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solve by graphing. Tell whether each system has one solution, infinitely many solutions, or no solution. x-2y=3 y=-2x+6

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Chak W. | Experienced Mechanical Engineering, Science, and Math TutorExperienced Mechanical Engineering, Scie...
4.9 4.9 (278 lesson ratings) (278)

To solve a system of linear equation graphically, it is best to write both equations in slope-intercept form first. After you put them in slope-intercept form, you should be able to graph them easily.

Now, to determine whether the system one solution, no solution, or infinitely many solution, you will have to see how the graph looks like.

For one solution, the two lines have to intersect at exactly one point, since both equations are satisfied at the intersection.

For no solution, the two lines have to never intersect, which means they have to be parallel lines. In other words, no solution will satisfy both equation.

For infinitely many solutions, the two lines have to overlap each other, which means one equation has to be a multiple of the other equations. When that happens, any point on the line will satisfy the equation, so you will have infinitely many solutions.

Now, for this particular question, if you rewrite the first equation into slope-intercept form, you will get y=1/2x - 3/2. If you put both line on the graph, you will see them intersect at exactly one location, and that will be your solution for the solution. (i.e. one solution)

George P. | Penn State Instructor exp.: Algebra, Trig., PreCalculus, and CalculusPenn State Instructor exp.: Algebra, Tri...
4.9 4.9 (587 lesson ratings) (587)

To solve a system of linear equations by graphing, you will need to graph both lines.

The 3 possible solutions as mentioned in the problem are: one solution, no solution, or an infinite number of solutions.  When you graph the 2 lines, you will have one of 3 different possibilities.  You will either have 2 lines that intersect, 2 lines that are parallel (don't intersect), or the 2 lines could actually just be the same line.

If the lines intersect, your solution is the (x,y) point where they intersect.  If you have parallel lines, there is no solution because they do not intersect and have no points in common.  If the 2 lines actually end up being the same line, then you actually have infinitely many solutions because ALL the points on the line are in common with the "two" lines.

When solving a system of equations with the graphing method, you should use graph paper or at least make sure your graph is to scale so that your spaces between each number on your x and y axes are the same.  This is important because if you have one solution, you are determining that solution by looking at where the 2 lines intersect.

To actually graph the 2 lines, you could convert both lines into slope-intercept form (y=mx+b), and use the y-intercept and slope to graph the lines.  If you get the same equation when you solve both for y, you will know that they are actually the same line, and you have infinitely many solutions.  If you get different equations, but the slope is the same, then you will have parallel lines.  If the slope is different, then you will have one solution, and you will need to see where the lines intersect after you graph them to find the solution.

Or, here is another method other than converting into slope-intercept form (y=mx+b). You could just find the x and y intercepts of each line, (0,b) and (a,0) and draw a straight line through them.  If the x and y intercepts involve fractions, you may want to scale your graph larger since you are actually using the graph to find your solution.  (Remember, if BOTH the x and y intercepts are the same, you will have the same line, and infinitely many solutions.)  If the lines look parallel (and thus you have no solution), you could check the slope of each of them to make sure they really are parallel.