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Find conditions on a and b such that the following system of linear equations has

(a) no solutions, (b) exactly one solution, and (c) an infinite number of solutions

x+2y=3

ax+by= -9

Questioner:Vanessa

Country:Quebec, Canada

Category:Advanced Math

Private:No

Subject:Linear Algebra

Question:

Find conditions on a and b such that the following system of linear equations has

(a) no solutions, (b) exactly one solution, and (c) an infinite number of solutions

x+2y=3

ax+by= -9

.................................

If you attempt to solve using the standard techniques (eliminate x from the equations) you will get:

3a + 9

y = --------

2a - b

(and use that to get an expression for x)

Now if 2a - b /= 0 all is well and you will have one solution.

But if 2a - b = 0 all is not so well

This happens if b = 2a

if 3a + 9 /= 0, then there is no solution at all.

if 3a + 9 = 0 you will have many solutions.

You can take it from there.

...............................

Suggestion: You have studied Cramer's rule?

D = the determinant:

| 1 2 |

| a b |

= b - 2a

And the solution for x will be:

| 3 2 |

| -9 b |

-------------

b - 2a

3b + 18

= --------

b - 2a

which leads to the same reasoning.

.......................................

Another:

These are equations of lines, mon ami.

They have slopes, no?

You know how to find the slope of a line, oui?

If the slopes are:

-- different, then these lines will intersect in a point.

-- the same, then either:

-- these are parallel lines

-- these are the same line.

Try a few numbers for a,b and see what happens.

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Answers by Expert:

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

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