Systems of Linear Equations Lesson #3: Solving Systems of Linear Equations by Inspection and by Substitution Method of Inspection In some simple cases, a system of linear equations can be solved by mentally trying different values for the variables until a correct solution is reached. This is called the method of inspection and is really only practical if the equations are very simple. Class Ex. #1 The sum of two numbers is 14 and the difference between the numbers is 2. Form two equations in two variables and determine the numbers by inspection. x & y = 2 numbers x + y = 14 8 + 6 = 14 x - y = 2 8 - 6 = 2 x = 8 y = 6 Class Ex. #2 Solve the system x + 2y = 12 and x + 3y = 17 by inspection. x = 2 y = 5 x + 2(5) = 12 x + 10 = 12 x = 2 x + 3(5) = 17 x + 15 = 17 x = 2 Complete Assignment Questions #1 - #3 Systems of Linear Equations Lesson #3: Solving Linear Systems by Inspection and by Substitution Method of Substitution If the equations are too complex to be solved by inspection, then algebraic procedures such as the method of substitution and the method of elimination (next lesson) may be used. When using the method of substitution, there are four steps which are shown in the flowchart below. [Flowchart Steps] Step 1: Choose the simpler equation and express one variable in terms of the other. Solve for x or y Step 2: Substitute the expression from step 1 into the other equation. Step 3: Solve the single variable equation. Step 4: Substitute the solution from step 3 into the equation in step 1 to find the value of the other variable. Class Ex. #3 Consider the following system of equations: x + 4y = 17 2x - y = 7 a) Solve the system using the method of substitution by rewriting the first equation in the form x = .... x + 4y = 17 x = 17 - 4y 2x - y = 7 2(17 - 4y) - y = 7 34 - 8y - y = 7 34 - 9y = 7 -9y = -27 y = 3 x = 17 - 4y x = 17 - 4(3) x = 17 - 12 x = 5 (5, 3) b) Solve the system using the method of substitution by rewriting the first equation in the form y = .... x + 4y = 17 4y = 17 - x y = (17/4) - (1/4)x 2x - y = 7 2x - (17/4 - 1/4x) = 7 2x - 17/4 + 1/4x = 7 8x - 17 + x = 28 9x - 17 = 28 9x = 45 x = 5 y = 17/4 - 1/4(x) y = 17/4 - 1/4(5) y = 17/4 - 5/4 y = 12/4 y = 3 c) Which method, a) or b), was simpler? d) Verify that the solution satisfies both equations. plug in x & y , and make sure both sides are equal

Solving Systems of Linear Equations by Inspection and by Substitution

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