Question

.Examine whether the equations x-3y+2z=4 ,2x+y-3z=-2 ,4x-5y+z=5 are

Consistent.​

Answer 1

Given:

x-3y+2z=4

2x+y-3z=-2

4x-5y+z=5

To find:

Whether the given system of equations is consistent.

Solution:

A consistent system of equations is a system of equations that has at least one solution.

The given system of equations can be written as

AX=B

where A= $\left[\begin{array}{ccc}1&-3&2\\2&1&3\\4&-5&1\end{array}\right]$,

X= $\left[\begin{array}{ccc}x\\y\\z\end{array}\right]$

and B= $\left[\begin{array}{ccc}4\\-2\\5\end{array}\right]$

For the given system of equations to be consistent, the determinant of A ≠0

Or, |A| ≠ 0

∴ Expanding the determinant of A, we get

|A| = [1×{1×1-(3×(-5))}-(-3)×{2×1-4×3}+2×{2×(-5)-4×1}]

Upon simplifying, we get

|A| = [1×{1+15}+3×{2-12}+2×{-10-4}]

⇒|A| = [1×16+3×(-10)+2×(-14)]

⇒|A| = [16-30-28]

⇒|A| = -42

∴ |A| ≠ 0

Hence, the given system of equations is consistent.

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