91Ó°ÊÓ

One of the core concepts in solving linear equations is isolating the variable. This means to get the variable by itself on one side of the equation. This helps in simplifying and solving the equation.
For example, if we have:
\(x-5y=-20\),
we start by moving all other terms except those with the variable we want to isolate (here, y) to the other side of the equation. In the provided solution, we accomplish this by subtracting \(x\) from both sides:
\[x - 5y - x = -20 - x\]
Simplifying, we get:
\[-5y = -20 - x\]
This is our intermediate step to make the variable easier to solve for in the next step.

Linear equations are equations where the highest power of the variable is 1. This makes them, quite literally, 'linear'. For instance, in the equation \(x - 5y = -20\), each term is either a constant or a constant multiplied by a variable, and there are no terms like \(y^2\) or \(x^3\).
Linear equations are particularly useful because they have straightforward methods for solving, like algebraic manipulation. Solving them often involves simple steps such as addition, subtraction, multiplication, or division. It's helpful to remember that whatever operation we perform on one side of the equation, we must also perform on the other side to keep the equality true.
In graph terms, solutions to linear equations represent straight lines.

Algebraic manipulation involves using algebraic rules to simplify or rearrange equations. The goal is often to solve for a specific variable.
In our example, after isolating the term with \(y\), we then perform algebraic manipulation to solve for \(y\). We start with:
\[-5y = -20 - x\]
To isolate \(y\), we divide every term by -5:
\[\frac{-5y}{-5} = \frac{-20 - x}{-5}\]
Simplifying, we obtain:
\[y = \frac{20 + x}{5}\]
This final step includes combining like terms and simplifying fractions. Let's break it down a bit more:
* Division: We divided each side of the equation by the coefficient of \(y\), which was -5.
* Simplification: Finally, we simplified both sides to arrive at the solution.
Practicing these steps regularly helps in mastering algebraic manipulation.