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## Solving Linear Systems by Substitution

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**Solving Linear Systems by Substitution**AII, 2.0: Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices. LA, 6.0: Students demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions**Solving Linear Systems by Substitution**Objectives Key Words Solve a system of linear equations in two variables by substitution Substitution Method**Prerequisite Check:If you do not know, you need to let me**know Solve the equation for the indicated variable Evaluate For For For For**Steps:**• Solve one equation for one of its variables • Substitute the expression from Step 1 into the other equation and solve for the other variable • Substitute the value from Step 2 into the revised equation from Step 1 and solve • Check the solution in each of the original equations Solving a Linear System by Substitution Step-by-Step**–**4x y 6 = y 2x = SOLUTION Substitute 2x for y in Equation 2. Solve for x. – Write Equation 2. 4x y 6 = – Substitute 2x for y. 4x 2x 6 = 2x 6 Combine like terms. = Solve for x. x 3 = Example 1 Use Substitution Solve the system using substitution. Equation 1 Equation 2**Write Equation 1.**Substitute 3 for x. ( ) 3, 6 Solve for y. y 2x = y 2 = y 6 = ANSWER The solution is . ( ) 3 Example 1 Use Substitution Substitute 3 for x in Equation 1. Solve for y. You can check your answer by substituting 3 for x and 6 for y in both equations.**SOLUTION**STEP 1 Solve Equation 2 for x. Choose Equation 2 because the coefficient of x is 1. – – x 2y 3 = – x 2y 3 Solve for x to get revised Equation 2. = Example 2 Use Substitution Solve the system using substitution. Equation 1 + 3x 2y 7 = – – Equation 2 x 2y 3 =**Write Equation 1.**+ 3x 2y 7 = Substitute 2y3 for x. – + 3 2y 7 = Use the distributive property. + 6y 9 – 2y 7 = Combine like terms. ( ) 2y 3 – 8y 9 – 7 = Add 9 to each side. 16 8y = Solve for y. y 2 = Example 2 Use Substitution STEP 2 Substitute2y3 for x in Equation 1. Solve for y. –**–**Write revised Equation 2. x 2y 3 = Substitute 2 for y. – x 2 3 = Simplify. x 1 = ( ) 2 STEP 4 Check by substituting 1 for x and 2 for y in the original equations. Equation 1 Equation 2 – – + 3x 2y 7 x 2y 3 = = Write original equations. Example 2 Use Substitution STEP 3 Substitute2 for y in revised Equation 2. Solve for x.**–**3 + 2 7 1 2 Simplify. – 1 4 3 + 4 7 Solution checks. 7 7 = = ( ) 1, 2 ANSWER The solution is . ( ( ( ) ) ) 1 2 2 ? ? ? ? – – – – 3 3 3 3 = = = = Example 2 Use Substitution Substitute for x and y.**Checkpoint**ANSWER + 2x y 3 = . Sample answer: The second equation; this equation had 0 on one side and the coefficient of y was 1, so I solved for y to obtain y3x. ( ) – 3 , 9 + 3x y 0 = – = Use Substitution Solve the system using substitution. Tell which equation you chose to solve and use for the substitution. Explain. 1.**Checkpoint**+ 2x 3y 4 = + x 2y 1 = ANSWER . Sample answer: The second equation; the coefficient of x in this equation was 1, so solving for x gave a result that did not involve any fractions. ( ) – 5, 2 Use Substitution Solve the system using substitution. Tell which equation you chose to solve and use for the substitution. Explain. 2.**Checkpoint**– 3x y 5 = + 4x 2y 10 = ANSWER . Sample answer: The first equation; the coefficient of y in this equation was 1, so solving for y gave a result that did not involve any fractions. ( ) 2, 1 – Use Substitution Solve the system using substitution. Tell which equation you chose to solve and use for the substitution. Explain. 3.**Conclusions**Summary Assignment • How do you solve a system of linear equations algebraically? • To use the substitution method sole for one variable and then substitute the value found into the other equations and solve for the second variable. Pg135#(12,18,22,30,31) Due by the end of the class.