text: First, put each pair of equations into "slope-intercept" form, which means that each equation is expressed as y = mx + b, where m and b are constants.\n\nPerforming this step for the three pairs of equations produces the following results:\n\n3x+y=5 and x=2y-3 \nbecomes y = -3x + 5 and y = (1/2)x + 3/2\n\ny=-x+4 and y=-x+2 is already in slope-intercept form\n\ny=-x+4 and 2x+2y=8\nbecomes y = -x + 4 and y = -x + 4\n\nNow you're ready to analyze them:\n1. if the two equations are identical, then they are "dependent." That is, for any value of x the two equations will produce the same value for y. The FULL name of this situation is "consistent, dependent." In other words, the two equations have at least one solution (i.e., they are "consistent"). And in fact they have an infinite number of solutions because they are identical (i.e., they are "dependent").\n\n2. If the two equations have the same value for m (the slope), but different values for b (the y-intercept), then they are inconsistent. That is, there are no combinations of x and y values that satisfy both equations. In graphical terms, they consist of two straight lines that do not intersect (if they intersected, that point of intersection would be a solution of the two equations). This is true because the two lines are parallel. We know that they are parallel because they have the same slope. We know that they do not meet because they have different y-intercepts (i.e., they are parallel lines that pass through the y axis at two different points).\n\nFinally, if the two equations have different slopes (different values of m), then they are consistent, independent equations. That is, they are consistent, in that there is at least one solution. And they are independent because they are not simply identical equations. How do we know this? Because the slopes are different, the two lines can not be parallel, so they must intersect. And being straight lines, they will intersect in exactly one point.\n\nThat covers all the possibilities:\n1. the lines have different slopes (different values of m), so they are not parallel (consistent, independent)\n2. the lines have the same slope (same value of m) and the same y-intercept (same value of b), so they are the same equation, and their lines exactly overlap (consistent dependent)\n3. the lines have the same slope (same value of m), but different y-intercept (different values of b), so they are parallel and non-intersecting (inconsistent)