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Midterm

161 Midterm Notes

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Department
Commerce
Course
COMM 161
Professor
Peter Thompson
Semester
Fall

Description
Systems of Equations Linear Functions Systems of Equations Linear Functions While a linear equation is in the form , its often more useful to remember the point-slope formula ( ) . This is particularly useful because the line goes through the point (a, b). Thus if you are given a slope and a point, you should write down the equation of the line using this formula. Solving a system of equations graphically: Notes: A unique solution is found when the # of variables = # of independent equations No solution is found when there exists a false statement in the matrix Unique Solution No Solution Infinite # of Solutions An infinite # of solutions are found when there are more parameters than non-zero rows Solving a system of equations algebraically through substitution: Steps: Example: 1. Choose the equation in which a variable can Equation 1: easily be isolated (integer coefficients, without Equation 2: fractions, etc) 2. Isolate one of the variables 1. 2. 3. Replace the value of the variable from equation 1 in equation 2 and solve for the remaining 3. 4. variable ( ) ( ) 4. Replace the given value of the variable found using equation 2 with the same variable in equation 1 and solve for the remaining variable 5. Verify your answer by plugging your value of 5. both variables in equation 2 ( ) ( ) Notes: The substitution method is good for a system of two equations with few terms. Use this method for simple linear functions where you feel you can rapidly isolate variables. 1 ExamBlitz | www.ExamBlitz.com Systems of Equations Solving a system of equations algebraically through elimination: Steps: Example: 1. Organize the equations so that the same Equation 1: variable are in the same column Equation 2: 2. Multiply the equations by a number that 1. 2. ( ) ( ) will give you the same coefficient for one variable ( ) ( ) 3. Subtract or add the equations together to eliminate one variable 4. Solve for the remaining variable 5. Replace your value for the variable you 3. 4. just solved in equation 1 and solve for the remaining variable 6. Verify your answer by plugging in your 5. ( ) 6. ( ) ( ) values for both variables in equation 2 Notes: The elimination method is good for a system of two equations even with many terms. Use this method when coefficients can easily be aligned. When it is difficult to match coefficients, multiply equation 1 by the coefficient in equation 2 and multiply equation 2 by the coefficient in equation 1 like we did in the above example (3 x equation 2; 2 x equation 1). Introduction to Matrices While other algebraic strategies may appear simpler, matrices allow for confident answers to questions containing multiple equations with multiple variables. It is important to understand how to manipulate matrices appropriately in order to answer complex questions. Matrix Size: m x n n = 3 Column Matrix: Row Matrix: Matrix with a single column Matrix with a single row Each Entry: A ij Where: m = # of rows [ ] m = 4 [ ] [ ] n = # of columns i = # of specific row j = # of specific column A 418 A 239 Matrix Addition and Subtraction When adding or subtracting matrices, you must ensure the dimensions of the participating matrices are equal. If the above criteria have been met, add/subtract corresponding entries to achieve the matrix solution. 2 ExamBlitz | www.ExamBlitz.com Systems of Equations Introduction to Matrices Example 1 Find the sum of Matrix A and Matrix B. [ ] [ ] Solution Given both A and B are 2 x 3 matrices, you may proceed by adding the corresponding entries. [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Example 2 Find the result of Matrix A + Matrix B Matrix C. [ ] [ ] [ ] Solution Given all participating matrices are 2 x 2 matrices, you may proceed by adding and subtracting the corresponding entries. [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Notes: If the participating matrices are not the same dimensions and they cannot be transposed (we will explore this concept soon) to become the same dimensions, you cannot solve for the sum (addition or subtraction) Scalar Multiplication When multiplying a matrix (A) by a constant (c), the entries of the resulting matrix are the product made up of the constant multiplied by each corresponding entry (c x A ) thus making ije resulting matrix the same dimensions. 3 ExamBlitz | www.ExamBlitz.com Systems of Equations Introduction to Matrices Example 1 Find the product of 6 and Matrix A [ ] Solution Multiply each entry of Matrix A by 6. [ ] [ ] [ ] [ ] [ ] [ ] Given Matrices A and B, find 4A 2B. Example 2 [ ] [ ] Solution Multiply each entry of Matrix A by 4 and each entry of Matrix B by 2 then perform the subtraction. [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Matrix Multiplication Matrix (m x n) x Matrix (n x p) = Matrix (m x p) In matrix multiplication, the number of rows in the first matrix (m) and the number of columns in the second matrix (p) will be the number of rows and columns in the resulting matrix (m x p). However, matrix multiplication can only take place if the number of columns in the first matrix (n) is equal to the number of rows in the second matrix (n). 4 ExamBlitz | www.ExamBlitz.com
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