Identifying Simultaneous Linear Equations in 2 Variables

Simultaneous linear equations (SLE) are 2 unique linear equations with the same 2 variables. Before learning how to solve them, you'll need to be able to identify them. Skip to the next section to get straight to the solving.




Here are some examples to help you identify SLEs.

Example 1 (SLE)
x + y = 0 ----- (1)
2x + 3y = 0 ----- (2)

These are SLE because they have the same variables (x and y).

Example 2 (SLE)
4x + 9y = 1 ----- (1)
y = 0 ----- (2)

These are SLE although x is "missing" from the 2nd equation. "x" is visible only when you rewrite the equation as "0x + y = 0". However, to simplify things a multiple of 0 is generally not written down, hence "y = 0".

Example 3 (Not SLEs)
x + y = 5 ----- (1)
a + y = 6 ----- (2)

They do not have the exact same 2 variables (there's a, x and y).

Example 4 (Not SLE)
x + y = 1 ----- (1)
2x + 2y = 2 ----- (2)

These are not SLE, as they are in fact the same equation. If you divide both sides of the 2nd equation by 2, you'll get the 1st equation.


Solving Simultaneous Linear Equations

There are 2 main techniques to solve SLE. Work with either the elimination method OR the substitution method.

Example 1 (Elimination method)
Solve the following SLE.
3x + 2y = 18 ----- (1)
5x – 2y = 6 ----- (2)

In the elimination method, 1 of the 2 variables are eliminated by adding or subtracting the equations together. Here's how.

Step 1: Line up the variables.



You'll notice that if we add both equations, we'll eliminate "y".

Step 2: Hence, add the equations, by variable.

Doing so, we'll get:



Step 3: Substitute the solved variable into EITHER equation.

If you substitute into equation (1):



If you substitute into equation (2):





Example 2 (Substitution method)
Solve the following SLE



The substitution method is much more easily understood by example. There are 3 steps to follow.

Step 1: Take EITHER one of the equations, and express one of the variables in terms of the other variable.

I'm choosing to express "x" in terms of "y" in equation 1. You can do express "y" in terms of "x", if you wish to, and reach the same solution.



Let's call this equation (3).

Step 2: Substitute the expression in Step 1 into the other equation so that it has only 1 variable and solve for that variable.

Substitute equation (3) into equation (2):



Step 3: Substitute the result in Step 2 into Equation (1), (2) OR (3) to solve for the remaining variable.

Substituting into equation (3), for example, results in:




Tip 1: You can check your answers by substituting both variables back into any of the simultaneous linear equations, preferably another one other that you didn't substitute a variable into. If the left of the equation = to the right of the equation, then you've obtained the correct value for each variable.

Tip 2: When working with word problems, the strategy to solve it is to form 2 unique equations from the information given. If you're not sure what's are unique equations, check out the first section of this page.



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