5.04 Systems of equations using elimination method
Ideas
Elimination method
We've already had a look at two ways of solving a system of equations: graphically and by the substitution method .
The last method of solving a system of equations that we'll be learning is the elimination method. It works by adding or subtracting equations from one another in order to eliminate one variable, leaving us with only the other variable to solve for.
We should use addition when the coefficients of the variable we want to eliminate are equal and opposite in sign, and subtraction when they're equal and the same sign. But what happens when they don't have the same coefficients at all?
When we don't have the same value coefficients for like terms we want to eliminate, we can multiply or divide the whole equation by a constant until it gets to the coefficient that we want.
Let's look at an example where we solve the following system of equations:2x - y = 1 \\ 5x + y = 2
Because the variable y has the same coefficient and opposite signs in the two equatons, we can add our two equations together to eliminate y and sovle for x:\begin{array}{c} & &2x &- &y &= &1 \\ &+ &5x &+ &y &= &2 \\ \hline \\ & &7x & & &= &3 \\ & &x & & &= &\dfrac{3}{7} \end{array}
Now, substitute the x-value into one of the equations from the system:
| \displaystyle 5x + y | \displaystyle = | \displaystyle 2 | Seconds equation |
| \displaystyle 5 \times \dfrac{3}{7} + y | \displaystyle = | \displaystyle 2 | Substitute the value of x |
| \displaystyle \dfrac{15}{7} + y | \displaystyle = | \displaystyle 2 | Evaluate the multiplication |
| \displaystyle \dfrac{15}{7}+y-\dfrac{15}{7} | \displaystyle = | \displaystyle 2- \dfrac{15}{7} | Subtract \dfrac{15}{7} from both sides |
| \displaystyle y | \displaystyle = | \displaystyle -\dfrac{1}{7} | Evaluate |
The solution of the system is \left(\dfrac{3}{7}, -\dfrac{1}{7}\right).
When solving systems of equations with eliminaton, it can be helpful to do the following:
- Being organized is key. Name your equations by writing (1) & (2) next to them, and whenever you create new equations out of one or both of them, you can name them (3), \, (4) , etc. When adding two equations you can write it in shorthand as (3) + (1), and similarly for all the other operations. This helps to keep your ideas in order and not get confused.
- When dealing with equations with big numbers, see if you can simplify them before beginning to solve them. eg. 2x - 4 = 6y can be simplified to x - 2 = 3y without changing the values of x & y.
- If you have time, remember to check your answers by substituting your x and y values back into the original two equations.
Examples
Example 1
How would we eliminate a variable in the following system of equations?
2x + 3y = 4 \\ 5x - y = 3
Example 2
Use the elimination method by subtraction to solve for x and y.
Equation 1: 3x - 2y = -3
Equation 2: 3x + 5y = 39
First solve for y.
Now solve for x.
Example 3
Use the elimination method to solve for x and y.
Equation 1: -6x - 2y = 46
Equation 2: -30x - 6y = 246
First solve for x.
Now solve for y.
Elimination method works by adding or subtracting equations from one another to eliminate one variable, leaving us with the other variable to solve on its own.
We should use addition when the coefficients of the variable we want to eliminate are equal and opposite in sign, and subtraction when they're equal and the same sign.
When we don't have the same value coefficients for like terms we want to eliminate, we can multiply or divide the whole equation by a constant to get the coefficient that we want.