# Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations’ right sides to make the new system inconsistent?

Question
Equations
Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations’ right sides to make the new system inconsistent?

2020-12-16
Step 1
According to the given information, a non-homogeneous system of six linear equations in eight unknown has a solution with two free variables.
It is required to check whether is it possible to change some constants on the equations on right side to make new system inconsistence.
dim(Nul(A)) is same as number of free variables
so, dim(Nul(A))=2
Step 2
Since, n = 8 as system has eight unknowns so, rank of coefficient matrix is:
rank(A)=8-2=6
Step 3
Further, col(A) is a six-dimensional subspace of $$\displaystyle{R}^{{6}}$$ because it has six linear equations so, $$\displaystyle{c}{o}{l}{\left({A}\right)}={R}^{{6}}$$
This means that Ax = b has a solution for every b and cannot change constant on right side to make system inconsistence.
Hence, it is not possible to change the constants.

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The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
$$\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}$$
where A is the cross-sectional area of the vehicle and $$\displaystyle{C}_{{d}}$$ is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity $$\displaystyle\vec{{{v}}}$$. Find the power dissipated by form drag.
Express your answer in terms of $$\displaystyle{C}_{{d}},{A},$$ and speed v.
Part B:
A certain car has an engine that provides a maximum power $$\displaystyle{P}_{{0}}$$. Suppose that the maximum speed of thee car, $$\displaystyle{v}_{{0}}$$, is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power $$\displaystyle{P}_{{1}}$$ is 10 percent greater than the original power ($$\displaystyle{P}_{{1}}={110}\%{P}_{{0}}$$).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, $$\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}$$, is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.