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This problem may become clearer if we think of graphing. If x = 1/2 is an equation to a line in the Cartesian (rectangular) coordinate system, then we know that by graphing the slope is undefined and the vertical line of that graph passes through 1/2 on the x-axis. We know that no matter how high or how low you go on this line, the x-value will always read 1/2. By algebra, we know that the x-value must remain 1/2 (simplified) in order for the equation x = 1/2 to be true. Therefore, in ordered pairs the x-value will remain 1/2. This will satisfy algebra and graphing. Now the y-values may make this problem more confusing. Looking at the line and equation, we can think of two different paths of logic to take to justify our solutions.\r
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\n" ); document.write( "\n" ); document.write( "Again, we know the line states that the x-value will always remain 1/2, but the y-value will change when we move on the graph. Logically, since x must remain 1/2 and y is going to change, our three solutions must have x as 1/2 and y as anything. The y-values in each of our three solutions must not repeat because that would mean we have used the same point over again. (If our three answers are (1/2, 9), (1/2, 5) and (1/2, 9), we have two answers that repeat and thus we only have two unique points. We must have three different points for our answer.) So, we could have answers as so:\r
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\n" ); document.write( "\n" ); document.write( "(1/2, -13)
\n" ); document.write( "(1/2, 85)
\n" ); document.write( "(1/2, 1/2)
\n" ); document.write( "(1/2, \"anything\").\r
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\n" ); document.write( "\n" ); document.write( "We can virtually plug anything (within reason) for y and satisfy the problem. This is because x must remain the same since the x-value of the line x = 1/2 never changes, and this is because the y-values can change if we go up or down. You may ask, \"Looking at our equation, how is it that we can pick any y-value and satisfy the equation x = 1/2? Where do the y-values go?\"\r
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\n" ); document.write( "\n" ); document.write( "The second logical method we could use to prove our point is by looking at our equation an breaking it down. We've said before that x must remain 1/2 because the equation literally states \"x is equal to 1/2.\" To answer the question about the y-values, I look at it like this: Since we've said that the y-value can be anything on the line x = 1/2, it would make things very complicated and complex to try to introduce every single y-value (even using the infinity notation) into an equation that can simply be written as \"x = 1/2.\" This equation satisfies the idea that the x-value remains the same, and since the x-value remains the same the line graphed goes straight up and down. Since the line goes on forever unless otherwise directed, the positive and negative y-values are being represented by this line. That is, the line passes by (not through) parallel to every y-value in the Cartesian coordinate system. We have fixed point at x = 1/2, so the line can't pass through the y-axis without bending. However, it still represents every point on the y-axis. Therefore, we really only need to look at the x-value for an equation reading \"x equals...\" in order to find other solutions to that line (y can virtually be anything). \r
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\n" ); document.write( "\n" ); document.write( "This is all explained with the idea that \"x = 1/2\" is an equation to a line in the Cartesian coordinate system, and that there are no other restrictions or opinions given by your teacher or course that I am unaware of. \n" ); document.write( "