Solving Linear Equations
If you've ever needed to figure out an unknown , how many hours to work to pay off a bill, what price makes two deals equal, how long until two trains meet , you were setting up a linear equation whether you knew it or not. This is the skill that makes those questions answerable.
To solve for x, get it alone on one side. Whatever you do to one side of the equation, do to the other. Add or subtract to move numbers around, multiply or divide to get rid of coefficients. Check your answer by plugging it back in.
First, what actually is a linear equation?
A linear equation has one variable , usually x , raised to the first power. No x squared, no square roots, just x on its own. Something like 3x + 5 = 20. When you graph it you get a straight line, which is where "linear" comes from.
Solving it means finding the specific value of x that makes the equation true. In 3x + 5 = 20, the answer is x = 5, because 3(5) + 5 = 20 checks out. Any other value of x and the equation falls apart.
The reason this matters in real life: any time you have a situation where something changes at a constant rate and you want to find a specific value, you're dealing with a linear equation. A plumber charging $75 plus $50/hour , how many hours until the bill hits $275? That's 75 + 50h = 275. Same structure, different numbers.
The one rule that makes all of this work
Think of an equation as a scale that's perfectly balanced. The equals sign is the middle. As long as whatever you do to the left side you also do to the right side, the scale stays balanced and the equation stays true.
That's it. That's the whole game. You're just doing the same thing to both sides over and over until x is alone.
The operations you'll use are addition, subtraction, multiplication, and division , and you always use the opposite of what's happening to x. If 7 is being added to x, you subtract 7 from both sides to cancel it. If x is being multiplied by 4, you divide both sides by 4.
Starting simple: one-step equations
These only need one operation to solve. Good for getting the pattern down before things get messier.
Two-step equations
Most equations you'll see have two things happening to x. The trick is doing them in the right order: deal with addition and subtraction first, then multiplication and division. This is basically PEMDAS in reverse.
When x appears on both sides
This is where people start to feel uncertain, but the approach is the same. You just need to collect all the x terms on one side first, then solve as usual.
Where people go wrong
The most common mistake is not applying an operation to the entire side. If you have 3x + 5 = 20 and you subtract 5, you need to subtract 5 from the whole right side , not just part of it. That sounds obvious but under pressure people do things like write 3x = 20 − 5 correctly but then also subtract 5 from 3x by accident.
The second big one is sign errors when moving negative numbers. If you have x − 7 = 12 and you add 7 to both sides, you get x = 19. Fine. But if you have −x − 7 = 12 and you add 7, you get −x = 19, not x = 19. You still need to divide by −1. Negative signs in front of the variable trip people up constantly.
The fix for both: always check your answer by plugging it back in. Takes ten seconds and tells you immediately if something went wrong. There is no excuse for skipping this step on a test.
Unlike inequalities (where dividing by a negative flips the sign), in equations you just divide normally. −3x = 12 gives x = −4. That's it. No flipping. Equations and inequalities have different rules on this point.
Practice Problems
Sources & Further Reading
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