If two linear equations have the same slope and the same y intercept, how many common solutions would they share?

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If two equations have the same slope and the same y-intercept, then they are the same line, therefore meaning that they connect an infinite number of times on the graph. These lines have an infinite number of solutions, meaning that the same numerical values one uses for x and y in the first solution will also work for the second.

In order to test this special system of equations, one should write both equations in slope-intercept form. The equation is as follows: y=mx+b where "m" serves as the slope and "b" serves as the y-intercept. In some textbooks this is also called solving for y in terms of x. If one's equation is not already written like this, it is probably presented in standard form, which is Ax+By=C where A, B, and C are constants (numbers). In order to take a standard equation from standard to slope-intercept form, one should subtract the "Ax" value from both sides of the equation. It is important to do this from both sides, as this will leave the equation balanced. When one does this, one is left with the following: By=-Ax+C. Next, divide by B on both sides. This will isolate the "y" variable and should leave one with an equation that is y=mx+b. It is acceptable if "m" and/or "b" are negative values.

One should put both equations into y=mx+b form. If one remembers the rule that the same y-intercept and the same slope mean infinite solutions, one can usually stop there, writing the infinity symbol (`oo)` ` ` ` ` ` ` as this means that the solution can be anything. If one forgets the rule, one can graph the lines on graphing paper by first plotting the y-intercept and then using the slope in order to plot three additional points. If the student does not remember the first rule, seeing the graph of the two equations will often cue the memory that a system of equations with the same slope and the same y-intercept will have infinite solutions.

This is not to be confused with a system that has the same slope but different y-intercepts, which has no solution, since they are parallel lines.

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