Systems of Linear Equations and Problem Solving Chapter 8 Systems of Linear Equations and Problem Solving

Systems of Equations in Two Variables 8.1 Translating Identifying Solutions Solving Systems Graphically

System of Equations A system of equations is a set of two or more equations, in two or more variables, for which a common solution is sought.

T-shirt Villa sold 52 shirts, one kind at $8. 25 and another at $11 T-shirt Villa sold 52 shirts, one kind at $8.25 and another at $11.50 each. In all, $464.75 was taken in for the shirts. How many of each kind were sold? Set up the equations but do not solve. Solution 1. Familiarize. To familiarize ourselves with this problem, guess that 26 of each kind of shirt was sold. The total money taken in would be The guess is incorrect, now turn to algebra.

We have the following system of equations: 2. Translate. Let x = the number of $8.25 shirts and y = the number of $11.50 shirts. Kind of Shirt $8.25 shirt $11.50 shirt Total Number sold x y 52 Price $8.25 $11.50 Amount $8.25x $11.50y $464.75 x + y = 52 8.25x + 11.50y = 464.75 We have the following system of equations:

Identifying Solutions A solution of a system of two equations in two variables is an ordered pair of numbers that makes both equations true.

Determine whether (1, 5) is a solution of the system x – y = –4 1 – 5 –4 –4 = –4 TRUE 2x + y = 7 2(1) + 5 7 7 = 7 TRUE The pair (1, 5) makes both equations true, so it is a solution of the system.

Solving Systems Graphically One way to solve a system of two equations is to graph both equations and identify any points of intersection. The coordinates of each point of intersection represent a solution of that system.

Solve the system graphically. x – y = 1 Solution (3, 2) Graph both equations. x + y = 5 It appears that (3, 2) is the solution. A check by substituting into both equations shows that (3, 2) is indeed the solution.

Solve the system graphically. Solution Graph both equations. The lines have the same slope and different y-intercepts, so they are parallel. The system has no solution.

Solve the system graphically. Solution Graph both equations. The same line is drawn twice. Any solution of one equation is a solution of the other. There is an infinite number of solutions. The solution set is

When we graph a system of two linear equations in two variables, one of the following three outcomes will occur. The lines have one point in common, and that point is the only solution of the system. Any system that has at least one solution is said to be consistent. The lines are parallel, with no point in common, and the system has no solution. This type of system is called inconsistent. The lines coincide, sharing the same graph. This type of system has an infinite number of solutions and is also said to be consistent.

When one equation in a system can be obtained by multiplying both sides of another equation by a constant, the two equations are said to be dependent. If two equations are not dependent, they are said to be independent.