Introduction
In order to solve two equations with multiple variables simultaneously, we need a way to combine the information in both equations into a single equation with only one variable.
The substitution method achieves this by setting the values of the variables to be equal across both equations and then substituting the value of one variable into the other.
Another way to achieve this is using the elimination method, which takes a combination of the two equations such that all but one variable is eliminated.
Elimination method
Combining equations relies on the fact that the left and right-hand sides of any given equation are equal. Because they are equal, we can make expressions out of the two equations that still hold true.
For example: if we know that:
Equation 1: a=b
Equation 2: c=d
Then we can combine these two equations to also know that:
Equation 1 \,+\, Equation 2: a+c=b+d
Equation 1 \,-\, Equation 2: a-c=b-d
We made these two new equations by adding or subtracting one equation from the other.
We can also take multiples of equations before combining them. As such, these equations must also hold true:
3\, \times Equation 1\,-\, 4\, \times Equation 2: 3a-4c=3b-4d
\dfrac{1}{2}\, \times Equation 1 \,+ \,3\, \times Equation 2: \dfrac{1}{2}a+3c=\dfrac{1}{2}b+3d
To solve equations simultaneously, we want to combine equations such that the new equation only has one variable.
Examples
Example 1
For two numbers, x and y:
two times the first number is added to the second number to get 27, and
the difference between five times the first number and the second number is 43.
We want to find the values of these two numbers.
Set up two equations by letting x and y be the two numbers. Use x as the first of the two numbers.
Solve for x by eliminating the y-terms.
Solve for y.
To solve equations simultaneously, we want to combine equations such that the new equation only has one variable.
If the coefficient of the variable we want to eliminate in both equations has the same value but different sign, we can add the equations to eliminate the variable.
If the coefficients of the variable we want to eliminate in both equations had the same value and the same sign, we can subtract the equations to eliminate the variable.
Elimination with multiplication
However, not every pair of equations will have coefficients that nicely match up for easy elimination of variables. In some cases, we will need to also multiply the equations before combining them.
Examples
Example 2
Given the following equations, we want to solve for x and y using the elimination method.
| Equation 1 | 12x+3y=-30 |
|---|---|
| Equation 2 | 10x-6y=-76 |
Notice that Equation 1 has a 3y term and Equation 2 has a -6y term. How can we combine the equations to eliminate the y-terms?
Solve for x by adding Equation 2 to 2 times Equation 1.
Substitute x=-4 into either of the equations and solve for y.
In some cases, we will need to multiply one or both equations before combining them to eliminate one of the variables.