If p+pq is 4 Times p-pq, which of the following has exactly one value? Pq does not equal 0
A) p
B) q
C) pq
D) p+pq
E) p-pq
p+pq = 4(p-pq)
p(1+q) = 4p(1-q)
Since pq ≠ 0, then we can cancel out p on both sides.
=>
1+q=4(1-q)
Expand and solve for q
1+q=4-4q
5q=4-1=3
q=3/5
Check:
(1+q)=1+3/5=8/5
4(1-q)=4(1-3/5)=4(2/5)=8/5
Therefore the value of q is correct and unique.
To solve the equation p + pq = 4(p - pq), let's simplify step-by-step:
1. Distribute the number 4 to both terms inside the parentheses:
p + pq = 4p - 4(pq)
2. Multiply -4 by each term inside the parentheses:
p + pq = 4p - 4pq
3. Move all terms to one side of the equation:
p + pq - 4p + 4pq = 0
4. Combine like terms:
-3p + 5pq = 0
Now, we can determine which option has exactly one value:
A) p: There is no restriction on the value of p in the equation -3p + 5pq = 0, so it does not have exactly one value.
B) q: The variable q is not present in the equation -3p + 5pq = 0, so it does not have exactly one value.
C) pq: The equation -3p + 5pq = 0 does not restrict the value of pq, so it does not have exactly one value.
D) p + pq: The equation -3p + 5pq = 0 does have exactly one value since the equation is in terms of p and q, not p + pq.
E) p - pq: The equation -3p + 5pq = 0 does not restrict the value of p - pq, so it does not have exactly one value.
Therefore, the answer is D) p + pq, which has exactly one value.
To determine which of the given variables has exactly one value, let's solve the equation and see how many unique solutions we find.
Given equation: p + pq = 4(p - pq)
Let's simplify this equation step by step:
Distribute the 4 on the right side:
p + pq = 4p - 4pq
Rearrange the terms:
p + pq - 4p + 4pq = 0
Combine like terms:
p - 4p + pq + 4pq = 0
-3p + pq + 4pq = 0
Factor out variables:
(1 - 4) p + (1 + 4) pq = 0
-3p + 5pq = 0
Now let's solve this linear equation for p:
Factor out common terms:
p(-3 + 5q) = 0
We have two factors, p and (-3 + 5q), that multiply to give zero. For the whole equation to be true, either p = 0 or (-3 + 5q) = 0.
Case 1: p = 0
If p = 0, then p + pq = 0 and p - pq = 0. This means that both p+pq and p-pq would be zero, so it doesn't have exactly one value.
Case 2: -3 + 5q = 0
If -3 + 5q = 0, then q = 3/5. In this case, p + pq = 4p - 4pq becomes p + 3p/5 = 4p - 4(3p/5), which simplifies to 8p/5 = 8p/5. This means that p can take any real value, as long as q = 3/5.
Therefore, the only variable that has exactly one value is option B) q.