Linear Algebra Applications and Inequalities
Explore linear algebra concepts, graphical methods, solving systems of equations, and graphing linear inequalities in two variables. Learn how to find intercepts, sketch functions, and shade regions based on inequalities. Enhance your understanding of linear algebra with practical applications.
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Topic 7: Linear Algebra and Applications Week 12
Linear equality Solve the following 2x2 system by the graphical method: 2? ? = 5 ? + ? = 4 Recap: Provided that equations are not coincident (should not be multiples of one another) or not parallel (slope should not be same) solutions will exist To sketch linear functions: Find x intercept (make y=0) Find y intercept (make x=0)
For 2? ? = 5 , x intercept is 2.5 and y intercept is -5. (verify) For ? + ? = 4 , x intercept is 4 and y intercept is 4. Note that point of intersection of the two lines are at (3,1) therefore x=3 and y=1
Linear Inequalities: A linear equation in two variables x and y is written in the form ?? + ?? = ? Where A, B and C are real numbers. Also A & B both are not zero. If we replace the Equal Sign by an inequality symbol, namely, one of the symbols, < ,> , ?? we obtain a linear inequality in two variables x and y. Example: or
Graphs of linear inequalities The graph of a linear inequality in two variables x and y is the set of all points (x, y) for which the inequality holds. Rules, for > and < we will have dash lines (not included) and for ??? we will have solid lines (as it is also = and thus included). However for all inequalities we need shading.
Example Graph the inequality ? + ? > 1 Find x intercept by making x=0 and y intercept by making x=0. X intercept = -1 and y intercept is 1 we have two regions now. Which side to SHADE? Pick any two test points in these regions and see for which test point is the Inequality satisfied.
Let us pick test point (-1,1) in region 1 and TP (1,1) in region 2. Substitute this into our Inequality. TP1: -(-1) + 1 = 2 >1 (Okay) TP2: (-1)+1 = 0 (not okay) TP1 satisfies our inequality, thus shade in region 1
Exercise Shade the inequality 2? + 4? 6 X Intercept = 3 and y intercept is 1.5 (solid line or dash lines???) I have chosen TP1 = (0,0) and TP2 =(2,2) Which of these satisfy the inequality? Ans: TP2 thus shade region 2
Exercise Shade the inequality ? < 2? + 4 Note that this is not in the form ?? + ?? < ? , so we need to rearrange it first. 2? + ? < 4 Solution:
System of Linear Inequalities: A system of linear inequalities is a collection of two or more linear inequalities. To graph a system of two inequalities we locate all points whose coordinates satisfy each of the linear inequalities of the system. For a 2x2 system; we have 4 regions
Example Graph the following system of linear inequalities 2? 3? > 2 ? + 4? 6 Solution: For 2? 3? > 2 , x int = 1 and y int = -0.67 (dash lines) For ? + 4? 6 , x int=6 and y int= 1.5 (solid line) T1=(0,0) T2=(2,3) T3=(6,2) T4=(3,0) (YOUR CHOICES CAN BE DIFFERENT)
2? 3? > 2.eq 1 ? + 4? 6 ..eq 2 TP equation LHS Inequality RHS (0,0) 1 0 > 2 NO 2 0 6 OKAY (2,3) 1 -5 > 2 NO 2 14 6 NO (6,2) 1 6 > 2 OKAY 2 14 6 NO (3,0) 1 6 > 2 OKAY 2 3 6 OKAY Since TP in region 4 satisfies both inequalities shade REGION 4
Exercise Graph the following system of linear inequalities ? + 2? < 2 ? 1 Solution: For ? + 2? < 2 , x int = -2 and y int =-1 (dash lines) For ? 1 , this is a horizontal line with y int=1(solid line)
Pick Test points in these 4 regions and check which TP satisfies both the equations
