Question: We can solve linear systems by methods other than Gauss's. One often taught in high school is to solve one of the equations for a

We can solve linear systems by methods other than Gauss's. One often taught in high school is to solve one of the equations for a variable, then substitute the resulting expression into other equations. Then we repeat that step until there is an equation with only one variable. From that we get the first number in the solution and then we get the rest with back-substitution. This method takes longer than Gauss's Method, since it involves more arithmetic operations, and is also more likely to lead to errors. To illustrate how it can lead to wrong conclusions, we will use the system
x + 3y = 1
2x + y = -3
2x + 2y = 0
from Example 1.13.
(a) Solve the first equation for x and substitute that expression into the second equation. Find the resulting y.
(b) Again solve the first equation for x, but this time substitute that expression into the third equation. Find this y.
What extra step must a user of this method take to avoid erroneously concluding a system has a solution?

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