6 2 Solving Systems by Substitution Activating Prior

6 -2 Solving Systems by Substitution Activating Prior Knowledge – Solve each equation for x. 1. y = x + 3 2. y = 3 x – 4 x=y– 3 Tie to LO

6 -2 Solving Systems by Substitution Learning Objective Today, we will solve systems of linear equations in two variables by substitution. CFU

6 -2 Solving Systems by Substitution Concept Development – Notes #1 & 2 1. Sometimes it is difficult to identify the exact solution to a system by graphing. 2. In this case, you can use a method called substitution. CFU

6 -2 Solving Systems by Substitution Concept Development – Notes #3 3. Substitution is used to reduce the system to one equation that has only one variable. CFU

6 -2 Solving Systems by Substitution Concept Development – Notes #4 Solving Systems of Equations by Substitution Step 1 Step 2 Step 3 Step 4 Step 5 Solve for one variable in at least one equation, if necessary. Substitute the resulting expression into the other equation. Solve that equation to get the value of the first variable. Substitute that value into one of the original equations and solve for the other variable. Write the values from steps 3 and 4 as an ordered pair, (x, y), and check. CFU

6 -2 Solving Systems by Substitution Skill Development – Notes #5 Solve the system by substitution. y = 3 x y=x– 2 Step 1 y = 3 x Both equations are solved for y. y=x– 2 Step 2 y = x – 2 Substitute 3 x for y in the second 3 x = x – 2 equation. Step 3 –x –x Now solve this equation for x. 2 x = – 2 Subtract x from both sides and 2 x = – 2 then divide by 2. 2 2 x = – 1 CFU

6 -2 Solving Systems by Substitution Skill Development – Cont. Notes #5 Solve the system by substitution. Step 4 Step 5 y = 3 x y = 3(– 1) y = – 3 (– 1, – 3) Write one of the original equations. Substitute – 1 for x. Write the solution as an ordered pair. Check Substitute (– 1, – 3) into both equations in the system. y = 3 x y=x– 2 – 3 3(– 1) – 3 – 1 – 2 – 3 – 3 CFU

6 -2 Solving Systems by Substitution Skill Development – Notes #6 Solve the system by substitution. y=x+1 4 x + y = 6 The first equation is solved for y. Step 1 y = x + 1 Write the second equation. Step 2 4 x + y = 6 Substitute x + 1 for y in the 4 x + (x + 1) = 6 second equation. 5 x + 1 = 6 Simplify. Solve for x. Step 3 – 1 Subtract 1 from both sides. 5 x = 5 Divide both sides by 5. 5 5 x=1 CFU

6 -2 Solving Systems by Substitution Skill Development – Cont. Notes #6 Solve the system by substitution. Step 4 Step 5 y=x+1 y=1+1 y=2 (1, 2) Write one of the original equations. Substitute 1 for x. Write the solution as an ordered pair. CFU

6 -2 Solving Systems by Substitution Skill Development – Notes #7 Solve the system by substitution. x + 2 y = – 1 x–y=5 Step 1 x + 2 y = – 1 Solve the first equation for x by subtracting 2 y from both sides. − 2 y x = – 2 y – 1 Step 2 x – y = 5 (– 2 y – 1) – y = 5 – 3 y – 1 = 5 Substitute – 2 y – 1 for x in the second equation. Simplify. CFU

6 -2 Solving Systems by Substitution Skill Development – Cont. Notes #7 Step 3 – 3 y – 1 = 5 +1 +1 – 3 y = 6 – 3 y = – 2 Step 4 x – y x – (– 2) x+2 – 2 x =5 =5 =5 – 2 =3 Step 5 (3, – 2) Solve for y. Add 1 to both sides. Divide both sides by – 3. Write one of the original equations. Substitute – 2 for y. Subtract 2 from both sides. Write the solution as an ordered pair. CFU

6 -2 Solving Systems by Substitution Skill Development – Notes #8 Solve the system by substitution. Check your answer. x = 2 y – 4 x + 8 y = 16 The first equation is solved for x. Step 1 x = 2 y – 4 Step 2 x + 8 y = 16 Substitute 2 y – 4 for x in the second equation. (2 y – 4) + 8 y = 16 Step 3 10 y – 4 +4 10 y 10 y = 16 +4 = 20 20 = 10 =2 Simplify. Then solve for y. Add 4 to both sides. Divide both sides by 10. CFU

6 -2 Solving Systems by Substitution Skill Development – Cont. Notes #8 Solve the system by substitution. Check your answer. Step 4 x + 8 y = 16 x + 8(2) = 16 x + 16 = 16 – 16 x = 0 Step 5 (0, 2) Write one of the original equations. Substitute 2 for y. Simplify. Subtract 16 from both sides. Write the solution as an ordered pair. CFU

6 -2 Solving Systems by Substitution Skill Development – Cont. Notes #8 Solve the system by substitution. Check your answer. Check Substitute (0, 2) into both equations in the system. x = 2 y – 4 x + 8 y = 16 0 + 8(2) 16 0 2(2) – 4 0 0 16 16 CFU

6 -2 Solving Systems by Substitution Closure – Notes 1. What did we learn today? 2. Why is this important to you? 3. How can you tell whether an ordered pair is a solution of a given system? 4. Solve each system by substitution. y=x-2 y = 4 x + 1 (-1, -3) CFU