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).

In order to solve this system of equations, it will probably be best to use the elimination method, in which we add or subtract two equations. In general, if , and , then . In other words, we can combine equations.

We need to solve the following equations:

We want to look for a way to combine these two equations that will eliminate at least one variable. If, for example, we multiplied the first equation by  and then added this to the second equation, we could get rid of the  terms. Let's multiply the first equation by . Remember, we must multiply both sides.

Distribute the .

We can now add to the equation . The sum of the left sides of each equation will equal the sum of the right sides.

We can simplify both sides. Notice that the  terms on the left disappear, leaving only  terms.

Divide both sides by 3.

.

Now that we have , we can go back to either of our original two equations, substitute the value of  we obtained, and solve for . Let's use the equation .

Substitute  for .

Add  to both sides. We will need to rewrite 4 with a denominator of 3.

Divide by 2 on both sides. When dividing a fraction, we simply multiply by its reciprocal.

Point , which is the intersection of the lines, must have the coordinates .

The question ultimately asks for the slope of the line passing through  and the origin. We can use the equation of the slope between two points with coordinates and .

slope =

The answer is .

Finding the intersection of two lines is the same as solving simultaneous equations.  Here, both equations are already solved for , so the right sides of both equations are set equal to each other.  Solving for  results in a value of 11.  Setting  equal to 11 in either of the original equations gives a  value of 27.

Let x equal the number of hamburgers and y equal the number of hotdogs.

Solve for y in the first equation.

Substitute that solution into the second equation.

Let Amy's current age be equal to A and Jackie's age be equal to J.

Therefore, and

Substitute the first equation into the second.

Therefore, today Jackie is 6 years old.