The correct answer is 5 ¹/₃. The problem is solved by substitution. The first step is to substitute x – 4 into the second equation. Then we have 2x + 4(x – 4) = 16. Next step 2x + 4x – 16 = 16. Then 6x = 32. We then divide 32 by 6 for X and get 5 ¹/₃. 

The question asks us to find the value of –x – 5z, which doesn't include any y terms. Therefore, we need to eliminate y terms from our equations. One way to do this is to solve for y in the second equation and substitute this value into the first one. 

y – 2z = 6

Add 2z to both sides.

y = 6 + 2z

Now, we take 6 + 2z and substitute this in for y in the first equation.

x + 2(6 + 2z) + z = 5

Distribute. 

x + 12 + 4z + z = 5

x + 5z + 12 = 5

Subtract 12 from both sides.

x + 5z = –7

The original question asks for the value of –x – 5z, which is equal to –1(x + 5z). Let's multiply both sides of the equation x + 5z = –7 by negative one.

–1(x + 5z) = –7

–x – 5z = 7

The answer is 7. 

Let t and s represent Tom's and Susan's current ages, respectively.

We are told that six years ago, Tom's age is twice as large as Susan's. We could represent Tom's age six years ago as t – 6, and we could represent Susan's as s – 6. Because t – 6 is twice as large as s – 6, we could write the following equation:

t – 6 = 2(s – 6)

Additionally, we are told that thirteen years ago, Tom was three times as old as Susan. Thirteen years ago, Tom's age would be t – 13, and Susan's would be s – 13. We can then write the following equation: 

t – 13 = 3(s – 13)

We now have two equations and two unknowns. In order to solve this system of equations, we could solve for t in the first equation and substitute this value into the second equation.

t – 6 = 2(s – 6)

Distribute.

t – 6 = 2s – 12

Add six to both sides.

t = 2s – 6

Next, we will substitute 2s - 6 into the second equation.

(2s – 6) – 13 = 3(s – 13)

Distribute.

2s – 6 – 13 = 3s – 39

Combine constants.

2s – 19 = 3s – 39

Subtract 2s from both sides.

–19 = s – 39

Add 39 to both sides.

s = 20

Since t = 2s – 6, t = 2(20) – 6 = 34

This means that Tom is currently 34, and Susan is currently 20. The question asks us how many years older Tom is than Susan, which is 34 – 20 = 14 years.

The answer is 14.

We are asked to find (x – y)². Let's expand (x – y)² using the FOIL method.

(x – y)² = (x – y)(x – y) = x(x) – x(y) – y(x) + y(y) = x² – 2xy + y²

In other words, we need to find the value of x² – 2xy + y². We can use the given information to find the values of x² + y² and –2xy. Then, if we combine the values of x² + y² and –2xy, we will have the value of x² – 2xy +y², which is equal to (x – y)².

The problem states that (x² + y²)^(1/2) = 4. If we were to square both sides of the equation, we can find the value of x² + y². 

((x² + y²)^(1/2))² = 4² = 16

If we use the property of exponents that states that (a^b)^c = a^bc, then ((x² + y²)^(1/2))² becomes (x² + y²)^2(1/2) = x² + y².

Thus, x² + y² = 16.

The second piece of given information states that 4xy = 4. If we divide both sides of the equation by –2, we will have –2xy on the left side.

4xy = 4

Divide both sides by –2.

–2xy = –2

Finally, we will add x² + y²  + –2xy.

x² + y²  + –2xy = 16 + –2 = 14

The answer is 14. 

First, let's expand (x + y)² by using the FOIL method.

(x + y)² = (x + y)(x + y)

According to the FOIL method, we will multiply the first terms of the binomials, then the outer terms, then the inner terms, and then the last terms. We will then add these four products together.

(x + y)(x + y)= x(x) + x(y) + y(x) + y(y) = x² + 2xy + y²

We are told that (x + y)² = 15. Let's replace (x + y)² with x² + 2xy + y².

x² + 2xy + y² = 15

Rearranging, we can write the equation as follows:

x² + y² + 2xy = 15

The second part of the problem tells us that x² + y² = 27. Thus, we can replace x² + y² with 27.

27 + 2xy = 15

Subtract 27 from both sides.

2xy = –12

Divide by two.

xy = –6

The question asks us for x²y². According to one of the properties of exponents, (xy)² = x²y². Thus, if we square both sides of the equation xy = –6, we will obtain the value of x²y².

(xy)² = (–6)²

x²y² = 36

The answer is 36.

First, we need to find points A and B, which we are told form the points of intersection between the graphs y = 9 – x² and y = 3 – x. In order to solve for these two equations, we can set the value of y in the first equation equal to the value of y in the second and then solve for x.

9 – x² = 3 – x

Add x² to both sides.

9 = 3 – x + x²

Subtract 9 from both sides. Then rearrange so that the powers of x are in descending order.

-6 – x + x² = x² – x – 6 = 0

Factor x² – x – 6 by thinking of two numbers that multiply to give –6 and add to give –1. Those two numbers are –3 and 2. 

x² – x – 6 = (x – 3)(x + 2) = 0

Set each factor equal to zero and solve.

x – 3 = 0

x = 3

x + 2 = 0

x = –2

Thus, the points of intersection occur where x = –2 and 3. We can find the y values of the points of intersection by substituting –2 and 3 into either equation. Let's use the equation y = 3 – x.

When x = –2, y = 3 – (–2) = 5. One point of intersection is (–2,5).

When x = 3, y = 3 – 3 = 0. The other point of intersection is (3,0).

Let's assume point A is at (–2,5) and that B is at (3,0). We are told that C is located at (p,0), where p < 0. Let's draw triangle ABC with the information we have so far.

In the figure above, the orange line represents the height from side BC to A.

The area of any triangle is (1/2)bh, where b is the length of the base, and h is the length of the height. We will use BC to represent the base, and the orange line to represent the height.

The length of BC will be equal to 3 – p, since both points lie on the x-axis. The length of the orange line is the distance from CB to point A, which is 5. We can now find a formula for the area and set it equal to 50.

Area of ABC = (1/2)(3 – p)(5) = 50

Multiply both sides by 2.

(3 – p)(5) = 100

Divide by 5.

3 – p = 20

Subtract 3 from both sides. 

–p = 17

Multiply both sides by –1.

p = –17.

The answer is –17.

Since the number has three digits, there must be a digit in the hundreds place, a digit in the tens place, and a digit in the ones place. Let's let h represent the digit in the hundreds place, t represent the digit in the tens place, and let u represent the digit in the ones place. Thus, we can represent our mystery number as follows:

mystery number = 100h + 10t + u

First, we are told that the sum of the digits is 13. Thus, we can write the following equation.

h + t + u = 13

Next, we are told that reversing the digits decreases the value of the original number by 594. We already established that the original number can be represented as 100h + 10t + u. We can then represent the reversed number as 100u + 10t + h. We can now write the following equation:

original number = 594 + reversed number

100h + 10t + u = 594 + 100u + 10t + h

Let's get the h, t, and u terms on the same side. Subtract h from both sides.

99h + 10t + u = 594 + 100u + 10t

Subtract 10t from both sides.

99h + u = 594 + 100u 

Subtract 100u from both sides.

99h – 99u = 594

Since 99 and 594 are divisible by 9, we will divide both sides by 9.

11h – 11u = 66

Divide both sides by 11.

h – u = 6

We are told that the first digit, represented by h, must be prime. Let's solve for h in terms of u by adding u to both sides.

h = 6 + u

Since h is a digit, it can't be larger than 9. (The possible values for a digit are 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.) The only prime numbers less than 9 are 2, 3, 5, and 7. However, if h were 2, 3, or 5, then u would be negative, and we can't have a negative digit. In short, the only value that h can be is 7.

7 = 6 + u

Since h is 7, u must be 1.

If h = 7 and u = 1, we will solve for t by using the first equation h + t + u = 13.

7 + t + 1 = 13

Subtract 8 from both sides.

t = 5

Since h = 7, t = 5, and u = 1, the number must be 751. We are told to find the squares of the digits.

sum of squares of digits = 7² + 5² + 1² = 49 + 25 + 1 = 75.

The answer is 75. 

To be a solution to the system, the point must satisfy BOTH inequalities.  All five points satisfy the inequality .  On the other hand,  does not satisfy .

Let x equal the number of children that played minature golf and let y equal the number of children who bowled.

Based on that, we can conclude that:

and

Let's solve for x in the first equation by subtracting y from each side.

Now, let's substitute that expression for x in the second equation:

Now, distribute the 4:

Simplify:

Subtract 80 from both sides:

Divide both sides by 2:

Therefore, 8 children bowled. By substituting y into the first equation, we know that  

Solve for x by subtracting 8 from both sides: 

In order to find this point, we must find the solution to the system of equations. we will use substitution, setting the two expressions for y equal to one another.

Then we plug this value back into either expression for y, giving us  

So the point is (1, 6).