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Explanation

x + y = 30 ... (Equation 1)

x - y = 8 ... (Equation 2)

We can solve these equations by adding them together, which eliminates y:

2x = 38

Dividing both sides by 2 gives us:

x = 19

Now that we have x, we can find y by substituting x into Equation 1:

19 + y = 30

Subtracting 19 from both sides gives us:

y = 11

Now that we have both x and y, we can find their product:

Product = x * y

= 19 * 11

= 209

Explanation

Let's solve for x:

6 + 5x = 30 - x

Add x to both sides:

6 + 6x = 30

Subtract 6 from both sides:

6x = 24

Divide both sides by 6:

x = 24/6

x = 4

So, the correct answer is 4.

Explanation

3x - 15 - 6 = 0

3x = 21

X = 21/3

X = 7

Explanation

Let the two-digit number be xy, where x is the tens digit and y is the units digit.
The sum of its digits is x + y.
The number is three times the sum of its digits, so we can write an equation: xy = 3(x + y).
Simplify the equation: 10x + y = 3x + 3y.
Rearrange the equation to get: 7x = 2y.
Now, if 45 is added to the number, the digits are reversed, so we can write another equation: xy + 45 = yx.
Simplify the equation: 10x + y + 45 = 10y + x.
Rearrange the equation to get: 9x = 9y - 45.
Divide both sides by 9 to get: x = y - 5.
Now we have two equations:

1. 7x = 2y

2. x = y - 5

Substitute the second equation into the first equation:
7(y - 5) = 2y
Expand and simplify:
7y - 35 = 2y
5y = 35
y = 7
Now that we have found y, substitute it back into one of the original equations to find x:
x = y - 5
x = 7 - 5
x = 2
So, the original number is 27.

Explanation

(7^a) (7^b) = (7^c) (7^d)

Using the property of exponents, we get:

7^(a+b) = 7^(c+d)

Equating the exponents, we get:

a + b = c + d

Rearranging the terms, we get:

d = a + b - c

Explanation

The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

In this case, 4 × 4 × 4 = 64, so 4 is the cube root of 64.

Explanation

The correct inequality that represents the relationship between Jerry's estimate and the actual number of marbles in the jar is m - 10 ≤ n ≤ m + 10.

This inequality states that Jerry's estimate, m, is within 10 marbles (inclusive) of the actual number, n. By subtracting 10 from Jerry's estimate and adding 10 to his estimate, you create a range within which the actual number must fall.

For example, if Jerry estimates there are 50 marbles (m = 50), the actual number of marbles (n) must be between 40 (50 - 10) and 60 (50 + 10), inclusive.

Explanation

Explanation

The pattern follows the format n² + 1:

• 2² + 1 = 5
• 3² + 1 = 10
• 4² + 1 = 17
• 5² + 1 = 26

Given these, the correct 8th line should follow the same formula:

9² + 1 = 82

Explanation

Given: a/b = 3/5 ... (1)

When 2 is added to both numbers, the ratio becomes 2/3, so we can write:

(a+2)/(b+2) = 2/3 ... (2)

Cross-multiplying (2), we get:

3(a+2) = 2(b+2)

Expanding and simplifying, we get:

3a + 6 = 2b + 4

Subtracting 6 from both sides gives:

3a = 2b - 2

Now, substituting (1) in the above equation, we get:

3(3/5)b = 2b - 2

Multiplying both sides by 5, we get:

9b = 10b - 10

Subtracting 10b from both sides gives:

-b = -10

Dividing both sides by -1, we get:

b = 10

So, the correct answer is: 10