Question - Solving a System of Linear Equations using Elimination
Solution:
The image shows a system of linear equations. Here they are rewritten for clarity:1) 8x + 16y - 9z - 8w = -402) -2x - 2y + 3z + 2w = 43) 2x + 4y - 3z - 2w = -104) 8x + 16y - 9z - 9w = -42To solve the system, we can use different methods such as substitution, elimination (also known as linear combination), matrix methods, etc. Given the complexity of this system, it's often easiest to use matrix methods such as Gaussian elimination or using an augmented matrix. However, since it is a 4x4 system, I'll outline the process using elimination here to get you started:First, we can add equations (2) and (3) to eliminate x and y because they have equal coefficients with opposite signs:2x - 2x + 4y - 2y + (-3z + 3z) + (-2w + 2w) = -10 + 4, which simplifies to 0 = -6, which is a contradiction. The fact that we have obtained a contradiction implies that there is an error either in the recording or transcription of the equations or there is no solution to the system as it's written.Please double-check the coefficients and terms in the original equations to ensure they are correct before attempting to solve. If the system is correct and leads to a contradiction such as this, it implies that the system has no solution. This means the planes represented by the equations do not intersect at a common point in four-dimensional space.
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