There is really no need to alter this equation using algebra. Simply find that x = 2 and plug in. We see that 4(4)(1) – (1)=15. Remember to use correct order of operations here (parentheses, exponents, multiplication, division, addition, subtraction).

If x² + 5x – 24 = 0, 

(x – 3)(x + 8) = 0 or x = –8 or +3. 

y² – 9y + 20 = 0, then 

(y – 5)(y – 4) = 0, or y = +4 or +5. 

y is always greater than x.

We can factor the equation x² + kx - 12 = 0, knowing that we will have (x - 3) as one of the parentheses since the root is equal to 3.

x - 3 = 0

x = 3

We also know that the other root will be -4, because we multiply the 4 and -3 in (x + 4)(x - 3) to get our constant, -12.

This means that kx is equal to 4x - 3x = x. Therefore k = 1, and quantity A > quantity B.

Quantity A is greater.

We know that Quantity A = y / 3 = 15 /3 = 5.

If we plug in 15 for y, we can solve for x, for Quantity B.

y = x² - 10

y = 15

15 = x² - 10 (Add 10 to both sides.)

25 = x²

x = 5 or -5

Since 5 is equal to 5 but is greater than -5, we cannot determine the relationship between Quantities A and B.

The point of intersection for two lines is the same as the values of x and y that mutually solve each equation.  Although you could solve for one variable and replace it in the other equation, use elementary row operations to add the two equations since you have a 4y and -4y:

3x + 4y = 6

15x - 4y = 3

18x = 9; x = 0.5

You can now plug x into the first equation:

3 * 0.5 + 4y = 6; 1.5 +4y = 6; 4y = 4.5; y = 1.125

Therefore, our point of intersection is (0.5, 1.125)

The cars have a distance from each other of 25 + 120t miles, where t is the number of hours, 25 is their initial distance and 120 is 50 + 70, or their combined speeds.  Solve this equation for 400:

25 + 120t = 400; 120t = 375; t = 3.125

However, the question asked for minutes, so we must multiply this by 60:

3.125 * 60 = 187.5 minutes.

Start by setting up an equation using Quantity A and Quantity B. In other words, you can solve an inequality where Quantity A > Quantity B. You would have one of four outcomes:

1. Quantity A = Quantity B: the two quantities are equal.
2. The inequality is always satisfied: Quantity A is always larger. 
3. The inequality is never satisfied (but the two are unequal): Quantity B is always larger. 
4. The inequality is not always correct or incorrect: the relationship cannot be determined. 

So solve:

 –5x + 4 > 8 – 2x (Quantity A > Quantity B)

    +2x      +2x

  –3x + 4 > 8

    –4      –4

  –3x > 4 or x < –4/3

*remember to switch the direction of the inequality when you divide by a negative number

As the inequality [x < –4/3] is always false for [x>0], Quantity B is always larger.

Set up a system of linear equations based on our data:

40,000P + 45,000A = 40,415,000

P/A = 4/3

To make things easiest, solve the second equation for A in terms of P:

A = (3/4) P

Replace this value into the first equation:

40,000P + 45,000 * (3/4)P = 40,415,000

Simplify:

40,000P + 33,750P = 40,415,000

73,750P = 40,415,000

P = 548 (The number of professors)

To show that c is of the profit of the transaction, we must represent the profit as the difference between the price and the value of the car, or "(p – v)"

To show that Abby's commission in dollars is a percentage of the profit, we use 0.01 * c to convert the commission she earns to a percent.

To shift the earnings from Abby to the dealership (which is what the question requires of us), we must take 1 – 0.01c since this will accommodate for the remaining percentage. For example, it shifts 75% (0.75) to 25% (1 – 0.75 or 0.25).

Putting this all together, we get a final expression of:

(p – v)(1 – 0.01c) = dealership earnings

Check answer with arbitrary values: letting p = 300, v = 200, and c = 20, we get a value of 80 which makes sense as the $100 profit must be distributed evenly between Abby ($20) and the dealership ($80).

To simplify the word problem, express the ages in terms of variables in a system of equations. Note that we want to compare S with D:

S = A – 2

D = T + 5

A = T + 6

By substituting A for T in the first equation, we can get S in terms of T, which will let us directly compare the values of S and D.

S = (T + 6) – 2 = T + 4

If D = T + 5, and S = T + 4, D must be the greater value and Daisy is one year older than Sally.  B is the correct answer.