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When solving linear equations, isolating the variable is an essential step. This means we want to get the variable, typically represented by letters like \(a\), \(x\), or \(y\), alone on one side of the equation. By doing this, we can find its specific value.
To isolate the variable, follow these key steps:

• Remove any constants: Use addition or subtraction to move constants (numbers without variables) to the other side of the equation.
• Eliminate coefficients: If the variable is multiplied by a number (the coefficient), use division to cancel it out.

For example, in the equation \[4a + 4 = 40\], we first subtract 4 from both sides to remove the constant, then divide both sides by 4 to isolate \(a\). This process allows us to conclude that \[a = 9\].

Linear equations are mathematical statements of equality involving linear expressions. A linear expression is an algebraic expression where each term is either a constant or the product of a constant and a single variable. The form typically looks like this: \ y = mx + b \.
In simpler terms, a linear equation translates to a straight line when graphed. The fundamental operations used in linear equations are addition, subtraction, multiplication, and division.
For example, in the linear equation \[4a + 4 = 40\], we see that it takes a standard form without any exponents or higher-degree variables. By manipulating this equation using basic algebra (subtracting 4 and dividing by 4), we can find the value of \(a\).

Basic algebra is the foundation of many mathematical concepts and involves understanding and manipulating mathematical symbols. The fundamental operations include:

• Addition and Subtraction: Moving terms across the equals sign using these operations while maintaining balance in the equation.
• Multiplication and Division: Used to eliminate coefficients and solve for variables.

Let's take a closer look at an example: Solving \[\frac{a}{6} = 7\]. Here, we need to isolate \(a\) by performing the opposite operation of what is being done to it. Since \(a\) is being divided by 6, we multiply both sides by 6 to reverse the operation. This gives us \[ a = 42 \].
Understanding these steps in basic algebra allows you to tackle more complicated equations with ease.