SOLUTION: In presidential elections, each state has a designated number of votes in the Electoral College, which are generally all cast for the candidate who won the popular vote in the stat

Question 1176244: In presidential elections, each state has a designated number of votes in the Electoral College, which are generally all cast for the candidate who won the popular vote in the state. The number of members of the Electoral College from each state is based on the number of Senators and House of Representatives.
a. If a presidential candidate is 42 votes ahead of his opponent before the votes for the state of California are added, what absolute value equation would represent the margin of votes between the candidate and his opponent after California’s 55 votes are cast?
b. What absolute value equation can be used to determine the minimum number of votes needed to change the outcome of the election in question 4?
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Let's define variables to set up the absolute value equations for both parts of the problem.
### Given Information:
- A presidential candidate is **42 votes ahead** before California's **55 electoral votes** are added.
- Electoral votes are generally awarded in full to the winner of the state's popular vote.
#### **Part (a)**: Absolute Value Equation for the Margin After California's Votes Are Added
Let \( x \) represent the final margin after California's votes are added. If the leading candidate wins California, their lead increases by 55 votes. If the opposing candidate wins California, their lead decreases by 55 votes.
The equation representing this scenario is:
\[
| x - 42 | = 55
\]
This equation accounts for both possibilities:
- If California's votes go to the leading candidate, the margin increases: \( x = 42 + 55 = 97 \).
- If California's votes go to the opposing candidate, the margin flips: \( x = 42 - 55 = -13 \).
#### **Part (b)**: Absolute Value Equation for the Minimum Number of Votes Needed to Change the Outcome
To change the outcome, the opposing candidate must at least tie the election. This means overcoming the 42-vote lead by shifting votes. Since electoral votes are won in full per state, the smallest shift would be flipping just enough votes from the winning candidate to the losing candidate.
Let \( y \) represent the number of votes that must be shifted (moved from one candidate to the other). Since each vote switched affects the margin by **2** (one candidate loses a vote while the other gains one), the equation is:
\[
| 2y | = 42
\]
Solving for \( y \):
\[
y = \frac{42}{2} = 21
\]
So, the minimum number of electoral votes that must be flipped to change the outcome is **21**.
These absolute value equations properly express the given electoral vote conditions in the problem.
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