91Ó°ÊÓ

The basic principle of solving linear equations is to isolate the variable you're solving for. A linear equation can usually be written in the form of \(ax+b=0\), where \(a\) and \(b\) are constants. The goal is to manipulate the equation so that you end up with \(x\) on one side of the equal sign and a constant on the other.

To do this, you perform operations equally on both sides of the equation, ensuring its balance is maintained. For the exercise given, the equation is \(3(x+2)=5x+4\). You start by expanding the equation and follow the steps of combining like terms and isolating the variable \(x\).

Once \(x\) is isolated, as shown in Step 3 of the solution where \(x=1\), you have found a possible variable value that satisfies the equation. This step is crucial because it converts the equation from an abstract formulation into a specific solution or set of solutions.

After solving for \(x\) in a linear equation, it's important to verify that the proposed solution is correct. We do this by replacement or substitution: taking the found value of \(x\) and substituting it back into the original equation.

Verification checks if the left-hand side (LHS) equals the right-hand side (RHS) after substitution. If they are equal, the solution is correct for that particular equation; if not, either the solution is incorrect or the equation may not be consistent (no solutions exist) or it may have many solutions. In our example, verification is conducted in Step 4. The verification process shows us that the equation \(3(x+2)=5x+4\) is a conditional equation - it's only true under certain conditions, specifically when \(x=1\).

Algebraic identities are equations that are true for all values of the variables involved. Common examples include \(a^2-b^2=(a+b)(a-b)\) and \(a^3+b^3=(a+b)(a^2-ab+b^2)\). In contrast to conditional equations, which are only true under specific conditions, algebraic identities hold universally.

When assessing whether an equation is an identity or not, like in the provided exercise, you'd try to verify the equation with several values of \(x\). If the equation is true for any value of \(x\) you substitute, it is an identity. However, if you find at least one value of \(x\) for which the equation doesn't hold, as we did when verifying for \(x=2\), you confirm the equation is not an identity, but rather a conditional equation dependent on the value of \(x\).