OBJECTIVE 5 SOLVING SYSTEMS 5/19/2016 SOLVING SYSTEMS OF TWO EQUATIONS BY SUBSTITUTION:

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1 /9/ OBJECTIVE Sstems & Matrices SOLVING SYSTEMS OF TWO EQUATIONS BY SUBSTITUTION:. Solve one of the equations for one of the variables in terms of the other.. Substitute this epression into the nd equation, which will ield an equation in one variable.. Solve the new equation.. Use the result of step to find the other variable. SOLVING SYSTEMS Solve the sstem of linear equations:

2 /9/ SOLVING SYSTEMS Solve the sstem of linear equations:.. Not ever sstem of equations has a unique solution. There are three possibilities:. The graphs ma be the same line. We sa this is a dependent sstem.. The graphs ma be parallel but distinct lines. We sa this is an inconsistent sstem.. The graphs ma intersect in one and onl one point. The sstem is said to be consistent and independent. From Precalculus with Modeling and Visualiation rd ed. b Rockswold,, p.. SOLVING A SYSTEM OF TWO EQUATIONS BY ELIMINATION. Choose one of the variables to eliminate. Multipl each equation b a suitable factor so that the coefficients of that variable are opposite.. Add the two new equations termwise.. Solve the resulting equation for the remaining variable.. Substitute the value found in step into either of the original equations and solve for the other variable. This is also called the method of linear combinations.

3 /9/ SOLVING A SYSTEM BY ELIMINATION Solve the sstem of linear equations:... SOLVING A SYSTEM BY ELIMINATION Solve the sstem of linear equations: SOLVING A SYSTEM A jeweler is commissioned to create a piece of artwork that will weigh o and consist of % gold. She has on hand two allos that are % and % gold, respectivel. How much of each should she use?

4 /9/ SOLVING A SYSTEM An airplane fling due south from St. Louis, Missouri, to Baton Rouge, Louisiana, uses a strong, stead tailwind to complete the trip in onl. hr. On the return trip, the same wind slows the flight and it takes hr to get back. If the flight distance between these cities is 9 km, what is the cruising speed of the airplane (speed with no wind)? How fast is the wind blowing? SOLVING A SYSTEM OF EQUATIONS & UNKNOWNS. Clear each equation of fractions and put it in standard form.. Choose two of the equations & eliminate one of the variables b forming a linear combination.. Choose a different pair of equations & eliminate the same variable.. Form a sstem with the equations found in steps &. Eliminate one of the variables from this sstem using a linear combination.. Use back-substitution to solve. SOLVING A SYSTEM Solve the sstem:....

5 /9/ SOLVING A SYSTEM Solve the sstem:. SOLVING A SYSTEM A small business borrowed $, from three different lenders to epand their product line. The interest rates were %, %, and %. Find how much was borrowed at each rate if the annual interest came to $, and twice as much was borrowed at the % rate than was borrowed at the % rate. PARTIAL FRACTION DECOMPOSITION Decompose the epression into partial fractions:...

6 /9/ PARTIAL FRACTION DECOMPOSITION Write out the partial fraction template (DO NOT SOLVE!):... PROCEDURE FOR SOLVING INEQUALITIES IN TWO VARIABLES:. If possible, solve for. Graph the curve described b the equation.. If the form is shade below the line. If the form is shade above the line. Otherwise, use a test point to determine where to shade.. Use a dotted line for a strict inequalit! SOLVING INEQUALITIES Graph the solution set to the inequalit:..

7 /9/ SOLVING A SYSTEM OF INEQUALITIES Solve the sstem:.. SOLVING A SYSTEM OF INEQUALITIES As part of their retirement planning, James and Lil decide to invest up to $, in two separate investment vehicles. The first is a bond issue paing 9% and the second is a mone market certificate paing %. A financial adviser suggests the invest at least $, in the certificate and not more than $, in bonds. What various amounts can be invested in each? MATRICES A matri is a rectangular arra of elements: a a a a a a a a a A matri that has dimensions m n has m rows and n columns. A square matri has the same number of rows and columns (n n).

8 /9/ EXAMPLE Represent the sstem b an augmented matri and state the dimensions of the matri:.. ROW-ECHELON FORM A matri is in row-echelon form if it has ones down the main diagonal and eros beneath: A matri is in reduced rowechelon form if it has ones down the main diagonal and eros above and below: EXAMPLES Write the equations that correspond to the augmented matri, then solve the sstem:...

9 /9/ 9 GAUSSIAN ELIMINATION From Precalculus with Modeling and Visualiation rd ed. b Rockswold,, p. SOLVING A SYSTEM WITH A MATRIX Solve the sstem:.... SOLVING A SYSTEM WITH A MATRIX Use technolog to find the solution. Approimate values to the nearest thousandth. 9

10 /9/ From Precalculus with Modeling and Visualiation rd ed. b Rockswold,, p. ADDING MATRICES For the given matrices A and B, find each of the following:. A+B. B+A. A-B A B EXAMPLE For the given matrices A and B, find each of the following:. A+B. A. A-B A B

11 /9/ From Precalculus with Modeling and Visualiation rd ed. b Rockswold,, p. MULTIPLYING MATRICES If possible, determine the matri products AB and BA:... A B A B A B From Precalculus with Modeling and Visualiation rd ed. b Rockswold,, p.

12 /9/ INVERSES Find the inverse of A without a calculator:. A 9. A EXAMPLE Write the sstem in the form AX = B and solve the sstem b computing X = A - B From Precalculus with Modeling and Visualiation rd ed. b Rockswold,, p.

13 /9/ EXAMPLES Determine if the inverse eists b computing the determinant of the matri A:.. A A Eample: Find the minor M and the cofactor A for the matri A From Precalculus with Modeling and Visualiation rd ed. b Rockswold,, p. Eample: Find det A for A From Precalculus with Modeling and Visualiation rd ed. b Rockswold,, p.