Linear equations are equations where the highest power of the variable is one. In the GMAT problem we are exploring, we see a direct relationship between the number of stories written and the payment. The formula for payment after a certain number of stories involves linear components like \(a\), \(b\), and \(n\). For example, the equation for total earnings for \(n + a\) stories was derived as: \(na + a^2 + ab\). This kind of equation is linear because each term combines coefficients and variables linearly, without any variable being squared or having a higher power.
Understanding linear equations is crucial since they form the foundation for more complex algebraic manipulations and problem-solving techniques. They are easy to solve and interpret, providing clear answers in many practical scenarios.
To solve linear equations, remember to:
- Isolate the variable by combining like terms on each side of the equation.
- Use basic operations like addition, subtraction, multiplication, and division to simplify the equation.
- Always keep the equation balanced by performing the same operation on both sides.
- Check your solutions by substituting them back into the original equation.