4. In yesterday's swim meet Roosevelt High dominated in the individual events, with 24 individual swimmers scoring a total of 56 points for Roosevelt High. A first place finish scores 5 points, a second place finish scores 3 points and a third place finish scores 1 point There were as many third place finishers as first and second place finishers combined. How many first place, second place, and third place finishers were there? Write, then solve a system of equations to model the information above.

To solve this problem, we need to set up a system of equations based on the information given.<br /><br />Let's define:<br />- \( x \) as the number of first place finishers,<br />- \( y \) as the number of second place finishers,<br />- \( z \) as the number of third place finishers.<br /><br />From the problem, we have the following information:<br /><br />1. The total number of swimmers is 24:<br /> \[<br /> x + y + z = 24<br /> \]<br /><br />2. The total points scored is 56, with first place scoring 5 points, second place scoring 3 points, and third place scoring 1 point:<br /> \[<br /> 5x + 3y + z = 56<br /> \]<br /><br />3. There were as many third place finishers as first and second place finishers combined:<br /> \[<br /> z = x + y<br /> \]<br /><br />Now, we have a system of three equations:<br /><br />\[<br />\begin{align*}<br />1. & \quad x + y + z = 24 \\<br />2. & \quad 5x + 3y + z = 56 \\<br />3. & \quad z = x + y<br />\end{align*}<br />\]<br /><br />We can substitute equation (3) into equations (1) and (2):<br /><br />Substitute \( z = x + y \) into equation (1):<br />\[<br />x + y + (x + y) = 24 \\<br />2x + 2y = 24 \\<br />x + y = 12<br />\]<br /><br />Now substitute \( z = x + y \) into equation (2):<br />\[<br />5x + 3y + (x + y) = 56 \\<br />5x + 3y + x + y = 56 \\<br />6x + 4y = 56 \\<br />3x + 2y = 28<br />\]<br /><br />Now we have a simpler system of two equations:<br />\[<br />\begin{align*}<br />1. & \quad x + y = 12 \\<br />2. & \quad 3x + 2y = 28<br />\end{align*}<br />\]<br /><br />We can solve this system using substitution or elimination. Let's use substitution:<br /><br />From equation (1), express \( y \) in terms of \( x \):<br />\[<br />y = 12 - x<br />\]<br /><br />Substitute \( y = 12 - x \) into equation (2):<br />\[<br />3x + 2(12 - x) = 28 \\<br />3x + 24 - 2x = 28 \\<br />x + 24 = 28 \\<br />x = 4<br />\]<br /><br />Now substitute \( x = 4 \) back into \( y = 12 - x \):<br />\[<br />y = 12 - 4 = 8<br />\]<br /><br />Finally, use equation (3) to find \( z \):<br />\[<br />z = x + y = 4 + 8 = 12<br />\]<br /><br />Thus, there are 4 first place finishers, 8 second place finishers, and 12 third place finishers.