Algebraic Versus Graphical Solution of Linear Equations

There is a big difference. Algebraic methods are much more powerful in general. Whereas graphical methods can only be used to solve systems of linear equations involving up to 3 variables, algebraic methods can be used with an arbitrary number of variables. Also, graphical methods in general only yield an approximate solution, whereas algebraic methods can always yield an exact solution. For example, consider the following linear system:

x + 4y = 3
5x - y = 3.

First we solve this system algebraically. Multiplying the second equation by 4, we obtain

20x - 4y = 12.

Adding this to the first equation yields

21x =15.

Dividing this equation by 21, we obtain

x = 15/21 = 5/7.

Substituting this value into the second equation yields

25/7 - y = 3 = 21/7,

whence

y = 4/7.

Thus we have found the following exact solution:

x = 5/7, y = 4/7.

Next let us solve the system graphically. The first equation corresponds to the line with x-intercept 3 and y-intercept 3/4, while the second corresponds to the line with x-intercept 3/5 and y-intercept -3 (see the attachment). The solution then corresponds to the point of intersection of these two lines, which as we can see is approximately (0.7, 0.6), but it is very difficult to obtain an exact solution graphically.

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