tub.

The proportion of pentagons in the sample is:

4 / (3 + 2 + 2 + 4 + 2) = 4 / 13 ≈ 0.31

We can assume that this proportion is representative of the entire tub of geometric shapes, and we can apply it to the total number of geometric shapes in the tub:

0.31 x 250 ≈ 77.5

Rounding to the nearest whole number, we can estimate that there are approximately 78 pentagons in the tub.

Therefore, the estimated number of pentagons in the tub is 78.

Answer:

58 m

Step-by-step explanation:

The correct answer is 58.

To get the perimeter you add all the sides of the image, however, you are missing 2 values from the image.

If you make the shape into a square where all opposite sides are the same length, then you will see that one missing length is 6 m =(10m-4m).

The other missing number is 12 m which is base of 19 m - 7m that you are given on the top.

So you add the measurements (going clockwise starting at the top, 7+6+12+4+19+10=58 m

Answer:

  58 m

Step-by-step explanation:

You want the perimeter of the L-shaped figure shown.

The perimeter is the sum of the side lengths. Here, a couple of lengths are missing from the diagram, but that doesn't prevent us finding the perimeter.

The horizontal lengths at the top have the same total length as the length at the bottom marked 19 m. This means the sum of all of the horizontal lengths is ...

  2 × 19 m = 38 m

The vertical lengths at the right side have the same total length as the vertical length at the left side, marked 10 m. This means the sum of all of the vertical lengths is ...

  2 × 10 m = 20 m

The perimeter is the sum of the horizontal and vertical lengths:

  P = 38 m + 20 m = 58 m

The perimeter of the figure is 58 m.

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Answer:

x=3, y=7

Step-by-step explanation:

Substituting the first equation y=x+4 into the second:

3x+(x+4)=16

Simplifying:

3x+x+4=16

4x+4=16

Subtracting 4 from both sides:

4x=12

Dividing both sides by 4:

x=3

We can now substitute x=3 into our first equation, y=x+4.

Substituting:

y=3+4

y=7
So, x=3, y=7

Answer:  Gross income is the total amount of money earned before any deductions or taxes are taken out. Net income, on the other hand, is the amount of money left after all deductions and taxes have been taken out. In other words, net income is what you actually take home after all expenses have been accounted for.

The two solutions for the given equation in the interval are:

x = 0°

x = 159.1°

Here we have the equation:

|1 + 3sin(2x)| = 1

Breaking the absoulte value part, we will get two equations, these are:

1 + 3sin(2x) = 1

1 + 3sin(2x) = -1

Now we need to solve these two, the first one gives:

3sin(2x) = 1 - 1

3sin(2x) = 0

Then we know that:

2x = 0°

x = 0°/2 = 0

the other equation gives:

1 + 3sin(2x) = -1

3sin(2x) = -1 - 1

3sin(2x) =-2

sin(2x) = -2/3

2x = Asin(-2/3)

2x = 318°

x = 318.2°/2 = 159.1°

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The t-distribution becomes flatter as the sample size increases.

The t distribution, compared to the z distribution, is more peaked for small samples but increasingly like the z distribution as n increases.

The t-distribution :

A t-distribution is a symmetric distribution that is similar to the standard normal distribution, also known as the z-distribution.

It is a continuous probability distribution that is used to estimate population parameters when the sample size is small (less than 30) or when the population standard deviation is unknown.

The t-distribution has more variability than the standard normal distribution due to the smaller sample size.

For a given number of degrees of freedom, the t-distribution has more density in the tails than the standard normal distribution.

The z-distribution :

The standard normal distribution is also known as the z-distribution. It is a continuous probability distribution that has a mean of zero and a standard deviation of one.

The z-distribution is bell-shaped, with a median of zero, a mode of zero, and a mean of zero.

In conclusion, the t-distribution has a peakier curve than the z-distribution, especially for small sample sizes.

However, as the sample size increases, the t-distribution approaches the z-distribution.

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The coordinates of R' when rotated 90° about their center of rotation is (-6, -3).

Geometry is a branch of mathematics that deals with the study of shapes, sizes, relative positions of objects, and the properties of space.

The point R(3, -6) is located in the fourth quadrant as both the x-coordinate and y-coordinate are positive.

When the sails rotate 90 degrees clockwise, the point R will move to a new location R' in the third quadrant where both the x-coordinate and y-coordinate are negative.

To find the coordinates of R', we need to swap the x-coordinate and y-coordinate of point R and then negate the new x-coordinate.

That is, if R' has coordinates (x, y), then x = -6 and y = -3.

Hence, the coordinates of R' are (-6, -3).

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Answer:

Step-by-step explanation:

(1,-2) (-3,1)

1.     y--1=  -- 3/4 (x+3)

2.    y=  --3/4x--5/4

The measures are: [tex]MK = 30[/tex]

[tex]NL = 13[/tex]

[tex]m/MKL = 41^\circ[/tex]

[tex]m/NLK = 41^\circ[/tex]

[tex]y \approx 17.79^\circ[/tex]

In a rhombus, all sides are equal in length and opposite angles are congruent. Let's denote the measures as follows:

MK = 30 (Given)

NL = 13 (Given)

m/MKL = 41° (Given)

m/NLK = x (To be determined)

m/MKL = y (To be determined)

Since JKLM is a rhombus, we know that JM = KL and JK = ML. We can use the properties of a rhombus to find the values of x and y.

First, we can use the fact that opposite angles in a rhombus are congruent. Since   [tex]m/MKL = 41^\circ[/tex] , then m/NLK (opposite angle) is also 41°.

Next, we can use the fact that diagonals of a rhombus bisect each other at a 90° angle. Since MK = 30 and NL = 13, the diagonals JL and KM intersect at their midpoint, which we can denote as O.

Using the properties of right triangles, we can form a right triangle JOM with JO = JK/2 = ML/2 (since JL is the diagonal and KM is its midpoint), OM = MK/2 = 30/2 = 15, and JM = JK = ML (since JK = ML in a rhombus).

Now we can use trigonometry to find the value of y. In right triangle JOM, we have:

[tex]sin(y) = OM/JM = 15/JM[/tex]

Since JM = KL (by the properties of a rhombus), and KL is a diagonal bisecting NL, we can use Pythagoras' theorem to find KL:

[tex]KL^2 = NL^2 + LK^2 = 13^2 + (MK/2)^2 = 13^2 + 15^2 = 169 + 225 = 394[/tex]

[tex]KL = \sqrt394[/tex]

So, we can substitute JM = KL = √394 in the equation for sin(y):

[tex]sin(y) = 15/\sqrt394[/tex]

Taking the inverse sine of both sides:

[tex]y = sin^(-1)(15/\sqrt394)[/tex]

Using a calculator, we can find the approximate value of y to be  [tex]y ≈ 17.79^\circ.[/tex]

Therefore, the measures are: [tex]MK = 30[/tex]

[tex]NL = 13[/tex]

[tex]m/MKL = 41^\circ[/tex]

[tex]m/NLK = 41^\circ[/tex]

[tex]y \approx 17.79^\circ[/tex]

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The probability of having a boy is 0.51, and the probability of having a girl is 1 - 0.51 = 0.49. To find the probability of having three boys, we use the binomial probability formula: P(X = k) = (n choose k) * pk * (1-p)(n-k).

Probability is an estimate of how likely an event is to occur. It is a value ranging from zero to one, with 0 indicating an impossible event and 1 indicating a certain event. The higher the likelihood, the more probable the event will occur, and the lower the probability, the less likely the event will occur.

The probability of having a boy is 0.51, which means that the probability of having a girl is 1 - 0.51 = 0.49.

We want to find the probability that a family of five children has exactly three boys. We can use the binomial probability formula:

P(X = k) = (n choose k) * [tex]p^k * (1-p)^(n-k)[/tex]

where:

P(X = k) is the probability of getting k successes (in this case, having k boys)

n is the number of trials (in this case, the number of children born)

p is the probability of success (in this case, the probability of having a boy)

(n choose k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials

So, for this problem:

n = 5

k = 3

p = 0.51

P(X = 3) = (5 choose 3) * [tex]0.51^3 * 0.49^(5-3)[/tex]

= (10) * [tex]0.51^3 * 0.49^2[/tex]

= 0.234

Therefore, the probability that a family of five children has exactly three boys is 0.234, or about 23.4%.

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A word sentence that represents this inequality is "A number x is more than 1."

What is Inequality ?

In mathematics, an inequality is a statement that compares two values, expressing that one value is greater than, less than, greater than or equal to, or less than or equal to the other value. Inequalities are represented using symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">

Inequality: x > 3

Word sentence: The value of x is greater than 3.

Explanation: Based on the graph, we can see that the shaded area represents all the values of x that are greater than 3. Therefore, the inequality that represents this graph is x > 3, which means that x is any number greater than 3.

Question 2:

Word sentence: A number x is more than 1.

Explanation: The inequality represented by the given graph is x > 1, which means that x is any number greater than 1.

Therefore, a word sentence that represents this inequality is "A number x is more than 1."

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Answer:

It would be the last one.

Step-by-step explanation:

hope this is correct

The hypotenuse of the triangle is x = 10.

We can see that we have a right triangle, then we can use the trigonometric relation:

cos(a) = (adjacent cathetus)/hypotenuse

Replacing the known values:

cos(60°) =5/x

Solving for x we will get:

x = 5/cos(60°) = 10

That is the value of x.

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Answer:

Therefore, 2 + 2 with a square root of 9 is equal to 8.

Step-by-step explanation:

We can simplify the expression "square root of 9" to just 3. Then, we have:

2 + 2(3) = 2 + 6 = 8

Therefore, 2 + 2 with a square root of 9 is equal to 8.

Tnx   :)  :)  ;)

the solution of a system of linear equations is (4,36) with the help of the substitution method.

The algebraic approach to solving simultaneous linear equations is known as the substitution method. The value of one variable from one equation is substituted in the second equation in this procedure, as the name implies. By doing this, a pair of linear equations are combined into a single equation with a single variable, making it simpler to solve.

The given system of equations is

[tex]y=9x\\y=8x+4[/tex]

We shall solve it with the help of the substitution method

Substitute the value of y in Equation 2

[tex]9x=8x+4\\9x-8x=4\\x=4[/tex]

Put the value of x in Equation 1

[tex]y=9*4=36[/tex]

Hence the solution of a system of linear equations is (4,36) with the help of the substitution method.

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