How do you determine whether a linear system has one solution, many solutions, or no solution when given 8x + 3y= -9 and -8x + y = 29?

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Answer 1 · 2017-02-16T11:53:13

The lines will intersect at only one point, so there is only one solution.

The first thing to realise is that these are both equations for straight lines.

For two straight lines there are 3 possibilities:

$1$ . They do not intersect at all. This would mean they are parallel.
If you find the slope of each line you can determine whether this is so.

$2$ . They intersect only once. Any two lines which are not parallel will intersect exactly once.

$3$ . Two straight lines can interest many times only if they are the same line. Sometimes the equations might be in different forms,

Write each equation in the form $y = mx +c$

$8x +3y = -9color(white)(............)and -8x+y = 29$

$3y = -8x -9color(white)(......................)y = color(red)(8)x +29$

$y= color(red)(-8/3)x -3$

We can see that they are NOT parallel because their slopes are different $color(red)(-8/3 and 8)$

Therefore these lines will intersect at only one point and there is ONE solution for the system of equations.