Solving a Linear Equation in One Variable
Solving a linear equation means finding the one number that makes both sides equal, by doing the same operation to both sides until x stands alone.
An equation is a balance scale: both sides weigh the same. If you add, subtract, multiply, or divide one side by something, the scale tips unless you do the exact same thing to the other side. That's why any operation applied equally to both sides never changes which number makes the equation true — you're not changing the equation, just rewriting it in a simpler form that hides the same answer.
The skill is picking the order that gets you to x fastest. Distribute any parentheses first. Then decide which side to collect the x terms on: pick the side with the larger coefficient so you subtract a smaller x term away and never end up with a negative one to divide by. Move all constants to the other side by adding or subtracting. What's left is (coefficient)x = (number), so divide both sides by the coefficient to isolate x. Checking your answer means plugging it back into the original equation, not the simplified one, to catch arithmetic slips.
This is the base case for every harder equation type on the test. Systems of equations are just two linear equations solved together, and many function and inequality questions reduce to this exact same isolate-x routine once you substitute in values or expressions. Get this fast and clean, and everything built on top of it gets easier.
Key terms
- linear equation
- An equation where the variable appears only to the first power, with no exponents, roots, or products of variables.
- coefficient
- The number multiplied by a variable, such as the 5 in 5x.
- isolate the variable
- To rearrange an equation step by step until the variable sits alone on one side.
- inverse operation
- An operation that undoes another, like subtraction undoing addition or division undoing multiplication.
Watch it explained
- How it worksHow does a balance scale show what really happens when you subtract a term from both sides?
- How it worksHow do you decide which side to collect the x terms on so you never fight a negative coefficient?
- Worked exampleFind x in 5x - 7 = 3x + 11 the way the SAT wants it done in under 40 seconds.
- Worked exampleWalk me through solving 3(x - 2) = 5x + 8 for x, distributing first and then checking the answer.
- Common mistakeWhy isn't it legal to divide both sides of 4x = 4x + 3 by x to make the problem simpler?
Check yourself
End of topic.