Sunday, July 19, 2020

Solving Systems of Linear Equations - Notes

Here are the notes for this section.  We will learn how to solve systems of linear equations.  Let me know if you have any questions.

Substitution method: Step 1. Solve for one of the variables on one of the equations. Step 2. Substitute the result from Step 1 into into the other equation and solve. Step 3. Find the solution for the other variable. Example: Solve the system of equations by the substitution method. 4x + 5y = 3, 2y + x = 5. Step 1: Note that 2y + x = 5 implies that x = 5 - 2y. Step 2: Substitute x = 5 - 2y into 4x + 5y = 3 to get 4(5 - 2y) + 5y = 3 implies that 20 - 3y = 3. Step 3: Solve 20 - 3y = 3 for y to get y = 17/3. Put y = 17/3 into one of the equations in the original system. Without loss of generality, put y = 17/3 into 2y + x = 5 to get 2(17/3) + x = 5 implies that (34/3) + x = 5 implies that x = -19/3. Thus, the solution is (x, y) = (-19/3, 17/3). You can check your answer by substituting x = -19/3 and y = 17/3 back into the original system: 4(-19/3) + 5(17/3) = 3 and 2(17/3) + (-19/3) = 5, so our answer checks out and is indeed correct!

Elimination method. Step 1: Multiply both sides of one or both of the equations by a constant so that the result is the coefficients of one of the variables are opposites. Step 2: Add the two equations together. This should result in the elimination of one of the variables. Solve the equation for the variable that remains. Step 3: Substitute the solution from Step 2 into one of the original equations and solve for the other variable. Example: Solve the system 4x + 3y = -1, 2x - y = -13. Step 1: Multiply the second equation by 3 to get 6x - 3y = -39. Step 2: Add 4x + 3y = -1 to 6x - 3y = -39 to get 10 x = -40 which implies that x =-4. Step 3: Put x = -4 back into one of the equations in the original system of equations. Without loss of generality we will choose 2x - y = -13. So, put x = -4 in 2x - y = -13 to get 2(-4) - y = -13 which implies that y = 5.

Thus, the solution is (x, y) = (-4, 5). Check: Put (x, y) = (-4, 5) back into the original system to get: 4(-4) + 3(5) = -1 and 2(-4) - 5 = -13 which is correct. Thus, the answer is indeed correct. Solve applications involving systems of linear equations. Example: Mike and Ron are throwing a big party. Mike buys 20 bottles of whiskey and 600 bottles of beer. Mike spends $900. Ron buys 10 bottles of whiskey and 1200 bottles of beer. Ron spends $1350. What is the cost of each bottle of whiskey? What is the cost of each bottle of beer? Answer: Let w be the cost of a bottle of whiskey. Let b be the cost of a bottle of beer. We will solve the system: 20w + 600b = 900, 10w + 1200b = 1350. We will use elimination. Multiply 10w + 1200b = 1350 by -2 to get -20w - 2400b = -2700. Then add 20w + 600b to -20w - 2400b = -2700 to get -1800b = -1800 which implies that b = 1.

Put b = 1 into 20w + 600b = 900 (We could have also used 10w + 1200b = 1350.) to get 20w + 600 = 900, i.e. 20w = 300 which implies that w = 15. Thus, each bottle of whiskey costs $15 and each bottle of beer costs $1.

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