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6-2 Solving Systems by Substitution Activating Prior Knowledge Simplify each expression. 1. 2(x – 5) 2x – 10 2. 12 – 3(x + 1) 9 – 3x What is the difference between simplifying and solving? Tie to LO 6-2 Solving Systems by Substitution Learning Objective Today, we will solve systems of equations by substitution. CFU 6-2 Solving Systems by Substitution Concept Development Review – Notes #1 A system of linear equations is a set of two or more linear equations containing two or more variables. CFU 6-2 Solving Systems by Substitution Concept Development Review – Notes #2 & 3 2.A solution of a system of linear equations with two variables is an ordered pair that satisfies each equation in the system. 3. So, if an ordered pair is a solution, it will make both equations true. CFU 6-2 Solving Systems by Substitution Concept Development Review Notes #4 & 5 4. Sometimes it is difficult to identify the exact solution to a system by graphing. 5. In this case, you can use a method called substitution. CFU 6-2 Solving Systems by Substitution Concept Development 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 Concept Development Helpful Hint You can substitute the value of one variable into either of the original equations to find the value of the other variable. CFU 6-2 Solving Systems by Substitution Skill Development – Notes #6 Solve the system by substitution. y = 3x y=x–2 Step 1 y = 3x Both equations are solved for y. y=x–2 Step 2 y = x – 2 Substitute 3x for y in the second 3x = x – 2 equation. Step 3 –x –x Now solve this equation for x. 2x = –2 Subtract x from both sides and 2x = –2 then divide by 2. 2 2 x = –1 CFU 6-2 Solving Systems by Substitution Skill Development – Cont. Notes #6 Solve the system by substitution. Step 4 Step 5 y = 3x 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 = 3x y=x–2 –3 3(–1) –3 –1 – 2 –3 –3 –3 –3 CFU 6-2 Solving Systems by Substitution Skill Development – Notes #7 Solve the system by substitution. y=x+1 4x + y = 6 The first equation is solved for y. Step 1 y = x + 1 Write the second equation. Step 2 4x + y = 6 Substitute x + 1 for y in the 4x + (x + 1) = 6 second equation. 5x + 1 = 6 Simplify. Solve for x. Step 3 –1 –1 Subtract 1 from both sides. 5x = 5 5x = 5 Divide both sides by 5. 5 5 x=1 CFU 6-2 Solving Systems by Substitution Skill Development – Cont. Notes #7 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 #8 Solve the system by substitution. x + 2y = –1 x–y=5 Step 1 x + 2y = –1 Solve the first equation for x by subtracting 2y from both sides. −2y −2y x = –2y – 1 Step 2 x – y = 5 (–2y – 1) – y = 5 –3y – 1 = 5 Substitute –2y – 1 for x in the second equation. Simplify. CFU 6-2 Solving Systems by Substitution Skill Development – Cont. Notes #8 Step 3 –3y – 1 = 5 +1 +1 –3y = 6 –3y = 6 –3 –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 Concept Development – Notes #9 & 10 9. Sometimes you substitute an expression for a variable that has a coefficient. 10. When solving for the second variable in this situation, you can use the Distributive Property. CFU 6-2 Solving Systems by Substitution Concept Development Caution When you solve one equation for a variable, you must substitute the value or expression into the other original equation, not the one that had just been solved. 6-2 Solving Systems by Substitution Skill Development – Notes #11 Solve y + 6x = 11 by substitution. 3x + 2y = –5 Solve the first equation for y Step 1 y + 6x = 11 by subtracting 6x from each – 6x – 6x side. y = –6x + 11 Step 2 3x + 2y = –5 Substitute –6x + 11 for y in the second equation. 3x + 2(–6x + 11) = –5 3x + 2(–6x + 11) = –5 Distribute 2 to the expression in parentheses. CFU 6-2 Solving Systems by Substitution Skill Development – Cont. Notes #11 Solve y + 6x = 11 by substitution. 3x + 2y = –5 Simplify. Solve for x. Step 3 3x + 2(–6x) + 2(11) = –5 3x – 12x + 22 = –5 –9x + 22 = –5 – 22 –22 Subtract 22 from –9x = –27 both sides. –9x = –27 Divide both sides by –9. –9 –9 x=3 CFU 6-2 Solving Systems by Substitution Skill Development – Cont. Notes #11 Solve Step 4 y + 6x = 11 by substitution. 3x + 2y = –5 y + 6x = 11 y + 6(3) = 11 y + 18 = 11 –18 –18 Write one of the original equations. Substitute 3 for x. Simplify. Subtract 18 from each side. y = –7 Step 5 (3, –7) Write the solution as an ordered pair. CFU 6-2 Solving Systems by Substitution Skill Development – Notes #12 Solve –2x + y = 8 3x + 2y = 9 by substitution. Check your answer. Step 1 –2x + y = 8 + 2x +2x y = 2x + 8 Solve the first equation for y by adding 2x to each side. Step 2 3x + 2y = 9 3x + 2(2x + 8) = 9 Substitute 2x + 8 for y in the second equation. 3x + 2(2x + 8) = 9 Distribute 2 to the expression in parentheses. CFU 6-2 Solving Systems by Substitution Skill Development – Cont. Notes #12 Solve –2x + y = 8 3x + 2y = 9 by substitution. Check your answer. Step 3 3x + 2(2x) + 2(8) = 9 3x + 4x + 16 = 9 7x + 16 = 9 –16 –16 7x = –7 7x = –7 7 7 x = –1 Simplify. Solve for x. Subtract 16 from both sides. Divide both sides by 7. CFU 6-2 Solving Systems by Substitution Skill Development – Cont. Notes #12 Solve Step 4 –2x + y = 8 3x + 2y = 9 –2x + y = 8 –2(–1) + y = 8 y+2=8 –2 –2 y Step 5 by substitution. Check your answer. Write one of the original equations. Substitute –1 for x. Simplify. Subtract 2 from each side. =6 (–1, 6) Write the solution as an ordered pair. CFU 6-2 Solving Systems by Substitution Skill Development – Cont. Notes #12 Solve –2x + y = 8 3x + 2y = 9 by substitution. Check your answer. Check Substitute (–1, 6) into both equations in the system. 3x + 2y = 9 –2x + y = 8 3(–1) + 2(6) –3 + 12 9 9 9 9 –2(–1) + (6) 2 + (6) 8 8 8 8 CFU 6-2 Solving Systems by Substitution Closure 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 the system by substitution. x = 6y – 11 3x – 2y = –1 CFU