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- 1. Objective The student will be able to: solve systems of equations using substitution.
- 2. Solving Systems of Equations You can solve a system of equations using different methods. The idea is to determine which method is easiest for that particular problem. These notes show how to solve the system algebraically using SUBSTITUTION.
- 3. Solving a system of equations by substitution Step 1: Solve an equation Pick the easier equation. The goal for one variable. is to get y= ; x= ; a= ; etc. Put the equation solved in Step 1 Step 2: Substitute into the other equation. Step 3: Solve the equation. Get the variable by itself. Step 4: Plug back in to find Substitute the value of the variable the other variable. into the equation. Step 5: Check your Substitute your ordered pair into solution. BOTH equations.
- 4. 1) Solve the system using substitution x+y=5 y=3+x Step 1: Solve an equation The second equation is for one variable. already solved for y! Step 2: Substitute x+y=5 x + (3 + x) = 5 2x + 3 = 5 Step 3: Solve the equation. 2x = 2 x=1
- 5. 1) Solve the system using substitution x+y=5 y=3+x x+y=5 Step 4: Plug back in to find the other variable. (1) + y = 5 y=4 (1, 4) Step 5: Check your solution. (1) + (4) = 5 (4) = 3 + (1) The solution is (1, 4). What do you think the answer would be if you graphed the two equations?
- 6. Which answer checks correctly? 3x – y = 4 x = 4y - 171. (2, 2)2. (5, 3)3. (3, 5)4. (3, -5)
- 7. 2) Solve the system using substitution 3y + x = 7 4x – 2y = 0 It is easiest to solve the first equation for x. Step 1: Solve an equation for one variable. 3y + x = 7 -3y -3y x = -3y + 7 Step 2: Substitute 4x – 2y = 0 4(-3y + 7) – 2y = 0
- 8. 2) Solve the system using substitution 3y + x = 7 4x – 2y = 0 -12y + 28 – 2y = 0 Step 3: Solve the equation. -14y + 28 = 0 -14y = -28 y=2 4x – 2y = 0 4x – 2(2) = 0 Step 4: Plug back in to find the other variable. 4x – 4 = 0 4x = 4 x=1
- 9. 2) Solve the system using substitution 3y + x = 7 4x – 2y = 0 Step 5: Check your (1, 2) solution. 3(2) + (1) = 7 4(1) – 2(2) = 0 When is solving systems by substitution easier to do than graphing? When only one of the equations has a variable already isolated (like in example #1).
- 10. If you solved the first equation for x, whatwould be substituted into the bottom equation. 2x + 4y = 4 3x + 2y = 22 1. -4y + 4 2. -2y + 2 3. -2x + 4 4. -2y+ 22
- 11. 3) Solve the system using substitution x=3–y x+y=7 Step 1: Solve an equation The first equation is for one variable. already solved for x! Step 2: Substitute x+y=7 (3 – y) + y = 7 3=7 Step 3: Solve the equation. The variables were eliminated!! This is a special case. Does 3 = 7? FALSE! When the result is FALSE, the answer is NO SOLUTIONS.
- 12. 3) Solve the system using substitution 2x + y = 4 4x + 2y = 8 Step 1: Solve an equation The first equation is for one variable. easiest to solved for y! y = -2x + 4 4x + 2y = 8 Step 2: Substitute 4x + 2(-2x + 4) = 8 4x – 4x + 8 = 8 Step 3: Solve the equation. 8=8 This is also a special case. Does 8 = 8? TRUE! When the result is TRUE, the answer is INFINITELY MANY SOLUTIONS.
- 13. What does it mean if the result is “TRUE”? 1. The lines intersect 2. The lines are parallel 3. The lines are coinciding 4. The lines reciprocate 5. I can spell my name

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