![]() Substitution is not the only algebraic technique for solving a system of linear equations. The solution is x = 3, y = -1 or the point (3, -1). Then find the corresponding y-value from either of the two original equations. We have reduced the problem of solving two equations in two variables to solving one equation in one van'able. We now have one equation in only one variable, x. Would you put -2x + 55 for y in the second equation? Try this and see what happens. How would you substitute? The first equation is already solved for y. We simply substitute an expression for a variable from one equation into the other. When the system has one equation that can easily be solved for one of the variables, the algebraic technique of substitution is convenient. Solving a system of linear equations graphically can be time-consuming and does not always yield accurate results. For example, (4, 1) is a point on the line x + 2y = 6 since 4 + 2(1) = 6. All points that lie on one line also lie on the other line. The lines are parallel with the same slope, 3, and there are no points of intersection. We will be able to locate this point precisely using the techniques of the next section. The actual point of intersection is (3/5,-1/5). ![]() or (3/4,-1/3) some such point, your answer is acceptable. For example, if you estimated (1/2,-1/4). ![]() In such cases, any reasonable estimate will be acceptable. This example points out the main weakness in solving a system graphically. The system is consistent, but the point of intersection can only be estimated. If the system is consistent, find the point of intersection. Infinite number of points lines are the same (&hey coincide)ĭetermine graphically whether the fallowing systems are (a) consistent, (b) inconsistent, or (c) dependent. The following table summarizes the basic ideas and terminology. If a system has an infinite number of solutions (the lines coincide), the system is dependent. If a system has no solution (the lines are parallel with no point of intersection), the system is inconsistent. If a system has a unique solution (one point of intersection), the system is consistent. The term simultaneous is frequently used to emphasize the idea that the solution of a system is the point that satisfies both equations at the same time, or simultaneously. When two linear equations are considered together, they are called a system of linear equations, or a set of simultaneous equations. These three examples constitute all three possible situations involving the graphs of two linear equations. Any point that satisfies one equation will also satisfy the other This can be seen easily by putting both equations in the slope-intercept form :īoth equations are identical when written in the same form. The lines not only intersect they are the same line. The reason there is just one Line is that both equations represent the In this case, we could have anticipated the result by writing both equations in the intercept form and noted that both lines have the same slope, -1:ĭo the lines y=-x+4 and 2y + 2x = 8 intersect? If so, where do they intersect? If not, why not? The graphs of both lines are shown in Figure 9.3. The lines do not intersect because they are parallel. ![]() This intersection can be checked by substituting x = 1 and y = 3 into both equations:ĭo the lines y = -x + 4 and x + y = 2 intersect? If so, where do they intersect? If not, why not? The answers are in the graphs in Figure 9.2. The lines appear to intersect at the point (1, 3), or where x = 1 and y = 3. What do you think they are’? Stop to think about your lines and those of the other people in class before you read further.ĭo the lines y = -x + 4 and y = 2x + 1 intersect (cross each other)? If so, where do they intersect? If not, why not? We can answer these questions by graphing both equations as in Figure 9.1. What two lines did you see‘? No matter what specific lines you envisioned, there are only three basic positions for the lines relative to each other. Visualize two straight lines on the same graph.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |