Algebraic Expressions

To find the value of x, we can use the given information.

We are given that x = y = 2z and xyz = 256.

Since x = y, we can substitute x with y in the second equation:

xy * z = 256

(x * x) * z = 256

x^2 * z = 256

Now, we know that x = 2z, so we can substitute x in terms of z:

(2z)^2 * z = 256

4z^2 * z = 256

4z^3 = 256

To solve for z, we can divide both sides of the equation by 4:

z^3 = 64

Taking the cube root of both sides gives us:

z = 4

Now that we know the value of z, we can substitute it back into the equation x = y = 2z:

x = y = 2 * 4

x = y = 8

Therefore, the value of x is 8.

The answer is: C. 8

Given the operation x ^ y = 2x + y, we need to find a such that 2 ^ a = a ^ 3. This means 2(2) + a = 2(a) + 3, which simplifies to 4 + a = 2a + 3. Solving gives a = 1. Both sides equal 5 when a=1.

Starting with 3x + x = 2x + 2y, simplify step by step: 4x = 2x + 2y, subtract 2x from both sides: 2x = 2y, divide both sides by 2: x = y. Therefore, statement D (x = y) is correct. Statements A, B, and E are false. Statement C is false because we CAN determine the relationship - x equals y.

For divisibility by 36 = 4 × 9: 34x5y must be divisible by 4 (so 5y divisible by 4, y ∈ {2,6}) AND divisible by 9 (sum 3+4+x+5+y = 12+x+y divisible by 9). Testing options: Only 3462 works - 62 divisible by 4, sum 3+4+6+2=15 (not divisible by 9). Wait, that fails. Let me recalculate: For 34x5y divisible by 36, need divisible by 4 and 9. y must be 2 or 6 for divisibility by 4. Sum 12+x+y must be divisible by 9. Option D (4,2): 34,4252? That doesn't make sense. The number is 34x5y, so x=4, y=2 gives 34452. Sum = 3+4+4+5+2=18 (divisible by 9). 52 is divisible by 4. Correct.

If 2x - y = 4, then 6x - 3y = 3(2x - y) = 3 × 4 = 12. We can factor out 3 from 6x - 3y to get 3 times the given expression.

Given x ^ y = 2x + y, we have 2 ^ a = a ^ 3. This means 2(2)+a = 2a+3, which simplifies to 4+a = 2a+3, giving a=1. Verifying: 2^1 = 2(2)+1 = 5 and 1^3 = 2(1)+3 = 5, both equal.

Multiply the first equation by 3: 3(2x-y) = 3(4) gives 6x-3y = 12. This is a direct application of the distributive property and multiplication property of equality.

Since x=y=2z, all three variables are equal. Let x=2z, y=2z, then xyz = (2z)(2z)(z) = 4z^3 = 256, so z^3 = 64 and z = 4. Then x = 2z = 8. This requires recognizing that x=y, then substituting and solving.

The answer is 10 because Java uses short-circuit evaluation for the && operator. Since y > 10 is false (y equals 10), the second condition (x++ > 10) is never evaluated, so the increment x++ doesn't execute. This means x remains 10. Option C (11) would only occur if the increment executed, and options A and D are nonsensical in this context.

From r=3y and 15y=30z, we get y=2z, so r=3(2z)=6z. From 2z=p/3, we get p=6z. Therefore r=p. This is a chain of substitutions through y and z.

From 15y=30z, we get y=2z. From 2z=p/3, we get z=p/6. Substituting: y=2×(p/6)=p/3. Then r=3y=3×(p/3)=p. The key is working backwards from the final equation to express all variables in terms of p.

Using the exponent rule 2^(x + y) = 2^x * 2^y, we substitute the given values: 7 * 38 = 266. The other options are incorrect applications of algebraic rules.

Given 2x - y = 4. To find 6x - 3y, notice that 6x - 3y = 3(2x - y). Substituting the given value: 3(4) = 12. This is a direct application of multiplying both sides of an equation by a constant.

The pattern in numerators is odd numbers (1, 3, 5, 7, 9) and denominators double each time (2, 4, 8, 16, 32). Therefore X=9 and Y=32.