Solve the following differential equation by Laplace transforms. The function is subject to the given conditions. y" - 9y = 6 e 21, y(0) = 7, y'(0) = 0
Answers
The solution to the given differential equation\(y(t) = -1/4 e^(21t) + 5/8 cos(3t) - 7/8 sin(3t)\)
How we solve the following differential equation by Laplace transforms?To solve the differential equation y'' - 9y = 6e^21 using Laplace transforms, we follow these steps:
Take the Laplace transform of both sides of the equation using the differential property of Laplace transform and the fact the\(t L(e^at) = 1 / (s - a)\):\(Ly'' - 9 Ly = 6 Le^(21t)\)
\(s^2 Y(s) - s y(0) - y'(0) - 9 Y(s) = 6 / (s - 21)\)
\(s^2 Y(s) - 7 - 9 Y(s) = 6 / (s - 21)\)
Solve for Y(s) by manipulating the equation:\(s^2 Y(s) - 7 - 9 Y(s) = 6 / (s - 21)\)
\(Y(s) = 6 / [(s - 21)(s^2 - 9)] + 7 / (s^2 - 9)\)
Use partial fraction decomposition to write Y(s) in a form that can be inverted using the Laplace transform table:\(Y(s) = A / (s-21) + (Bs + C) / (s^2 - 9)\)
Multiplying both sides by the denominator of Y(s) gives:
\(6 = A(s^2 - 9) + (Bs + C)(s - 21) + 7(s - 21)\)
Setting s = 21, 3, -3, and solving for the coefficients A, B, and C, we get:
A = -1/4, B = 5/8, C = -7/8
Therefore, Y(s) can be written as:
\(Y(s) = -1/(4(s-21)) + (5s - 7)/(8(s^2-9))\)
Use the table of Laplace transforms to find the inverse Laplace transform of Y(s) to obtain the solution y(t):\(y(t) = L^-1Y(s) = -1/4 e^(21t) + 5/8 cos(3t) - 7/8 sin(3t)\)
Therefore, the solution to the given differential equation with initial conditions y(0) = 7 and y'(0) = 0 is:
\(y(t) = -1/4 e^(21t) + 5/8 cos(3t) - 7/8 sin(3t)\)
This is the final answer.
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Related Questions
Ira walked 20 miles last week and 22 miles this week. Raj walked 1/3 the distance that Ira walked. Write an expression for the distance Raj walked
Answers
Answer:
Multiply your pace by your distance; If your pace is 9.5 minutes per mile and you ran 3 miles: 9.5 min per mi × 3 mi = 28.5 minutes = 28 minutes, 30 seconds How to calculate running speed. Divide your run distance by your run time; If you ran 2.5 miles and you ran for 20 minutes
Step-by-step explanation:
Customers arrive at the Wendy's drive-thru at random, at an average rate of 17 per hour. During a given hour, what is the probability that 10 or fewer customers will arrive at the drive-thru? 0.049 0.588 0.026 0.023
Answers
The probability that 10 or fewer customers will arrive at the Wendy's drive-thru during a given hour is approximately 0.049 or 4.9%.
To solve this problem, we will use the Poisson distribution formula. The Poisson distribution is used to model the number of events (in this case, customers arriving) in a fixed interval of time (one hour) given the average rate (17 customers per hour). The formula is:
P(X=k) = (e^(-λ) * λ^k) / k!
Where:
- P(X=k) is the probability of having k events (customers arriving)
- e is the base of the natural logarithm (approximately 2.71828)
- λ (lambda) is the average rate of events (17 customers per hour)
- k is the number of events we are interested in (10 customers)
- k! is the factorial of k (the product of all positive integers from 1 to k)
To find the probability that 10 or fewer customers will arrive, we need to calculate the sum of the probabilities for 0 to 10 customers:
P(X≤10) = P(X=0) + P(X=1) + ... + P(X=10)
Now, we can use the Poisson distribution formula for each value of k (0 to 10) and sum the probabilities:
P(X≤10) = Σ [(e^(-17) * 17^k) / k!] for k = 0 to 10
After calculating each term and summing them up, we get:
P(X≤10) ≈ 0.049
So, the probability that 10 or fewer customers will arrive at the Wendy's drive-thru during a given hour is approximately 0.049 or 4.9%.
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a(n) ____ is a transformation in which the size or the shape of a geometric figure is changed.
Answers
Dilations involve scaling the figure uniformly along a given center and factor. This process results in an enlarged or reduced version of the original figure, while maintaining the same proportions and shape.
A dilation is a type of geometric transformation that alters the size or shape of a figure while preserving its proportions. It involves scaling the figure uniformly in all directions from a specific center of dilation. The scaling factor determines whether the figure will be enlarged or reduced.
When dilating a figure, each point is moved along a line that passes through the center of dilation. The distance between the original point and the center is multiplied by the scaling factor to determine the new position of the point. If the scaling factor is greater than 1, the figure will be enlarged, while a scaling factor between 0 and 1 will result in a reduction.
The center of dilation can be any point on the plane, and it serves as the reference point from which the scaling occurs. If the center of dilation is outside the figure, the shape will change along with the size. However, if the center is inside the figure, only the size will be affected, and the shape will remain the same.
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Which triangles are similar?
20°
100°
100°
60°
А
B
20°
60°
с
O A. Triangles A and B are similar to each other.
B. Triangles A, B, and C are similar to each other.
C. Triangles A and C are similar to each other.
D. Triangles B and C are similar to each other.
Answers
Answer:
B because when at B angels are the same qnd A and c are also same
How much of a 8% salt solution must combined with a 20% salt solution to make 4 gallons of a 15% salt solution? Step 1 of 3 : Identify the variable. Let x= number of gallons of salt solution to add. A
Answers
In response to the supplied query, we may state that Consequently, to equation create 4 gallons of a 15% salt solution, we must combine 1.67 gallons of the 8% salt solution with 2.33 gallons of the 20% salt solution.
What is equation?A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.
We know that 0.08x gallons of salt, or 8% of x gallons, is in the 8% salt solution.
Since we are adding x gallons of the 8% salt solution to a total of 4 gallons of the final solution, we know that 20% of (4 - x) gallons is salt, or 0.2(4 - x) gallons of salt, for the 20% salt solution.
0.08x + 0.2(4 - x) = 0.15(4) (4)
We can condense things and find x:
0.08x + 0.8 - 0.2x = 0.6
-0.12x = -0.2
x = 1.67
Consequently, to create 4 gallons of a 15% salt solution, we must combine 1.67 gallons of the 8% salt solution with 2.33 gallons of the 20% salt solution.
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ezra determined that the graph shown below is vertically compressed by a factor of 1/3 from the graph of y=|x| do you agree or disagree? why?
Answers
Answer:
Step-by-step explanation:
no graph was shown
Which of the following is a factor of 7x2 + 42x - 49
x2 + 49
x+7
X-7
x+ 1
Answers
Answer:
\(x + 7\)
is the answer because 7×2+42x-49
Recall that the cigarette industry requires that models in cigarette ads must appear to be at least 25 years old. Also recall that a sample of 50 people is randomly selected at a shopping mall. Each person in the sample is shown a "typical cigarette ad" and is asked to estimate the age of the model in the ad. The p -value for testing H0 versus Ha can be calculated to be 0.0038.
Determine whether H0 would be rejected at each of ? = .10, ? = .05, ? = .01, and ? = .001. (Round your answer to 4 decimal places.)
p-value Reject H0 at ? =
Answers
The p-value for testing H0 (null hypothesis) versus Ha (alternative hypothesis) is 0.0038. We need to determine whether H0 would be rejected at different significance levels: α = 0.10, α = 0.05, α = 0.01, and α = 0.001.
To make the decision, we compare the p-value with the chosen significance level. If the p-value is less than or equal to the significance level, we reject H0. Otherwise, we fail to reject H0.
At α = 0.10: Since the p-value (0.0038) is less than 0.10, we reject H0.
At α = 0.05: Since the p-value (0.0038) is less than 0.05, we reject H0.
At α = 0.01: Since the p-value (0.0038) is greater than 0.01, we fail to reject H0.
At α = 0.001: Since the p-value (0.0038) is greater than 0.001, we fail to reject H0.
In summary:
H0 would be rejected at α = 0.10 and α = 0.05.
H0 would not be rejected at α = 0.01 and α = 0.001.
Please note that the decision to reject or fail to reject the null hypothesis depends on the chosen significance level, and different levels of significance can lead to different conclusions.
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The equation N(t)=(650)/(1+49e^(-0.7t)) models the number of people in a town who have heard a rumor after t days. How many people started the rumor?
Answers
The initial number of people who started the rumor is 13.
Given, The equation N(t)=(650)/(1+49e^(-0.7t)) models the number of people in a town who have heard a rumor after t days. To find, how many people started the rumor? We know that, As per the given equation, the number of people in a town who have heard a rumor after t days is given by, N(t)=(650)/(1+49e^(-0.7t))
We need to find the initial number of people who started the rumor. The given equation is in the form of N(t)= a / (1 + be^(-ct)). Let, initial number of people who started the rumor is a. Now, at t=0, N(0) = a / (1 + b) According to the given equation, N(t)=(650)/(1+49e^(-0.7t))
Therefore, at t=0, N(0) = (650) / (1 + 49e^(-0.7*0))= (650) / (1 + 49e^0)= (650) / (1 + 49)= (650) / 50= 13 Therefore, the initial number of people who started the rumor is 13. Hence, the correct option is (c) 13.
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One of the legs of a right triangle measures 5 cm and the other leg measures 8 cm.
Find the measure of the hypotenuse. If necessary, round to the nearest tenth.
Answers
Answer:
9.4cm
Step-by-step explanation:
by pythagoras theorem,
a^2 + b^2 = c^2
5^2 + 8^2 = c^2
c^2 = 89
c = √89 = 9.433981 ≈ 9.4
Hank is going on a trip. He must pack two complete outfits consisting of one shirt, one pair of pants, and one pair of shoes. In his closet, Hank has 4 shirts, 2 pairs of pants, and 2 pairs of shoes.
How many different combinations of two complete outfits can Hank pack from his closet? (Assume that the outfits have no common elements.)
A. 48
B. 13
C. 16
D. 256
Answers
Answer:
C. Is i think the answer( im not perfect dont judge)
Step-by-step explanation:
안녕하세요 나쁜 소식은 나를 위해하지 아니 아니 나를 위해 감사f요 나쁜 소식 yiu 나를 위해 하지 감사
Answer:
A. 48
Step-by-step explanation:
4x4=16
2x2=4
2x2=4
16+4+4=24
24x2=48
Therefore I think it is 48.
DISCLAIMER
(I could be wrong so this is not with 100% satisfaction)
HELP AND WILL GIVE BRAINLIEST!!! Solve for x
Answers
Answer:
22
Step-by-step explanation:
yevdywjebecgeh3bebebv3hruejjd÷×+jfur
Question 7 (10 points) Find the slopes of the lines and determine if they are perpendicular or not. y -2 = '/2 (x+4) y - 3 = -2(x + 5) m = m = Perpendicular? (yes or no)
Answers
m= 1/2x m=-2x
they are perpendicular because the slopes are the negative reciprocal of each other.
Lupe's family traveled 5 7 of the distance to her aunt’s house on Saturday. They traveled 1 2 of the remaining distance on Sunday. What fraction of the total distance to her aunt’s house was traveled on Sunday?
Answers
Answer:
3/14
Step-by-step explanation:
They traveled 5/7 of the distance on Saturday and 1/2 of the remaining
distance, 3/7, on Sunday so..
(1/2)(3/7) = 3/14
Hope this helps :)
Answer:Lupe's family traveled 5/7 of the total distance to her aunt's house on Saturday, so the remaining distance was 1 - (5/7) = 2/7.
On Sunday, they traveled 1/2 of the remaining distance, which was 2/7, so the total distance traveled on Sunday was 1/2 * 2/7 = 2/14 of the total distance.
Step-by-step explanation:
Lupe's family traveled 5/7 of the total distance to her aunt's house on Saturday, so the remaining distance was 1 - (5/7) = 2/7.
On Sunday, they traveled 1/2 of the remaining distance, which was 2/7, so the total distance traveled on Sunday was 1/2 * 2/7 = 2/14 of the total distance. :D
Choose all numbers that divisible by 3 and 5
90469
57857
13935
90806
94555
10093
Answers
Answer:
Divisible by 5
13935
94555
Divisible by 3
13935
Divisible by both 5 and 3
13935
Step-by-step explanation;
When you are dividing by 5, you know if it's divisible by 5 if the last digit is a 5 or a 0.
When dividing by 3, you know if it's divisible by 3 if the sum of the digits is divisible by three
Here is an example:
13935
1+3+9+3+5 = 21
21 is divisible by 3 so 13935 is divisible by 3.
Evaluate SIS 2 1 dV, where E lies between the spheres x2 + y2 + z2 25 and x2 + y2 + z2 = 49 in the first octant. x² + y² + z² = =
Answers
The value of the integral is 2π/3.
To evaluate the integral SIS 2 1 dV, where E lies between the spheres x² + y² + z² = 25 and x² + y² + z² = 49 in the first octant:
1. We first set up the integral in spherical coordinates. The volume element in spherical coordinates is given by dV = ρ²sin(φ)dρdθdφ, where ρ represents the radial distance, φ represents the polar angle, and θ represents the azimuthal angle.
2. Since we are interested in the first octant, the ranges of the variables are:
- ρ: from 1 to √25 = 5
- θ: from 0 to π/2
- φ: from 0 to π/2
3. The integral becomes:
∫∫∫E dV = ∫₀^(π/2) ∫₀^(π/2) ∫₁⁵ ρ²sin(φ)dρdθdφ
4. Integrating with respect to ρ, θ, and φ in the given ranges, we obtain:
∫∫∫E dV = ∫₀^(π/2) ∫₀^(π/2) ∫₁⁵ ρ²sin(φ)dρdθdφ = 2π/3
Therefore, the value of the integral SIS 2 1 dV, where E lies between the spheres x² + y² + z² = 25 and x² + y² + z² = 49 in the first octant, is 2π/3.
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please sombody help mee
Answers
Answer:
A) y=1500+150x
Step-by-step explanation:
The y-intercept (0,1500) matches up with the first equation.
Each time x increases by one, y increases by 150. Example- (1, 650) and (2, 1800)
Please give brainliest if I helped! :)
Answer:
y= 150 + 1500x
Step-by-step explanation:
the yintercept is 1500. each point on the y axis is 100 and each point on the y axis is 1
so i go up 3 which is 300 and to the right 2 points which is 2 to get to the next point.
300/2 is 150
Suppose you are interested in looking at the determinants of a ballplayer's salary, and use the following econometric model to do so: salary =β 0 +β 1 WAR+β 2 age+u where WAR= total number of wins above a replacement player age - age in years u= error term You take a sample of 120 individuals and collect data on each person's salary, WAR, and age. An unbiased, observable estimator of the variance of the error term (σ 2 ) is ∂ 2 =φ
Answers
The given econometric model is salary = β₀ + β₁WAR + β₂age + u where WAR represents the total number of wins above a replacement player and age is the age in years. Here, u denotes the error term, which cannot be measured directly.
A sample of 120 individuals is taken and data on each person's salary, WAR, and age are collected. ∂² = φ is an unbiased, observable estimator of the variance of the error term (σ²). which cannot be measured directly. A sample of 120 individuals is taken and data on each person's salary, WAR, and age are collected. ∂² = φ is an unbiased, observable estimator of the variance of the error term (σ²).
An econometric model is given below: Salary is a function of the player's WAR and age, as determined by the equation. The parameter β₀ represents the intercept. The slope of the salary curve with respect to WAR is represented by the parameter β₁. Similarly, the slope of the salary curve with respect to age is represented by the parameter β₂. Finally, the error term u captures the effect of all other determinants of salary not included in the model.
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A line is perpendicular to y = -1/5x + 1 and intersects the point negative (-5,1) what is the equation of this perpendicular line?
Answers
Answer: y = 5x + 26
Step-by-step explanation:
To find the equation of a line that is perpendicular to the given line y = -1/5x + 1 and passes through the point (-5, 1), we need to determine the slope of the perpendicular line. The given line has a slope of -1/5. Perpendicular lines have slopes that are negative reciprocals of each other. So, the slope of the perpendicular line will be the negative reciprocal of -1/5, which is 5/1 or simply 5. Now, we have the slope (m = 5) and a point (-5, 1) that the perpendicular line passes through.
We can use the point-slope form of a linear equation to find the equation of the line:
y - y1 = m(x - x1)
Substituting the values, we get:
y - 1 = 5(x - (-5))
Simplifying further:
y - 1 = 5(x + 5)
Expanding the brackets:
y - 1 = 5x + 25
Rearranging the equation to the slope-intercept form (y = mx + b):
y = 5x + 26
Therefore, the equation of the perpendicular line that passes through the point (-5, 1) is y = 5x + 26.
help plz plz will mark brainliest
Answers
Answer:
x = 52°
Acute angle = 64°
Step-by-step explanation:
Problem 1:
Since DN bisects <BDW, this means that m<BDN = m<NDW
Thus:
5v - 8 = 4v + 4
Collect like terms
5v - 4v = 8 + 4
v = 12
x = m<BDN = 5v - 8
x = 5v - 8
Plug in the value of v
x = 5(12) - 8
x = 52°
Problem 2:
First, find the value of y
(4y - 44) + (3y + 35) = 180 (linear pair)
4y - 44 + 3y + 35 = 180
7y - 9 = 180
7y = 180 + 9
7y = 189
Divide both sides by 7
y = 27
Acute angle = 4y - 44
Plug in the value of y
Acute angle = 4(27) - 44 = 64°
A function, f(x), has a domain of --10 ≤ 2 < 20 and range -40 < f(I) < -10.
f(1)=-13
f (-10) =-40
Select each statement that must be false about f(x).
1. f(0)= 0
2. f(1) = 13
3. f(-15) =- 20
4. f(-9) = 88
5. f(5)- 40
Answers
By using the given domain and range, we will see that the false statements are:
1. f(0)= 02. f(1) = 133. f(-15) =- 204. f(-9) = 88Which statement is false?
What we know about the function f(x) is:
The domain is: -10 ≤ x ≤ 20
The range is: -40 ≤ f(x) ≤ -10
Which statements are false:
1) f(0)= 0
This is false, because the maximum value of the range is -10, so f(0) can't be equal to zero.
2. f(1) = 13
Same reasoning than above, 13 is outside of the range.
3. f(-15) =- 20
The smallest value on the domain (possible inputs) is -10, so the input -15 is not in the domain.
4) f(-9) = 88
88 is outside the range of the function, so this is false.
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10340000000 in standard form
Answers
Answer:
1034 x 10⁷
Step-by-step explanation:
the seven just means to multiply by 10 seven times
Let me know if this helps.
11 of 1411 of 14 Items
Question
Brenda pours laundry detergent from a bottle into the cap and then into the washing machine.
Every time she fills the cap, the amount of detergent in the bottle changes by -0.26 L.
She fills the cap 314 times in all.
John has a different bottle of detergent that is the same size.
Every time he fills his cap, the amount of detergent in his bottle changes by -0.21 L.
He fills his cap 256 times in all.
Who has the greater total change in the amount of detergent in their bottle?
Answers
John has the greater total change in the amount of detergent in their bottle.
Given that, Brenda pours laundry detergent from a bottle into the cap and then into the washing machine.
What is the unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Every time she fills the cap, the amount of detergent in the bottle changes by -0.26 L. She fills the cap 314 times in all.
Now, the volume of bottle =314×0.26
= 81.64
≈ 82 L
Every time he fills his cap, the amount of detergent in his bottle changes by -0.21 L. He fills his cap 256 times in all.
Here, the volume of bottle =256×0.21
= 53.76
≈ 54 L
Hence, John has the greater total change in the amount of detergent in their bottle.
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math i need to know
Answers
Answer:
Try first and last
Step-by-step explanation:
what is the primitive function of 4x
Answers
Hi! To find the primitive function of 4x, you need to determine the antiderivative of the given function.
The primitive function of 4x is:
∫(4x) dx = 4∫(x) dx = 4(x^2/2) + C
So the primitive function of 4x is \(2x^2 + C\), where C is the constant of integration.
How can representing sentences and phrases with symbols help you determine whether to apply the Law of Detachment or the Law of Syllogism
Answers
In the Law of Syllogism , Long sentences and phrases can occasionally be perplexing and written imprecisely at first glance.
What is the purpose of a syllogism?
The most typical format for organizing premises into a strong argument is a syllogism. In a syllogism, the conclusion derives from the truth of two or more premises, which is a type of deductive reasoning.Long sentences and phrases can occasionally be perplexing and written imprecisely at first glance.
Therefore, using symbols to denote various statements is quite helpful.
because the number of various statements we have and the relationship between them will then be much more obvious.
To put it another way, it would be simpler to decide which logical law to apply to a certain issue.
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2. Carl pays $79.99 for a Disney Plus subscription every year. Because Carl
is such a Disney fanatic, he also purchases the extra $30 per movie to gain
early streaming access. If Carl only has $450.00 a year budgeted for his
Disney Plus account, how many movies can Carl purchase early access for?
Be sure to set up an equation, solve the equation, and tell which algebraic
properties you used to help you solve. (3 points)
Answers
Answer:
12.33
Step-by-step explanation:]
450-80=370/30=12.33
61-2xl-9+41-2y + 6)..
Simplify
Answers
Answer:
-2xl -2y + 99
Step-by-step explanation:
Answer:
\(-2xl -2y+99\)
Step-by-step explanation:
STEP 1: Subtract 9 from 61.
\(-2xl+52+41-2y+6\)
STEP 2: Add 52 and 41.
\(-2xl+93-2y+6\)
STEP 3: Add 93 and 6.
\(-2xl-2y+99\)
. The sum of the probability that an event will happen and the probability that the event will not happen is 1. Consider the experiment of selecting 1 card at random from a complete deck of 52 cards. Compute the probabilities of the events: a. Not drawing an ace b. Not drawing a face card c. Drawing a club, heart, or diamond d. Not drawing a black face card 9. Compute the probability and odds of selecting each of the cards at random from a complete deck of 52 cards. a. a club b. an 8 or a 9 c. a red card d. a spade or a heart
Answers
The amount in the account after 6 years will be approximately $3,331.46.
To calculate the future value of the account after 6 years, we can use the formula for compound interest:
\(A = P \times (1 + r)^n\)
Where:
A is the future value of the account
P is the annual deposit amount
r is the interest rate per compounding period
n is the number of compounding periods
In this case, Derek is depositing $2,705.00 per year, the interest rate is 4.00%, and the compounding period is annually. The number of compounding periods is also 6, since Derek will deposit for 6 years.
Substituting the given values into the formula, we have
\(A = 2,705 \times (1 + 0.04)^6\)
Calculating the expression inside the parentheses first:
\((1 + 0.04)^6 \approx 1.2490024\)
Now, we can calculate the future value of the account:
\(A = 2,705 \times 1.2490024 \approx $3,331.46\)
Therefore, the amount in the account after 6 years will be approximately $3,331.46.
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What's the a answer to this
Answers
Answer:
if you take 180-124 you would get 56 then you would take 56 divided by 2x which would give you 28
Step-by-step explanation:
Answer:
Step-by-step explanation:
4) ∠ELD + m∠ELG = 180 linear pair
5) 2x + 124 = 180 Substitution
6) 2x +124 -124 = 180 -124 Subtraction property
2x = 56
7) 2x/2 = 56/2 Division property
x = 28