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5.3 – Two-Variable Systems: Elimination

Key Terms

  • Elimination – The adding or subtracting of two equations to remove one of the variables in a system of equations.

 


Review

  • Three Forms of Linear Equations
    • Slope-Intercept Form: y=mx+b
    • Point-Slope Form: y-y_1=m(x-x_1)
    • Standard Form: Ax+By=C
  • To solve a system of equations, you have already learned how to graph and use substitution.
    • Graphing
      • Use slope-intercept form to graph lines and see where they meet.
      • Write your answer as an ordered pair (x, y)
    • Substitution
      • Solve one equation for a variable, and substitute that variable’s expression into the other equation.
      • Solve for both variables and write your answer as an ordered pair (x, y)

 

  • Solving for a Variable
    • When you “solve for x,” you find the value of x
    • When you “solve for y,” you find the value of y

 


Notes

  • Methods for Solving Systems of Linear Equations
    1. Use graphing
    2. Use substitution
    3. Use elimination

 

  • Elimination
    • The elimination method is useful when you can eliminate one of the variable terms from an equation by adding or subtracting another equation.
    • The elimination method works best if the equations are in standard form: Ax + By = C.

 

  • Rules for Using Elimination
    • If the variables have equal coefficients, use subtraction.
      • Subtracting an equation is like multiplying every term in the subtracted equation by -1, and then adding the two equations.
    • If the variables have opposite coefficients, use addition.

 

  • Steps for Solving with Elimination
    • 1. Find the variable with equal or opposite coefficients.
    • 2. If equal, subtract the equations. If opposite, add the equations.
    • 3. Solve the resulting one-variable equation.
    • 4. Substitute the known value into the other equation and solve for the remaining variable.

 

  • Example: solve using elimination
    • 15x + 7y = 4
    • 5x + 7y = 2

Alg1B 5.3 Elimination Ex

 

    • So, subtracting the y-terms is going to help us eliminate the y’s.

Alg1B 5.3 Elimination Ex Distribution

Alg1B 5.3 Elimination Ex Distribution Contd

    • Substitute x=\frac{1}{5} into either of the original equations to solve for y.
      • 1st equation: 15(\frac{1}{5})+7y=4
      • Simplify the fraction: \frac{15}{5}+7y=4
      • Divide the fraction: 3+7y=4
      • Subtract 3 from both sides: 3+7y=4
      • Divide by 7 on both sides: \frac{7y}{7}=\frac{1}{7}
        • Answer: y=\frac{1}{7}
        • Check your answer using substitution

Alg1B 5.3 Elimination Ex Distribution Check

 


  • Coefficients that Do Not Match
    • If none of the variable terms have equal or opposite coefficients, multiply one or both equations by a number in order to create equal or opposite coefficients.
    • See Examples 2, 3, and 4 (below)

 

 


More Examples

  • Ex 1. Use elimination to solve
    • 3x – 4y = -30
    • 3x + 2y = 6
      • Step 1: x is the variable with equal coefficients
      • Step 2: Multiply the 2nd equation by -1 (using distribution)
      • Step 3: Combine (add) the equations together
      • Step 4: Since the x’s were eliminated, solve for y
      • Step 5: Substitute y = 6 back into either of the original equations to solve for x
      • Step 6: List the answer as an ordered pair (x, y)

Alg1B 5.3 Ex01

 

  • Ex 2. Use elimination to solve
    • 2x – 3y = 11
    • 5x + 4y = 16
      • Step 1: You will have to distribute a number into EACH equation to eliminate the x or y terms.  For this example, we’ll choose x.
      • Step 2: What is a multiple of 2 and 5?  10!  Let’s make 10x and -10x so we can eliminate the x-terms!
      • Step 3: Multiply the top equation by positive 5 and multiply the bottom equation by -2.
      • Step 4: Combine (add) the equations and solve for y.
      • Step 5: Substitute y into either of the original equations to solve for x.
      • Step 6: Write your answer as an ordered pair (x, y)

Alg1B 5.3 Ex04

 

  • Ex 3. What happens when  you eliminate both x and y?
    • 6x + 9y = 15
    • 2x + 3y = 5
      • Step 1: Pick a variable to eliminate.  Let’s choose x for this example.
      • Step 2: What is a multiple of 6 and 2?  6!  Let’s make 6x and -6x so we can eliminate the x-terms!
      • Step 3: Distribute -3 into the 2nd equation.
      • Step 4: Combine (add) both equations.
      • Step 5: Uh, oh!  Both x and y were eliminated!
      • Step 6: Since 0 = 0 (same answer on both sides), we have two lines on top of one another (same slope, same y-intercept)!  They have infinitely many solutions!  EVERY number you try for x or y will check out!

Alg1B 5.3 Ex02

 

  • Ex 4. What happens when you eliminate both x and y?
    • 6x + 9y = 15
    • 2x + 3y = 6
      • Step 1: Pick a variable to eliminate. Let’s choose x for this example.
      • Step 2: What is a multiple of 6 and 2? 6! Let’s make 6x and -6x so we can eliminate the x-terms!
      • Step 3: Distribute -3 into the 2nd equation.
      • Step 4: Combine (add) both equations.
      • Step 5: Uh, oh! Both x and y were eliminated!
      • Step 6: However, since 0 ≠ -3 (impossible), we have two parallel lines (same slope, different y-intercept)! They have NO solutions as they will never intersect!

Alg1B 5.3 Ex03

 


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