SLE - Solve Systems of Equations Using Elimination Lesson

Solve Systems of Equations Using Elimination

The final way we will learn how to solve systems of equations is by elimination. When we use elimination, we want to create a system of equations in which one variable has coefficients that are opposites. Opposites are two numbers that have the same value, but opposite signs. When you use elimination you ADD the two equations together:

2x + y = 9

3x - y = 16

Add the two equations together.

2x + y = 9

+3x - y = 16

5x = 25

x = 5

So now that we know that x = 5, we can substitute that back in to either equation to find y.

2(5) + y = 9

10 + y = 9

y = -1

So, the solution to this system of equations is (5, -1)!

The “Adding and Subtracting” Elimination method allows you to eliminate one variable by adding or subtracting the two equations together. It works like this:

  •  When two terms have the same coefficient, (like 2y and 2y) then SUBTRACT the two equations
  •  When two terms have opposite coefficients (like 5y and-5y) then ADD the two equations

This will allow you to eliminate one of the variables and solve for the remaining variable.

Example

Image of working out problems on a chalkboard.

Solving Systems with Multiplication

The Multiplication Method allows you to multiply one or both equations in your system by a number or numbers that you choose so you can set your system up to solve using the adding or subtracting method of elimination.

It works like this: Suppose you have a system of equations where all the like terms are in alignment like this one:

3y + 8x = 1

7y + 4x = 17

Since the like terms are not the same or opposite, you will need to multiply one or both equations by a number that gives you the same or opposite term so you can use the add or subtract method.    It does not matter if you change the first or the second equation – as long as you remember that you have to multiply the entire equation by the number you choose.  In this case, you could easily multiply the second equation by 2 and subtract the equations.

2(7y + 4x = 17) = 14y + 8x = 34  Now, using the new equation,  line up your equations and subtract. 

3y + 8x = 1

14y + 8x = 34

-11y = -33    y = 3 Substitute 3 for y in one of the equations and solve for x.

Or if you choose to multiply the second equation by -2 and add the equations:

    3y+ 8x = 1

-14y- 8x = -34

-11y = -33  y = 3 Substitute 3 for y in one of the equations and solve for x.

There will be times you will have to multiply both equations so you can make sure one pair of terms is either alike or opposite.

Solving Systems of Equations Using Elimination Practice

Solve these systems of equations using elimination.

  1. 16x - 10y = 10 and -8x - 6y = 6
  2. x + 2y = 3 and -3x - 6y = 1
  3. -3x + y = 6 and 9x - 3y = -18
  4. -x - 7y = 14 and -4x - 14y = 28

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

Application of the Methods

Now that you have learned three methods for solving systems, you can apply your method of choice to solving real world problems.  Remember that some methods work better than others, so try to make sure you use the one that requires less steps. 

Look at the following problem – Do you see how substitution would be the best fit for this problem? 

Nick has 100 trading cards and Jeremy has 20. Every day Nick gives Jeremy one card. Use the equations  c = 𝟏𝟎𝟎 − d and 𝒄 = 𝟐𝟎 + 𝒅 to model this situation. Solve the system by substitution. Explain what the solution means in this situation.

  • Set the equations equal to each other (c = c) and substitute:  100-d = 20+ d
  • Add d to each side to get 100 = 20 + 2d
  • Subtract 20 from each side to get 80 = 2d
  • Divide by 2 to get d = 40
  • Substitute 40 for d in either equation and solve for c.

(Did you get (60,40)?  Can you see that it means they will both have 60 cards in 40 days?

Look at the following problem –  Do you see how elimination works best for solving this system?

The perimeter of the triangle is 108 units. The perimeter of the rectangle is 36 units. Write and solve a system of linear equations to find the values of 𝒙 and 𝒚.

Picture of a triangle and square with side lengths.

Perimeter of a Triangle: 6x + 6x + 24y = 108; 12x + 24y = 108

Perimeter of a Rectangle: 2(2x) + 2(4y) = 36; 4x + 8y = 36

The system is: 12x + 24y = 108 (perimeter of a triangle) and 4x + 8y = 36 (perimeter of a rectangle)

Multiply the perimeter of a rectangle by 3 in order to be able to use the elimination method.

That would be 12x + 24y = 108.

Then subtract:

12x + 24y = 108

-12x -24y = -108

0 = 0

What does this mean?  This system of linear equations has infinitely many solutions!

Practice

If you would like to practice more problems and check your work, click here. Links to an external site. Make sure you check your answers and look at the course resources or ask your teacher if there are any problems you do not understand. Links to an external site.

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