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5.2 – Two-Variable Systems: Substitution

Key Terms

  • Expression – A combination of numbers, variables, and operations that does not contain an equals sign or inequality sign.
    • An expression that includes variables is often called an algebraic expression.
    • Ex. 4 + 3x
  • Substitution – The process of replacing a variable with a number or expression.
    • Ex. 2x + 8, when you substitute x = 4: 2(4) + 8

 


Review

  • Graphing is ONE way to locate the point where two lines meet (the solution)
    • Ex. Graph both lines to see where they meet
      • y = 2x + 1 (red line)
      • y = -3x + 6 (blue line)
      • They meet at point: (1, 3)

Alg1B 5.2 Lines Meet Ex2

 

  • Solutions
    • If you get an identity, such as 0 = 0 or 6 = 6, the system has an infinite number of solutions.
      • The lines are on top of one another, and have the same slope and the same y-intercept.
    • If you get a contradiction, such as 0 = 7 or 6 = -6, then the system has no solution.
      • The lines are parallel and will never cross.  They have the same slope and different y-intercepts.

 


Notes

  • Substitution
    • To solve a system of linear equations using the substitution method, you replace a variable with an equal value or expression.
    • Write the answer as an ordered pair: (x, y).  x is always first, then y.

Alg1B 5.2 Steps for Substitution

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  • Example

Alg1B 5.2 Substitution

Answer: (2, 3)

 

  • Verifying the Solutions (from example above)
    • There are two ways to verify that your solutions are correct
      1. Graph the equations and see where they meet
        1. Green Line: x = 2
        2. Red Line: 2x +3y = 13Alg1B 5.2 Substitution Check Graph
      2. Substitute back into both equations
        1. When you substitute y = 3 into the original equations, they both check out!Alg1B 5.2 Substitution Check

 


  • Watch and copy these videos to have notes on a few more complicated examples

 


More Examples

  • Ex 1.  Use the substitution method to solve the system of equations.
    • y = 8x – 3
    • x = 4
      • Setup with Substitution: y = 8(4) – 3
      • Evaluate: y = 12 – 3
      • Simplify: y = 9
        • Answer as an ordered pair (x, y): (4, 9)

 

  • Ex 2. Use the substitution method to solve the system of equations.
    • 2x + 4y = 14
    • x = 3
      • Setup with Substitution: 2(3) + 4y = 14
      • Simplify: 6 + 4y = 14
      • Subtract 6 on both sides: 4y = 8
      • Divide by 4 on both sides: y = 2
        • Answer as an ordered pair (x, y): (3, 2)

 

  • Ex 3. Use the substitution method to solve the system of equations.
    • y = 4x – 2
    • y = x + 13
      • Since both expressions equal y, set both equal to each other!
      • Setup by Substitution: 4x – 2 = x + 13
      • Add 2 to both sides: 4x = x + 15
      • Subtract x on both sides: 3x = 15
      • Divide by 3 on both sides: x = 5
      • Substitute x = 5 back into either of the original equations:
        • a. y = 4(5) – 2, which simplifies to y = 18
        • b. y = 5 + 13, which also simplifies to y = 18
          • Answer in ordered pair (x, y): (5, 18)

 

  • Ex 4. Use the substitution method to solve the system of equations.
    • 14x – 2y = 78
    • 2x – 2y = 6
      • Since we do not know what x or y equals, we need to choose an equation to simplify first.
        • The 2nd equation has smaller numbers, and all are divisible by 2; so, let’s use that one:
      • Divide all three terms by 2: \frac{2x}{2}-\frac{2y}{2}=\frac{6}{2}
      • Simplify: x – y = 3
      • Add y to both sides: x = 3 + y
      • Substitute 3 + y for x in the 1st equation: 14(3 + y) – 2y = 78
      • Distribute 14 into (3 + y): 42 + 14y – 2y = 78
      • Combine like terms (y’s): 42 + 12y = 78
      • Subtract 42 on both sides: 12y = 36
      • Divide both sides by 12: y = 3
      • Substitute y = 3 back in to either of the original equations.
        • The 2nd one is easier, as the numbers are smaller: 2x – 2(3) = 6
      • Simplify: 2x – 6 = 6
      • Add 6 to both sides: 2x = 12
      • Divide by 2 on both sides: x = 6
        • Answer in ordered pair (x, y): (6, 3)
        • You COULD also graph it and look for where the two lines meet

Alg1B 5.2 Lines Meet

 


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