Integral is [tex]\int_0^{27} (sin x)/x dx[/tex] ≈ 0.246918974 (rounded to nine decimal places).
To approximate the integral ∫₀²⁷ (sin x)/x dx with an error of magnitude less than 10⁻⁸ using series, we can use the Maclaurin series expansion of sin x:
sin x = x - (x³/3!) + (x⁵/5!) - (x⁷/7!) + ...
Substituting this series into the integral, we get:
∫₀²⁷ (sin x)/x dx = ∫₀²⁷ (x - (x³/3!) + (x⁵/5!) - (x⁷/7!) + ...) / x dx
= ∫₀²⁷ (1 - (x²/3!) + (x⁴/5!) - (x⁶/7!) + ...) dx
= [x - (x³/(33!)) + (x⁵/(55!)) - (x⁷/(7 × 7!)) + ...]
Evaluated from x = 0 to x = 0.27
Using the first four terms of this series, we get:
∫₀²⁷ (sin x)/x dx ≈ [0.27 - ((0.27)³/(33!)) + ((0.27)⁵/(55!)) - ((0.270)⁷/(7×7!))]
= 0.246918974
To estimate the error of this approximation, we can use the remainder term of the Maclaurin series:
|Rn(x)| ≤ M(x-a)ⁿ⁺¹/(n+1)!
M is an upper bound for the nth derivative of sin x, and a = 0 for the Maclaurin series.
The sin x Maclaurin series, we can use M = 1.
Using the fifth term of the series as the remainder term, we get:
|R5(0.27)| ≤ ((0.27)⁶)/(6!)
≈ 1.96 x 10⁻⁸
Since this is less than 10⁻⁸, we can conclude that our approximation is accurate to the desired level of precision.
[tex]\int_0^{27} (sin x)/x dx[/tex] ≈ 0.246918974 (rounded to nine decimal places).
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The value of the integral, to an error of magnitude less than 10^-8, is approximately 0.24618491.
To approximate the value of the integral with an error of magnitude less than 10^-8, we can use the Taylor series expansion of sin x/x about x=0. We have:
sin x/x = 1 - x^2/3! + x^4/5! - x^6/7! + ...
Integrating this series term by term from 0 to 0.27, we obtain:
integral 0.27 0 sin x/x dx ≈ 0.27 - 0.27^3/3!/3 + 0.27^5/5!/5 - 0.27^7/7!/7 + ...
We can use the alternating series estimation theorem to estimate the error in the approximation. The terms of the series decrease in magnitude and alternate in sign, so the error is less than the absolute value of the first neglected term, which is 0.27^9/9!/9. This is less than 10^-8, so we can stop here and round the approximation to nine decimal places:
integral 0.27 0 sin x/x dx ≈ 0.24618491
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What is called absolute number?
Any value inside an absolute value in a problem or equation indicates that the value is always positive.
What is called an absolute number?In a wide range of mathematical contexts, generalizations of the absolute value for real numbers occur. Complex numbers, quaternions, ordered rings, fields, and vector spaces are a few examples of objects for which an absolute value is also specified. In a variety of mathematical and physical situations, the absolute value is intimately related to the concepts of magnitude, distance, and norm. Without taking direction into account, absolute value describes how far away from zero a certain number is on the number line. A number can never have a negative absolute value. View some illustrations. 5 is 5 in its entirety.Any value inside an absolute value in a problem or equation indicates that the value is always positive. Absolute values are occasionally utilized with inequality problems and are frequently employed in situations concerning distance.To learn more about absolute number refer to:
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Rahul bought a weater and aved r20 when a dicount of 25% wa given. What wa the price of the weater before the dicount ?
The price of a sweater bought by Rahul before the discount is Rs. 80.
Let us assume the price of the sweater before the discount to be x.
Given that,
The money saved by Rahul will be equal to the discount given for the sweater.
Therefore, we can say that Rs. 20 will be equal to 25% of the price of the sweater before the discount.
The equation we would obtain from the language above would be,
25% of x = 20
Solving this equation further, we can see that,
=> 25% of x = 20
=> 25* 1/100 * x = 20
=> x/4 = 20
=> x = 80
As we assumed x to be the price of the sweater before the discount,
So, the price of a sweater bought by Rahul before the discount is Rs. 80.
Complete question: Rahul bought a sweater and saved Rs. 20 when a discount of 25% was given. What was the price of the sweater before the discount?
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Is 1 1 2 3 5 8 an arithmetic sequence?
No, 1 1 2 3 5 8 is not an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a constant to the previous term.
Thus, the difference between successive terms is the same. In the given sequence, the difference between terms 1 and 2 is 0, between terms 2 and 3 is 1, between terms 3 and 5 is 2, and between terms 5 and 8 is 3. Therefore, the difference between successive terms is not the same and hence, 1 1 2 3 5 8 is not an arithmetic sequence.
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Which explains why the commutative property of addition does not work for subtraction?
The commutative property of addition states that the order of the numbers being added does not affect the result. For example, the sum of 2 + 3 is the same as the sum of 3 + 2, which is 5.
The commutative property of addition does not work for subtraction because subtraction is not commutative. In subtraction, the order of the numbers being subtracted does affect the result. For example, the difference of 3 - 2 is 1, but the difference of 2 - 3 is -1.
The reason for this is that subtraction is the inverse operation of addition and the inverse operation of a commutative operation is not always commutative. In addition, subtraction is not commutative because when we subtract the smaller number from the larger number, we get a positive result but when we subtract the larger number from the smaller number we get a negative result.
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Is x³ an exponential function?
Yes, x³ is an exponential function. An exponential function is a function in which the variable (x) is in the exponent.
The general form of an exponential function is f(x) = a^x, where a is the base and x is the exponent. In the case of x³, the variable (x) is in the exponent. x³ represents x raised to the power of 3, where x is the base and 3 is the exponent.
Exponential functions are often used to model growth and decay in a variety of natural phenomena, such as population growth, radioactive decay, and compound interest. Understanding exponential functions is important for solving mathematical problems, especially in algebra, calculus, and applied fields such as physics and engineering.
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Can somebody help me on this one please I am so confused don’t know none of this stuff
rotation then reflection.
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Ruben measured how many minutes he meditated each night and the number of hours he slept. Plot the data in a scatter plot.
Meditation (minutes) 444 888 101010 444 181818 444
Sleep (hours) 4.54.54, point, 5 9.59.59, point, 5 8.58.58, point, 5 666 101010 777
The scatter plot for the given data (See full and correct details attached) is appended accordingly.
What is a scatter plot?A scatter plot is a collection of dots drawn on two axes, horizontal and vertical. Scatter plots are useful in statistics because they illustrate the amount, if any, of correlation between the values of observed quantities or events (called variables).
Scatter plots aid in the visual representation of correlations between economic factors such as employment and production, inflation and retail sales, and taxes and economic growth.
As you can see there is a slight positive correlation between the number of hours meditated and the numberr of hours slept.
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Please help me and explain
The value of x in the equation 8 (x - 3) + 7 = 2x (4 - 17) is x = 1/2.
What is an Equation?An equation is the statement of two expressions located on two sides connected with an equal to sign. The two sides of an equation is usually called as left hand side and right hand side
The given equation is,
8 (x - 3) + 7 = 2x (4 - 17)
Distributive property : p (q + r) = p q + p r
Applying the property here,
8x - (8 × 3) + 7 = 2x × (-13)
8x - 24 + 7= -26x
8x - 17 = -26x
8x + 26x = 17
34x = 17
x = 17/34
x = 1/2
The mistake of Lin happened in the second step where 4 - 17 = -13 but she wrote 13.
Hence the value of x is 1/2 not -1/2 as the solution of Lin.
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Contruct an iocele right-angled Triangle 'XYZ' in which angle Y=90° and two ide are 5cm each
Isosceles right angled triangle XYZ is constructed with the given measure of ∠Y =90° , YX = 5cm and YZ = 5cm each using steps of construction.
Constructed diagram is attached.
As given in the question,
Given measurement to construct a isosceles right angled triangle are:
Measure of ∠Y =90°
Measure of side length
YX = 5cm and YZ = 5cm
Constructed diagram is attached.
Step of construction :
Draw a ray.Take point Y on it and draw an arc of any measure.Take the measure more than half of the radius of the drawn arc.Take each end point as centre and draw an arc on the upper half plane of the ray.Draw ray perpendicular through point Y in the upper half plane.Now take measure of 5cm and point Y as centre and cut an arc on each of the ray.Name the point as 'X' and 'Z'.Join XZ .Isosceles right angled triangle is constructed with the given measurement.Therefore, construction of isosceles right angled triangle is done as per the above steps of construction.
The above question is incomplete, the complete question is :
Construct an isosceles right-angled Triangle 'XYZ' in which angle Y=90° and two side are 5cm each.
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Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar. Complete the table of inputs and outputs for the given function. g(x)=3−8x
The complete table is,
x 3/8 0 1 3
g(x) 0 3 -5 -21
What are linear equations?Each term in a linear equation has an exponent of 1, and when this algebraic equation is graphed, it always produces a straight line. It is called a "linear equation" for this reason.
Given equation g(x) = 3 − 8x
1. when g(x) = 0
0 = 3 - 8x
8x = 3
x = 3/8
2. when x = 0
g(x) = 3 − 8x
g(x) = 3 − 8(0)
g(x) = 3
3. when g(x) = -5
-5 = 3 - 8x
8x = 3 + 5
8x = 8
x = 1
4. when x = 3
g(x) = 3 − 8x
g(x) = 3 − 8(3)
g(x) = -21
Hence the missing values of x are 3/8 and 1,
missing values of g(x) are 3 and -21.
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Is rotation congruent or similar?
Answer:
a
Step-by-step explanation:
The graph of a linear function passes through the points (-10, -14) and (10, 11). Place (click and drag) a choice from each list into its corresponding b
describe the slope and y-intercept of the function.
Answer:
Slope = 1.25
Y-intercept = -1.5
Step-by-step explanation:
Slope:
[tex]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}=\frac{11-(-14)}{10-(-10)}=\frac{11+14}{10+10}=\frac{25}{20}=1.25[/tex]
Y-intercept:
[tex]y=mx+b\\\rightarrow y=1.25x+b\\\rightarrow 11=1.25(10)+b\\\rightarrow 11=12.5+b\\\rightarrow -1.5=b[/tex]
Final Equation:
[tex]y=1.25x-1.5[/tex]
How do you know if a domain is all real numbers?
A domain is all real numbers if it includes all of the values on the real number line (including negative and positive numbers, fractions, decimals, and irrational numbers).
1. A domain is the set of all possible values of an independent variable that produces a valid solution to a given equation.
2. To determine if a domain includes all real numbers, you must examine the set of values in the domain and see if it includes all the values on the real number line.
3. This includes negative and positive numbers, fractions, decimals, and irrational numbers.
4. If the domain includes all of these, then it is all real numbers.
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applied partial differential equations 7.7.2
The solution of the partial differential equation 7.7.2 is given by
u(x,t) = t^2 sin(2πx) + c,u(0.3,1.5) = 1.96 + c, where c is an arbitrary constant.
Partial differential equations are mathematical equations used to describe how a function changes as its inputs change. The equation 7.7.2 is a linear second order partial differential equation in two independent variables. It has the form of a generalized wave equation, in which a wave propagates through a medium with varying physical properties.
To calculate partial differential equations u(x,t) for given values of x and t:
Let x = 0.3 and t = 1.5
Substituting these values into the equation, we get:
u(0.3,1.5) = (1.5)^2 sin(2π(0.3)) + c
= 2.25 sin(0.6π) + c
= 2.25(0.87) + c
= 1.96 + c
Therefore, u(0.3,1.5) = 1.96 + c, where c is an arbitrary constant.
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Can someone help me out with this question fast!?!?!
Answer:
x=2
Step-by-step explanation:
2(4x-1)= -10(x-3)+4
(2×4x)-(1×2 )=(-10×x) -(3×-10) +4
8x -2 = -10x -(-30)+4
8x -2 = -10x +30+4
8x-2= -10x+ 34
8x+10x=34+2
18x=36 we do ÷18
x=2
Is the residual always 0?
The residual square sum may equal 0. Your model will fit your data more closely if the residual sum of squares is small; conversely, if the residual sum of squares is large, your model will match your data less closely. A value of 0 indicates that your model fits the data perfectly.
What Exactly Is Residual Value?
The estimated value of an asset that is fixed at the end of its useful life or lease term is its residual value, which is often referred to as salvage value. In lease agreements, the residual value is one of the lessor's main methodologies for calculating the lessee's periodic lease payments.
A good residual value is what?
Check out the Car Lease Ratings section of our Lease Kit for a complete list of high-residual vehicle brands and models.
A car is probably not a good lease deal if the lease-end residual value is less than 50% of the MSRP (for a 36-month lease). 55% to 65% of MSRP would be a nice residual.
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drew was shopping on amazon on black Friday. he purchased 5 DVDs all at the same price. the tax on his purchase was 6 bring the total cost to 61. what was the price of each DVD?
Based on the information provided, the price of each DVD is $11.
How to calculate the price of the DVD?To calculate the price of the DVD, two main variables must be considered:
Price of the DVD Total of units boughtTaxesWhat is the price of each DVD?Tax is a compulsory amount that is levied on goods and services by the government. Tax serves as a form of transfer of wealth from the private individuals to the government.
Taxes increase the cost of a good or service.
Now, let's calculate the cost of each DVD:
The price of each DVD = (total cost - tax) / number of DVDs bought
The price of each DVD = (61 - 6) / 5
The price of each DVD = 55 / 5
The price of each DVD = $11
Based on this, the price of each DVD was $11, you can double-check this below
$11 x 5 = $55 (prices for the DVDs) + 6 = $61 (total price)
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Evaluate the expression use calculator if needed
Answer:
780.
Step-by-step explanation:
I cant explain Its just the answer.
a right triangle is formed by fbe y-axis, the y = 2x + 4, and another line. if the legs of the right triangle intersect at (2, 8), what is the equation of the line other line of the triangle?
Replace -0.5 with the fraction -1/2 if needed.
======================================================
Explanation:
The legs of any right triangle are always perpendicular. They meet up to form a right angle, aka 90 degree angle.
The line y = 2x+4 goes through (2,8). This can be confirmed with a graph. Or note how plugging x = 2 leads to y = 8.
Therefore, y = 2x+4 forms one of the legs of this triangle.
The slope here is 2 = 2/1; the negative reciprocal is -1/2 = -0.5, which is the slope of the perpendicular line. We want this line to go through (2,8)
So,
y = mx+b
8 = -0.5*2+b
8 = -1+b
8+1 = b
9 = b
b = 9
The equation of the perpendicular line is y = -0.5x+9
Replace -0.5 with the fraction -1/2 if needed.
Check out the graph below.
What is the exponential of 10?
The term "exponential of 10" means the numbers in which the base is 10 and the exponent is an integer.
The Exponential of 10 or the "powers of 10" means when the number 10 is multiplied a certain number of times, then the product is expressed using exponents.
These numbers which are written as the exponents are exponents of 10.
For example : 10² , 10³ represents different exponents of 10 .
In general form of the exponent of 10 or power of 10 is represented as 10ⁿ .
For Example : 10 to the exponent of 5 can be written as 10⁵, and
it's value is calculates as :
⇒ [tex]10^{5}=10\times\ 10 \times 10\times 10\times 10[/tex]
⇒[tex]10^{5}= 100000[/tex] .
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What is f(-3) if f(X) =1/3x
Answer: 1/3x
Step-by-step explanation:
Hey there!
f(x) = 1/3x
f(x) = 1/3(-3)
f(x) = 1/3(-3/1)
f(x) = 1(-3)/3(1)
f(x) = -3/3
f(x) = -1
Therefore, your answer should be:
f(x) = -1
Good luck on your assignment & enjoy your day!
~Amphitrite1040:)
K
Find the measures of an interior angle and an exterior angle of a regular 15-gon.
The measure of an interior angle is
The measure of each interior angle of the polygon is 156°.
What is the interior angle of polygon?In a regular polygon, all interior angles are equal. The interior angle of a polygon = sum of interior angles and the number of sides is the formula for determining the size of an interior angle. A polygon's exterior angles add up to 360°.
Given that the number of sides of the polygon is 15. The sum of the interior angles of the polygon is calculated by the formula:-
Sum = ( n - 2 ) x 180
Sum = ( 15 - 2 ) x 180
Sum = 2340
The measure of each angle is calculated as:-
Angle = 2340 / 15
Angle = 156°
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Graph the image of A(7, −6)
after a reflection in y=x
followed by a reflection in x=−3.
The image of A After a reflection in y=x followed by a reflection in x=−3. is (0. 7).
What is a reflection?There are two ways of reflection.
Along x-axis:
(x, y) – (x, -y)
Along y-axis:
(x, y) - (-x, y)
We have,
A = (7, -6)
Reflection rules over y = x.
(x, y) → (y, x)
So,
(7, -6) = (-6, 7)
Now,
Reflection in x = -3
Reflection rules over the y-axis.
(x, y) → (-x, y)
So,
x = -3
There are 3 units between -3 and -6.
So,
3 + 3 = 6 units will be added during reflection over y = -3
i.e
3 units on the left side of y = -3
3 units on the right side of y = -3
So,
(-6, 7) = (-6 + 6, 7) = (0, 7)
Thus,
The image of A is (0. 7).
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The reflection of image of point in y = x is (-6, 7),
and reflection in x = −3 is (-13, -6).
What are transformations?The transformation, or f: X X, is the name given to a function, f, that maps to itself. After the transformation, the pre-image X becomes the picture X. Any operation, or a combination of operations, such as translation, rotation, reflection, and dilation, can be used in this transformation.
Given a point A(7, −6)
A. a reflection in y = x
Flipping is the term for reflection. A reflection is the shape's mirror image. The line of reflection is the path taken by an image when it reflects.
reflection in y = x
x changed to y and y changed to x
x = -6 and y = 7
coordinates are A'(-6, 7)
B. a reflection in x = −3.
let the coordinates be (a, b)
reflection in x = k
image coordinates will be A" (2k - a, b)
here a = 7, k = -3, and b = -6
2k - a = 2(-3) - 7
2k - a = -13
image coordinates will be A"(-13, -6)
Hence the coordinates are A'(-6, 7),
and reflection in x = -3,
coordinates aere A"(-13, -6).
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How do you find the value of n in exponents?
We can find the value of n in exponents through various rules based on the given equation.
If x is any real number and n is a positive integer, then [tex]x^n[/tex] is equivalent to repeated multiplication.
We can also write [tex]x^n[/tex] in the following way:
[tex]x^n[/tex] = x * x * x * x * x *...... * x (n times)
This can be referred to as "n raised to the power of x," "n to the power of n," or just "n." In this case, n is the exponent or power, and x is the base.
We may infer a few fundamental guidelines for exponentiation from this concept. There are various methods of finding the exponent based on various rules, namely,
Product ruleQuotient rulePower of a powerPower of a productPower of onePower of ZeroPower of negative oneChange the sign of exponentsFractional exponentsRead more about exponents from:
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To find the value of n in exponents, you can use the property that a^n * a^m = a^(n+m)
For example, if you have the expression (2^3)^n, you can use the property above to find the value of n by dividing both sides by 2^3, so (2^3)^n / 2^3 = 1^n. Now you can see that n = 1.
Another way to find the value of n in exponents is to use logarithms. The logarithm base "a" of a^n is n. Therefore, if you know the value of a^n, you can find the value of n by taking the logarithm base "a" of a^n.
For example, if a^n = 8 and a=2, then n = log2(8) = 3.
In summary, there are a couple of ways to find the value of n in exponents, but the most common way is to use the property of exponents or logarithms.
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Find the values of x and y.
y°
40°
x°
(x,, y,) is a point on the line in the point-slope formula, and m is the slope.
How should slope m be written?When you know the slope of the line to be investigated and the point supplied is also the y intercept, you may utilize the slope intercept formula y = mx + b. (0, b). The y value of the y intercept point is represented by b in the formula.
The equation of a line can be written in several different formats:
Standard form
Ax+ By = C
A, B, and C must be real numbers and A and B cannot both be zero.
Slope-intercept form
Y = MX + b
M is the slope and b is the y-intercept.
Point-slope form
y-y₁ = m(x-x1)
In the point-slope formula, (x,, y,) is a point on the line and m is the slope.
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An unusually wet spring has caused the size of a mosquito population in some city to increase by 7% each day. If an estimated 250,000 mosquitoes are in the city on May 11, find how many mosquitoes will inhabit the city on May 23. Use y250,000(1.07)x where x is number of days since May 11
The number of mosquitoes that will inhabit the city on May 23, given the increase in population, is 563, 048 mosquitos
How to find the number of mosquitoes ?The formula for finding the number of mosquitoes after May 11, is given as:
y = 250,000 ( 1. 07 ) ˣ
Given that x is the number of days since May 11, the number of days till May 23 is :
= 23 - 11
= 12 days
The number of mosquitoes is therefore:
= 250, 000 x ( 1. 07 ) ¹²
= 250, 000 x 2.252191588960823337718801
= 563, 048 mosquitos
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Twelve less the product of two numbers, ‘a’ and ‘b’. what does it mean.
Answer: (a * b) - 12
Step-by-step explanation:
We will pull details for the sentence given to write an expression representing this situation.
In this case, these words mathematically mean:
less ➜ subtraction
product ➜ multiplication
Using this information, we will write our expression.
(a * b) - 12
Jon has to choose which variable to solve for in order to be able to do the problem below in the most efficient manner.
6 x + 3 y = 27. 5 x + 2 y = 21.
Which variable should he choose so that he can use substitution to solve the system?
Jon should solve for y in the first equation because the coefficients can be reduced by a common factor to eliminate the coefficient for y.
Jon should solve for x in the first equation because the coefficients can be reduced by a common factor to eliminate the coefficient for x.
Jon should solve for y in the second equation because the coefficients can be reduced by a common factor to eliminate the coefficient for y.
Jon should solve for x in the second equation because the coefficients can be reduced by a common factor to eliminate the coefficient for x.
Jon should solve for y in the first equation because the coefficients can be reduced by a common factor to eliminate the coefficient of y.
What are Linear Equations?Linear equations are equation involving one or more expressions including variables and constants and the variables are having no exponents or the exponent of the variable is 1.
Given linear equations are,
6x + 3y = 27
5x + 2y = 21
The coefficients of the first equation are multiples of 3.
Hence dividing throughout the first equation by the common factor 3, coefficient of y will be eliminated.
6x/3 + 3y/3 = 27/3
2x + y = 9
y = 9 - 2x
Solving by substitution, the second equation becomes,
5x + 2(9 - 2x) = 21
5x + 18 - 4x = 21
x = 3
y = 9 - 2x = 3
Hence Jon should could solve the problem by eliminating the coefficient of y in the first equation by a common factor 3.
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how to know if a piecewise function is a function
Answer:
Step-by-step explanation:
i need help this is due at 12:10
Answer:
I believe the answer is 180°