Question 600156
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First thing to do is to decide which of your variables, <i>a</i> or <i>b</i> is your independent variable, i.e. the one you want to graph on the horizontal axis.  Then, in the discussion below replace <i>x</i> with your choice of independent variable and replace <i>y</i> with the other one.

<b>Step 1.</b>  Pick a value for <i>x</i>.  It can be anything you like, but 0, 1, or some other small integer usually works well and makes the arithmetic easier.

<b>Step 2.</b>  Substitute that value in place of <i>x</i> in your equation.  Do the arithmetic and determine the value of <i>y</i> that results.

<b>Step 3.</b>  Take the value of <i>x</i> that you selected for step 1 and the value of <i>y</i> that you calculated in step 2 and form an ordered pair (<i>x</i>,<i>y</i>).

<b>Step 4.</b>  Plot the ordered pair from Step 3 on your graph.  Remember that the <i>x</i> value is the distance right or left along the horizontal axis and the <i>y</i> value is the distance up or down along the vertical axis.

<b>Step 5.</b>  Repeat steps 1 through 4 with a different value for <i>x</i>.

<b>Step 6.</b>  Draw a line across your graph that passes through the two points that you plotted.

<b>Step 7.</b>  Repeat steps 1 through 6 using the other equation.

The point where the lines intersect is the solution, because the coordinates of that point will satisfy (read: make true) both of your equations.  You need to determine, by inspection of the graph, what the coordinates of that point are and report your answer as an ordered pair, (<i>x</i>,<i>y</i>), using those coordinates.  To check your answer, you should substitute this set of coordinates into each of your original equations and verify that you have a true statement for each of the equations.

If both lines graph to the same line, then the solution set is infinite, i.e. every ordered pair that satisfies one equation will satisfy the other.  If the lines are parallel, then the solution set is empty.

A <i>consistent</i> system has at least one solution.

An <i>inconsistent</i> system has no solutions.

An <i>independent</i> system has exactly one solution.

A <i>dependent</i> system has infinitely many solutions.

Therefore a system can be either consistent and independent, consistent and dependent, or inconsistent.

John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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