The Boltzmann equation: Seminar 1970 to 1971 by Grunbaum F.A. (ed.)

By Grunbaum F.A. (ed.)

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Two equations are said to be equivalent if and only if when one holds then so does the other. Equivalent equations, therefore, have precisely the same solutions if they have any at all. However, it is important that you never multiply or divide through an equation by 0. For example, take the equation 1 = 2, which is not valid, and multiply both sides by 0. Then we obtain the equation 0 = 0, which is true. So the two equations are not equivalent. If an equation contains a fraction, then the equation may be simplified by multiplying through by the denominator.

Consider the system of equations 2x + y = 10, 2x + y = 5. This system of equations does not have a solution. In fact, the equations are inconsistent. They cannot hold simultaneously since 10 = 5! We shall see later in this chapter that the solution of a system of simultaneous linear equations may be interpreted as the point of intersection of two straight lines. For the example under consideration, the two lines are parallel and therefore never intersect. Next consider the system of equations 2x + y −6x − 3y = 10, = −30.

6. Find the slope of the straight line passing through the points (−1, −3) and (4, 2). 7. Find the slope of the straight line passing through the points (0, 0) and (2, 1). 8. A person has e60 to spend on two goods, X and Y , whose respective prices are e6 and e4. a) Draw a budget line showing all the different combinations of the two goods that can be bought within the given budget. b) What happens to the original budget line if the budget is increased by 20%? c) What happens to the original budget line if the price of X is halved?

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