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In the realm of algebra, linear equations are the simplest type of equations to handle. They are called 'linear' because they represent a straight line when graphed on a coordinate plane. The standard form of a linear equation in one variable, like r, is \( ax+b=0 \), where \( a \) and \( b \) are constants. Solving a linear equation typically involves finding the value of the variable that makes the equation true. For example, the equation solved in the exercise \( \frac{3r+2}{5}=-1 \) is a form of a linear equation where the variable \( r \) needs to be isolated to find its value.

When solving linear equations, it's pivotal to properly perform operations such as addition, subtraction, multiplication, and division. The equation's solution, which in this case is \( r = \frac{-7}{3} \), gives us the value of r that satisfies the equation. It’s important that the constant \( a \) is never zero, as that would no longer make it a linear equation.

An algebraic fraction is similar to an ordinary fraction, but it includes variables, for instance, \( \frac{3r+2}{5} \). The strategies for solving an equation with algebraic fractions are similar to those used for numerical fractions. The primary approach is to eliminate the fraction to simplify the equation, which is accomplished by finding a common denominator. In the given problem, multiplying both sides of the equation by 5, which is the denominator of the fraction, achieves this simplification.

To ensure a smooth resolution, remember to apply the same operation on both sides of the equation to maintain balance. This is a key principle in algebra — whatever you do to one side, you should also do to the other. Once the algebraic fraction is eliminated, the equation becomes a straightforward linear equation, easing the process of finding the variable.

The technique of isolating the variable involves manipulating an equation to 'solve for' or to find the value of a solitary variable. Think of it as isolating the variable on one side to see what it equals to on the other side. In the example provided,

Step 1: Eliminate the Fraction

isolates the variable by first multiplying both sides of the equation by the denominator. After this step, you have a simpler equation, \( 3r + 2 = -5 \).

The next part of the isolation process involves getting r all by itself on one side of the equation, which is fulfilled by

Step 2: Solve for r

. This is done by subtracting 2 from both sides and then dividing by 3. The final result, \( r = \frac{-7}{3} \), indicates that r has been successfully isolated. Mastery of isolating the variable is crucial, as it is a fundamental operation in algebra that applies to nearly all types of equations.