Now let’s study the nature of the solution of a system of linear equations in more detail. Where a, b, c, d, h, and k are real constants and neither a and b nor c and d are both zero. Recall that such a system may be written in the general form ax by h cx dy k We begin by considering a system of two linear equations in two variables. In this chapter we take up a more systematic study of such systems. These are two examples of real-world problems that call for the solution of a system of linear equations in two or more variables. How should they use their budget of $12 million to meet the expected additional demand for compact and full-size cars? In Example 5, page 133, we will see how we can find the solution to this problem by solving a system of equations.Ģ SYSTEMS OF LINEAR EQUATIONS AND MATRICESĢ.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that in Section 1.4 we had to solve two simultaneous linear equations in order to find the break-even point and the equilibrium point. For his work in formulating this model, Wassily Leontief was awarded the Nobel Prize in 1973.įormulated in terms of systems of linear equations, and we alsoĬheckers Rent-A-Car is planning to expand its fleet of cars next quarter. Finally, we show how matrices can be used to describe the Leontief input–output model, an important tool used by economists.
We then go on to consider some real-life applications of matrices. In addition, we see how matrices (rectangular arrays of numbers) can be used to write systems of linear equations in compact form. In this chapter, we see how some real-world problems can beĭevelop two methods for solving these equations. For example, a linear equation in three variables represents a plane in three-dimensional space. HE LINEAR EQUATIONS in two variables studied in Chapter 1 are readily extended to the case involving more
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