Simple linear systems

Linear systems of algebraic equations consist of N equations and N unknowns where the unknown variables all appear to the first power. Linear systems of equations are an essential aspect of nearly all engineering disciplines. They very often represent a mathematical model of N balance relationships that characterize an engineering system. This might be currents entering and leaving a node in an electrical circuit, numbers of cars per hour entering and leaving an intersection of a network of roads, forces on a node in a structural lattice such as you find on a bridge, or it could be the result of a numerical method that relates finite difference quantities to one another. There are countless examples of linear systems. The mathematical underpinning of linear systems is called linear algebra. Consider the high school algebra problem of weighing fruit. The combined weight of four apples and three bananas is three pounds. The combined weight of two apples and nine bananas is four pounds. What are the weight of one apple and the weight of one banana? First build a mathematical model of this problem. 4 apples plus 3 bananas equals 3 pounds → $4a+3b=3$ 4a+3b=3.gif 2 apples plus 9 bananas equals 4 pounds → $2a+9b=4$ 2a+9b=4.gif With these two pieces of information you can determine the answer. You need to solve for a and b. multiply the second equation by two and subtract it from the first equation. $$\left(4a+3b\right)-2\left(2a+9b\right)=3-2\times 4\to -15b=-5$$ sol1.gif This gives $b=1/3$ b=33.gif or bananas weigh a third of a pound. To find the weight of an apple, substitute the banana weight into one of the equations and solve for a. $$2a+9\times \left(1/3\right)=4\to 2a=4-3=1\to a=1/2$$ sol2.gif An apple weighs ½ pound. The basic method is to combine the equations to obtain an equation with only one of the variables. After solving for that variable, use it in one of the equations to solve for the other variable.