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For certain values of k and m, the system
a + 2b = -3,
4a + 2b = k - a - mb
has infinitely many solutions (a,b). What are k and m?

Please include an explanation! Thanks!

Sagot :

Answer:

k  = -15, m = 8

Step-by-step explanation:

Systems of Equations with Infinite Solutions

When two or more equations have infinite solutions, they are either the same or multiples of each other.

Multiples?

Say we have the system,

                                        2x + 4y = 8

and

                                        4x + 8y = 16.

If we plug in (2, 1) we get,

                                       2(2) + 4(1) = 8

                                             8 = 8

                                      4(2) + 8(1) = 16

                                        8 + 8 = 16

                                            16 = 16.

Both are solutions to each equation thus making it a solution to the overall system.

Similarly, it works for (-2,3), (0,2), (4,0), (6,-1) and much more!

If that's not convincing, dividing the second equation by 2 will get us the same equation as the first meaning that the system must have infinite solutions!

[tex]\hrulefill[/tex]

Solving the Problem

Understanding the Problem

We're told that the equations

                                           a + 2b = -3

and

                                   4a + 2b = k - a - mb

have infinitely many solutions.

This means that they must be the same or multiples of each other.

Let's simplify the second equation

                                      5a + (2+m)b = k.

The equations cannot be the same since the coefficients on the a terms are different: 1 and 5 (a and 5a).

So, they're multiples!

Let's assume that the first equation is a multiple of the second to simplify calculations.

Then, if we multiply the equation by a factor c

                        [tex]c(a+2b=-3) \rightarrow ca+2bc=-3c[/tex]

                    [tex]ca+2bc=-3c \Longleftrightarrow 5a+(2+m)b=k[/tex]

(the equations should be equal to each other)

[tex]\dotfill[/tex]

Finding the Value of C

To find the value of factor c, we can inspect each term in the equations. They should be equal to their corresponding other.

So,

  • the a terms
  • b terms
  • and constant values (on the right side of the equations)

should all be equal to each other.

That means

ca = 5a,

thus, c = 5!

[tex]\dotfill[/tex]

Putting it Together

So,

                 [tex](5)a+2b(5)=-3(5) \Longleftrightarrow 5a+(2+m)b=k[/tex]

                   [tex]5a+10b=-15 \Longleftrightarrow 5a+(2+m)b=k[/tex]

where k = -15 and m = 8 (10 = 2 + m).

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