Dynamically change html Login form to Registration form

I have a login form that takes input for user credentials. At the bottom I have a link to a Register user form. It takes me to a whole new page to register a user. However is there a way with JQuery AJAX to change the login form inputs to take registration inputs instead and send to the respective correct processing php files?

My form for login:

<div id="backgroundLogin">
            <form name="login" method="post" action="./Login.php">
               <fieldset>
                  <p id="title">
                     Connect-a-Cutie Login
                  </p>
                  <p id="email">
                     <label> Email:</label>
                     <input type="text" name="email" placeholder="Type your email...">
                  </p>
                  <p id="password">
                     <label>Password:</label>
                     <input type="text" name="password" placeholder="Type your password...">
                  </p>
                     <button href="./LoginPage.php" type="submit">Submit!</button>
                  <p>
                     <a href="RegistrationPage.php"><p>Not yet a member? Register now!</p></a>
               </fieldset>
            </form>
         </div>

And my registration form:

 <form action="RegistrationPage.php" method="POST">
            Name: <input type="text" name="name" placeholder="Enter your name..."/><br/>
            Email: <input type="text" name="email" placeholder="Enter your email..."/><br/>
            Password: <input type="text" name="password" placeholder="Enter your password..."/> (make sure no one is looking...)<br/>
            Addess: <input type="text" name="address" placeholder="Enter your address..."/><br/>
            Do you want to be an admin? <input type="text" name="isAdmin" placeholder="TRUE or FALSE"/><br/><br/>
            <button type="submit">Submit</button>
        </form>