Question

18. The equations 3x - 2y +z = 5, 6x - 4y + 2z = 10, 9x - 6y +32 = 15 have
1) No solution
2) one solution
3) Infinitely many solutions
4) none​

Answer 1

Answer:

3) Infinitely many solutions

Step-by-step explanation:

                     

Answer 2

The correct answer is option 3) Infinitely many solutions.

Given:

The equations 3x - 2y +z = 5, 6x - 4y + 2z = 10, and 9x - 6y +3z = 15

To Find:

Nature of solution of the system of equations

Solution:

We have three equations,

3x - 2y +z = 5         .....................................(I)

6x - 4y + 2z = 10    .....................................(II)

9x - 6y +3z = 15     .....................................(II)

Writing the above equations in matrix form,

$\left[\begin{array}{ccc}3&2&1\\6&4&2\\9&6&3\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right] = \left[\begin{array}{c}5\\10\\15\end{array}\right]$

Taking the determinant of the above 3x3 matrix,

Δ = 3(12 - 12) - 2(18 - 18) + 1(36 - 36) = 0

Since the determinant is 0, the system of equations has infinite solutions.

Also, we can observe that equations (II) and (III) can be obtained by multiplying  equation (I) by 2 and 3 respectively.

So essentially, we have only one equation,

3x - 2y +z = 5

⇒ z = 5 -3x +2y is an equation consisting of two free variables x and y. These variables can take infinite values, giving us infinite values of z.

∴ The correct answer is option 3) Infinitely many solutions.

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