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Example system: 4x + 2y = 20 and 5y + 4 = 8x
First, pick one equation and solve for one variable as far as possible (we will solve for the y variable).
4x + 2y = 20 (divide everything by 2)
2x + y = 10 (subtract 2x from each side)
y = 10 - 2x (stop when you have a variable all on its own)
Then, plug this new equation into the other one for the variable you solved for (we solved for the y variable earlier, so now we will solve for the x variable).
5y + 4 = 8x
5 (10 - 2x) + 4 = 8x (substitution)
50 - 10x + 4 = 8x (distribute/simplify)
54 - 10x = 8x (simplify)
54 = 18x (add 10x to both sides)
x = 3 (divide both sides by 18, stop here)
Then, plug x = 3 back into either equation to solve for the value of y (before we solved for y, but it stayed in the form of an equation).
4x + 2y = 20
4(3) + 2y = 20 (substitute)
12 + 2y = 20 (simplify)
6 + y = 10 (divide both sides by 2)
y = 4 (subtract 6 from both sides, stop here)
Now, you know that x = 3 and y = 4. This means that this is the intersection point of these two equations if you were to graph them (3, 4). If solve the system using the method above but end up with an answer that doesn't make sense (such as 3=7), the lines do not intersect and there is therefore no answer.
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