Solving Equations

Solving an equation means finding the value or values that make both sides exactly equal. You are not after any number, only the ones that make the statement true. For most school work this is a linear equation like $3x - 7 = 11$; quadratic and fractional equations share the same core idea but need different methods, and there is no single universal formula for every equation.

The rule you lean on

The whole method rests on one move: do the same operation on both sides. Add $7$ to one side and you must add $7$ to the other. Symbolically, if $A = B$ then

and the new equation has exactly the same solutions as the old one.

Why doing both sides keeps the answer

An equation is a statement that two quantities are equal. As long as you apply an identical operation to both, that equality is preserved, so the set of values that satisfies it does not change; you have only rewritten the same truth in a simpler form. Touch only one side and the two expressions are no longer guaranteed equal, so the result becomes unreliable. This is also why some operations need care: dividing requires the divisor to be nonzero, and squaring both sides can introduce extraneous solutions that satisfy the new equation but not the original.

Worked example: step by step

Solve

Expand the left side:

Combine like terms:

Subtract $2x$ from both sides:

Add $1$ to both sides:

Verify by substituting $x = 10$ into the original equation. Left side: $3(10-2)+5 = 24+5 = 29$. Right side: $2\cdot 10 + 9 = 29$. Both equal $29$, so $x = 10$ is correct.

Try one yourself, then check

Solve $4(x + 1) = 3x + 11$ and verify by substitution. Once that feels routine, move to a quadratic equation, where you may get two solutions instead of one and the method changes.

Calculation traps

• Operating on only one side, which breaks the equivalence.
• Bracket errors: $3(x-2)$ is $3x - 6$, not $3x - 2$. A small slip here pulls the whole calculation off course.
• Skipping the final check. In fractional equations a denominator must never be $0$; after squaring, a value can appear that does not satisfy the original equation.

Choosing the right method

Linear equations: isolate the variable directly with add, subtract, multiply, divide. Quadratics: factoring, completing the square, or the quadratic formula (the Mitternachtsformel). Fractional equations: clear the denominators first, while keeping the domain in mind. So the first question before calculating is always: what type of equation is this? The answer sets the fastest, safest path. A clean solution is two parts: correct rearranging, then verification.