A and B are two candidates in an election. A defeated B by 75 votes. Had 20% of the voters who voted for A had voted for B,the latter would have won by 19 votes. Find the number of votes polled in favor of A and B. Generate LaTex code for the solution wit
To find the number of votes polled in favor of candidates A and B, we can set up the problem using algebra.
Let \( x \) be the number of votes for candidate A and \( y \) be the number of votes for candidate B. According to the problem, we have two key pieces of information:
1. A defeated B by 75 votes:
\[
x - y = 75 \tag{1}
\]
2. Had 20% of A's votes gone to B, then B would have won by 19 votes:
\[
(x - 0.2x) - (y + 0.2x) = -19
\]
This simplifies to:
\[
0.8x - y - 0.2x = -19
\]
\[
0.6x - y = -19 \tag{2}
\]
Now we have a system of equations:
1. \( x - y = 75 \)
2. \( 0.6x - y = -19 \)
We can solve this system to find \( x \) and \( y \).
### Step 1: Solve for \( y \) from Equation (1)
From equation (1):
\[
y = x - 75 \tag{3}
\]
### Step 2: Substitute (3) into (2)
Now substitute (3) into (2):
\[
0.6x - (x - 75) = -19
\]
### Step 3: Simplify and Solve for \( x \)
\[
0.6x - x + 75 = -19
\]
\[
-0.4x + 75 = -19
\]
\[
-0.4x = -19 - 75
\]
\[
-0.4x = -94
\]
\[
x = \frac{94}{0.4} = 235
\]
### Step 4: Find \( y \)
Now substitute \( x = 235 \) back into equation (3):
\[
y = 235 - 75 = 160
\]
Thus, the number of votes polled in favor of A and B are:
- Votes for A: \( 235 \)
- Votes for B: \( 160 \)
The following LaTeX code can be used to display this solution:
```latex
\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\begin{document}
Let \( x \) be the number of votes for candidate A and \( y \) be the number of votes for candidate B. According to the problem, we have two key pieces of information:
1. A defeated B by 75 votes:
\[
x - y = 75 \tag{1}
\]
2. Had 20\% of A's votes gone to B, then B would have won by 19 votes:
\[
(0.8x) - (y + 0.2x) = -19
\]
This simplifies to:
\[
0.6x - y = -19 \tag{2}
\]
Now we have a system of equations:
1. \( x - y = 75 \)
2. \( 0.6x - y = -19 \)
Now solving this system:
\textbf{Step 1:} Solve for \( y \) from Equation (1):
\[
y = x - 75 \tag{3}
\]
\textbf{Step 2:} Substitute (3) into (2):
\[
0.6x - (x - 75) = -19
\]
\textbf{Step 3:} Simplify and Solve for \( x \):
\[
0.6x - x + 75 = -19
\]
\[
-0.4x + 75 = -19
\]
\[
-0.4x = -19 - 75
\]
\[
-0.4x = -94
\]
\[
x = \frac{94}{0.4} = 235
\]
\textbf{Step 4:} Find \( y \):
Substituting \( x = 235 \) back into equation (3):
\[
y = 235 - 75 = 160
\]
Thus, the number of votes polled in favor of A and B are:
- Votes for A: \( 235 \)
- Votes for B: \( 160 \)
\end{document}
```
This code provides a complete solution using LaTeX for the given election problem. You can compile it in a LaTeX editor to see the formatted output.