<p>from the blue book:</p>
<p>A four digit integer, WXYZ, in which W, X, Y, and Z each represent a different digit, is formed according to the following rules.</p>
<p>X=W+Y+Z
W=Y+1
Z=W-5</p>
<p>what is the four digit number?</p>
<p>I have no idea how to solve this... Can someone help solve these systems of equations for me please?</p>
<p>This question just wants you to be comfortable with playing around with numbers.
Let’s start.</p>
<p>X = W + Y + Z
This means that X is the biggest number;
Also it means that W + Y + Z <= 9;</p>
<p>Z = W - 5;
This means W >= 5, since you cannot have a negative number;</p>
<p>W = Y + 1;
Z = (Y + 1) - 5; <strong><em>I substituted for W in the equation Z = W - 5;</em></strong>
Z = Y - 4;
Y >= 4;</p>
<p>The ONLY possible combination in which all these conditions are met and W + Y + Z does not exceed 9 is when W = 5; Y = 4; Z = 0; Subsequently, X = 9;</p>
<p>Now put it all together and we find that the answer is 5940;</p>
<p>First, you have to understand the problem. It is not a system of equations asking you for the value of one of the variables. It is really not much different from the kinds of questions that start “Points A,B,C, and D are on line m, but not necessarily in that order…” but instead of asking for the value of a single variable, they are asking you to put all the values into a certain order to form a new number.</p>
<p>Second, all the information is there for a reason. They aren’t going straight up tell you that X is the greatest, that X and Y are consecutive and that W>Y. Instead, they give you these little equations; it is a math test after all. ;)</p>
<p>X=W+Y+Z tells you that X is the largest digit.</p>
<p>W=Y+1 tells you that w and Y are consecutive AND that W>Y</p>
<p>Z=W-5 tells you that the W > Z and that they are five away from each other.</p>
<p>SO… combining this, we know that Z<Y<W<X </p>
<p>The smallest that Z could be is 0.<br>
That would make make W=5
If W=5, then Y=4
guess what? 0+5+4=9, which satisfies the first clue. X=9</p>
<p>Now we answer the question by putting them in the order that they want: WXYZ = 5904</p>
<p>because the values
W=5
X=9
Y=0
Z=4
satisfy all of the rules set forth in the problem.</p>
<p>I hope I helped!</p>
<p>Yes that definitely helps.</p>
<p>I was trying to use my systems of equation skills. I’m such an idiot for not noticing that WXYZ have to be integers.</p>
<p>We know that W >= 5 because Z = W-5 is a digit. Do casework based on W:</p>
<p>Case 1: W = 5
Automatically this implies Y = 4, Z = 0. Since X = W+Y+Z, X = 5+4+0 = 9. Therefore 5940 works.</p>
<p>(This is a correct answer but for sake of completeness we can try W = 6).</p>
<p>Case 2: W = 6
Notice that W = Y+1 → Y = 5. Since X = W+Y+Z = 6+5+Z, X would have to be at least 11, so this is impossible.</p>
<p>This also eliminates the possibilities W = 7,8,9.</p>
<p>Therefore the only answer is 5940.</p>