91Ó°ÊÓ

In algebra, equations often have constants on both sides. A constant is a fixed number that doesn't change. For example, in the equation \(9x - 3 = 60\), both \textquoteleft\(-3\)\textquoteright on the left side and \textquoteleft\(60\)\textquoteright on the right side are constants.
To solve such equations, we need to isolate the variable by removing these constants systematically. The first step involves making sure that the variable terms are on one side of the equation, and all constants are on the other.
In our example, we do this by adding 3 to both sides:
\[ 9x - 3 + 3 = 60 + 3 \] Simplifying, this gives us:
\[ 9x = 63 \]

Understanding how to work with constants on both sides is vital, as it sets the foundation for isolating variables and solving the equation.

Isolation of variables is a fundamental concept in solving algebraic equations. It involves manipulating the equation so that the variable of interest is by itself on one side of the equation.
This is typically done through a series of inverse operations. In the equation \(9x = 63\), our goal is to get \textquoteleft\(x\)\textquoteright by itself. To do this, we need to perform the inverse of multiplication, which is division:
\[ \frac{9x}{9} = \frac{63}{9} \] The equation simplifies to:
\[ x = 7 \]

Set up a habit of systematically applying these steps: move constants first (often by adding or subtracting), then deal with coefficients by multiplying or dividing.
This process helps in clearly solving for the unknown variable.

Simple linear equations are equations of the first degree, which means they contain variables raised only to the first power. An example is \(9x - 3 = 60\). These equations are called linear because their graphs are straight lines.
When solving simple linear equations, always follow these steps:

• Combine like terms, if necessary;
• Isolate the variable terms on one side;
• Eliminate constants from the variable's side;
• Solve for the variable by performing inverse operations.

Returning to our example, after moving the constants and isolating the variable, we solved \(9x = 63\). Then, we divided both sides by 9 to get \(x = 7\).

Understanding the steps involved and practicing these methods are essential to mastering simple linear equations.