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First, we can factor out a 21​ from all the terms:

(a + b)/2 - a/2 - b/2 = \frac{1}{2} (a + b - a - b)

Then, we can combine the a and b terms:

\frac{1}{2} (a + b - a - b) = \frac{1}{2} \cdot 0 = \boxed{0}

Cost of 1 taco: x Cost of 1 burrito: y

Equation 1: 3x+2y=7.40 Equation 2: 4x+y= 6.45

To solve for x and y, we can use elimination.

Multiply the top equation by -1:

-3x-2y=-7.40

Add this equation to the bottom equation:

x-y=-1

Divide both sides by 1 to solve for x:

x=1

Substitute 1 for x in the top equation:

3(1)+2y=7.40

3+2y=7.40

2y=4.40

y=2.20

Therefore: Cost of 1 taco: 1 Cost of 1 burrito: 2.20

To calculate the cost of 3 tacos and 4 burritos:

(1)(3)+(2.2)(4)=11.80

Therefore, the cost of 3 tacos and 4 burritos is $11.80.

To solve the equation 3z+4izˉ=1−8i, we can use the following steps:

Rewrite the equation in terms of the real and imaginary parts of z. Let z=x+yi, where x is the real part of z and y is the imaginary part of z. Then, the conjugate of z is zˉ=x−yi. Substituting these into the original equation, we get:

3(x + yi) + 4i(x - yi) = 1 - 8i

Expanding the parentheses and multiplying through by i, we get:

(3x + 4y) + (3y - 4x)i = 1 - 8i

Separate the real and imaginary parts of the equation. Equating the real and imaginary parts on both sides of the equation, we get the following system of equations:

3x + 4y = 1 \\ 3y - 4x = -8

Solve the system of equations for x and y. We can solve the system of equations for x and y using any method we like. For example, we can use elimination to solve for x and then back-substitution to solve for y.

Eliminating x from the system of equations, we get:

12y = -7

Dividing both sides by 12, we get:

y = -\frac{7}{12}

Substituting this value of y into the first equation, we get:

3x + 4\left(-\frac{7}{12}\right) = 1

Solving for x, we get:

x = \frac{1}{3}

Therefore, the complex number z satisfying the equation 3z+4izˉ=1−8i is z = 1/3 - 7/12*i.

Here is the answer to part (f);

Let's call (a,b,c) a smiley face if b is less than a and b is less than c, because when we plot the graph, we get a happy face!  And if b is greater than a and b is greater than c, that's a frowny face, because we get a frowny face when we turn a smiely face up-side-down.

There are other kinds of faces like smirks (like a is less than b and b is less than c) and neutral faces (like when a is equal to b and b is equal to c).  If the face is neutral, then a equals b and b equals c, so when we choose a, b, and c are also chosen, and there are n choices for a, so there are n neutral faces.

Now we count the number of smirks.  There are n ways to choose a, and there are n - 1 ways to choose b.  We also multiply by 3, because the value that we chose for a could have also been the value of b, or the value of c.  So there are 3n(n - 1) smirks.

Now we count the number of smiley faces.  There are n ways to choose a, then n - 1 ways to choose b, then n - 2 ways to choose c.  So there are n(n - 1)(n - 2) = 3C(n,3) smiley faces.  By symmetry, there are 3C(n,3) frowny faces.

Therefore, the total number of faces is n^3 = n + 3n(n - 1) + 6C(n,3).

To find the probability that intervals I and J both have length greater than 3/4, we can use the following steps:

Calculate the probability that interval I has length greater than 3/4. Since x1​ and x2​ are chosen independently at random in the interval [0,1], the length of interval I is uniformly distributed between 0 and 1. Therefore, the probability that interval I has length greater than 3/4 is 1−01−3/4​=1/4​.

Calculate the probability that interval J has length greater than 3/4. Using the same logic as in step 1, we can calculate that the probability that interval J has length greater than 3/4 is also  1/4.

Multiply the two probabilities to find the probability that both intervals I and J have length greater than 3/4. Since x1​, x2​, x3​, and x4​ are chosen independently at random, the events that interval I has length greater than 3/4 and interval J has length greater than 3/4 are independent. Therefore, the probability that both intervals I and J have length greater than 3/4 is the product of the two probabilities we calculated in steps 1 and 2, which is 1/4 * 1/4 = 1/16.