Solving Linear Equations in One Variable
Solving a linear equation means doing identical operations to both sides until x stands alone, since equal quantities stay equal under the same change. Every step undoes whatever was done to x, in reverse order.
An equation is a balance: both sides have the same value. If you add, subtract, multiply, or divide both sides by the same thing, that balance holds, so the new equation has the same solution as the old one. Solving for x is just a sequence of these legal moves chosen to peel away everything attached to x, until x sits alone on one side.
The order matters, and it runs backward from how the expression was built. If x was multiplied, then had something added, you undo the addition first, then the multiplication. Take 3(x - 2) = 5x + 8. Distribute to get 3x - 6 = 5x + 8. Gather x-terms on one side: subtract 3x from both sides to get -6 = 2x + 8, then subtract 8: -14 = 2x, so x = -7. Check it: 3(-7-2) = -27, and 5(-7)+8 = -27. Balanced.
Equations with fractions work the same way, but clearing denominators first makes the algebra cleaner. For (x+4)/3 - (x-1)/2 = 1, multiply every term by 6, the least common denominator: 2(x+4) - 3(x-1) = 6. That gives 2x + 8 - 3x + 3 = 6, so -x + 11 = 6, meaning x = 5. This topic sits right before quadratics and rational equations in the same unit, which reuse this exact isolating logic but add extra steps like factoring or checking for extraneous roots.
Key terms
- Linear equation
- An equation where the variable appears only to the first power, with no exponents or roots on it.
- Isolating the variable
- Performing inverse operations on both sides until the variable stands alone on one side.
- Extraneous step check
- Verifying a solution by substituting it back into the original equation to confirm both sides match.
- No solution
- The case where no value of the variable makes the equation true, shown when variables cancel leaving a false statement.
- Identity
- An equation true for every value of the variable, shown when variables cancel leaving a true statement.
Watch it explained
- How it worksHow does isolating x undo each operation in the reverse order it was done?
- Worked exampleWalk me through solving 3(x - 2) = 5x + 8 for x in under 40 seconds.
- Worked exampleFind x when (x + 4)/3 minus (x - 1)/2 equals 1, by clearing denominators first.
- Common mistakeWhy isn't an equation with no solution the same as an equation whose answer is zero?
Check yourself
End of topic.