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Solving Systems of Equations using Substitution

Learn how to solve systems of equations using the substitution method. Find the solution by graphing and substitution. Practice with example problems.

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Solving Systems of Equations using Substitution

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  1. Objective I can solve systems of equations using substitution.

  2. Find the solution by graphing 1. 3x – 2y = 11 -x + 6y = 7 2. 6x – 3y = -15 2x + y = -3 3. -5x + y = 19 x – 7y = 3 (5,2) (-2,1) (-4,-1)

  3. Solving a system of equations by substitution Pick the easier equation. The goal is to get y= ; x= ; a= ; etc. Step 1: Solve an equation for one variable. Put the equation solved in Step 1 into the other equation. Step 2: Substitute Step 3: Solve the equation. Get the variable by itself. Substitute the value of the variable into the equation. Step 4: Plug back in to find the other variable. Substitute your ordered pair into BOTH equations. Step 5: Check your solution.

  4. 1) Solve the system using substitution x + y = 5 y = 3 + x The second equation is already solved for y! Step 1: Solve an equation for one variable. x + y = 5x + (3 + x) = 5 Step 2: Substitute 2x + 3 = 5 2x = 2 x = 1 Step 3: Solve the equation.

  5. 1) Solve the system using substitution x + y = 5 y = 3 + x x + y = 5 (1) + y = 5 y = 4 Step 4: Plug back in to find the other variable. (1, 4) (1) + (4) = 5 (4) = 3 + (1) Step 5: Check your solution. The solution is (1, 4). What do you think the answer would be if you graphed the two equations?

  6. 2) Solve the system using substitution 3y + x = 7 4x – 2y = 0 It is easiest to solve the first equation for x. 3y + x = 7 -3y -3y x = -3y + 7 Step 1: Solve an equation for one variable. 4x – 2y = 0 4(-3y + 7) – 2y = 0 Step 2: Substitute

  7. 2) Solve the system using substitution 3y + x = 7 4x – 2y = 0 -12y + 28 – 2y = 0 -14y + 28 = 0 -14y = -28 y = 2 Step 3: Solve the equation. 4x – 2y = 0 4x – 2(2) = 0 4x – 4 = 0 4x = 4 x = 1 Step 4: Plug back in to find the other variable.

  8. 2) Solve the system using substitution 3y + x = 7 4x – 2y = 0 (1, 2) 3(2) + (1) = 7 4(1) – 2(2) = 0 Step 5: Check your solution. 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).

  9. If you solved the first equation for x, what would be substituted into the bottom equation. • -4y + 4 • -2y + 2 • -2x + 4 • -2y+ 22 2x + 4y = 4 3x + 2y = 22

  10. 3) Solve the system using substitution x = 3 – y x + y = 7 The first equation is already solved for x! Step 1: Solve an equation for one variable. x + y = 7 (3 – y) + y = 7 Step 2: Substitute 3 = 7 The variables were eliminated!! This is a special case. Does 3 = 7? FALSE! Step 3: Solve the equation. When the result is FALSE, the answer is NO SOLUTIONS.

  11. 3) Solve the system using substitution 2x + y = 4 4x + 2y = 8 The first equation is easiest to solved for y! y = -2x + 4 Step 1: Solve an equation for one variable. 4x + 2y = 8 4x + 2(-2x + 4) = 8 Step 2: Substitute 4x – 4x + 8 = 8 8 = 8 This is also a special case. Does 8 = 8? TRUE! Step 3: Solve the equation. When the result is TRUE, the answer is INFINITELY MANY SOLUTIONS.

  12. What does it mean if the result is “TRUE”? • The lines intersect • The lines are parallel • The lines are coinciding • The lines reciprocate • I can spell my name

  13. Which answer checks correctly? • (2, 2) • (5, 3) • (3, 5) • (3, -5) 3x – y = 4 x = 4y - 17

  14. Closure Why do we need more than one way to solve a system? When should you use graphing? When should you use substitution? Homework: Pg 408 (14-30)

  15. Bell Work • Write an equation of the line that passes through the point (0,2) and is perpendicular to y = -4x + 6 • Write an equation of the line that passes through the point (4,5) and is perpendicular to y = 1/2x + 3 16

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