PHP is loading, but not sending the email to me - php

Question

I will admit that I am not the best at PHP, but what I am doing usually works well enough for me because the clients just need the information sent to their emails, and the security question works well enough to keep spam away from their inbox. But for some reason, this code will not work and I cannot figure out why. This same code with different variables is on another site I have and is tested and working. Does anyone have any advice?
<form action="submit_form.php" method="POST" >
<p>Name<br /><input type="text" name="name" required /></p>
<p>Email Address<br /><input type="text" name="email" required /></p>
<p>Phone Number<br /><input type="text" name="phone" required /></p>
<p>County<br /><input type="text" name="county" required /></p>
<p>Annual Income<br /><input type="text" name="income" required /></p>
<p>What is 4 + 1? (anti-spam)<br /><input type="text" name="answer" required /></p>
<input type="submit" value="Take The First Step" />
</form>
<?php # BOOST
$name = $_POST['name'];
$email = $_POST['email'];
$phone = $_POST['phone'];
$county = $_POST['county'];
$income = $_POST['income'];
$correct_answer = '5';
$answer = $_POST['answer'];
if ($correct_answer != $answer) {
die("You're not a valid user of this site!");
}
else {
$to = "glenn#boostbizseo.com";
$subject = "USDA LOANS";
$message = "USDA LOANS:\n
Name: $name
Email: $email
Phone: $phone
County: $county
Annual Income: $income";
$from = "USDA Loans";
mail($to,$subject,$message,$headers);
echo "Thank you for getting in contact with us. We will be in contact with you soon regarding your USDA Loan! <a href=http://www.usdaloansmo.com>Click Here</a> to go back to our website!";
}
?>

Answer 1

$from = "someonelse#example.com"; //email ID
$headers = "From:" . $from;
mail($to,$subject,$message,$headers);
I believe this will work.

Answer 2

This is wrong:
$from = "USDA Loans";
mail($to,$subject,$message,$headers);
You don't use $from and $headers is undefined.

Answer 3

Here is how you send email using php.
$to="some#domain.com";
$subject="Your subject";
$header="from: ABCName <me#mydomain.com>";
$message="Message \r\n";
$sentmail = mail($to,$subject,$message,$header);
if($sentmail)
{
echo "Done";
}
else
{
echo "Oops, Something went wrong";
}