91Ó°ÊÓ

A linear equation is an algebraic expression where each variable is raised to the power of one. In other words, no variable is squared, cubed, or higher. For example, equations like:
- \(3x + 2 = 11\)
- \(4y - 5 = 19\)
These are all linear because the variables (\(x\) and \(y\)) are to the first power. Linear equations result in straight lines when graphed. Solving linear equations involves:

• Isolating the variable on one side
• Performing the same operation on both sides

A common technique is to combine like terms, which means adding or subtracting coefficients of the same variable. For instance, in the given problem:
\(3x + 4x = 7x\)
we add the coefficients of \(x\) to simplify it.

Factoring is a technique used to simplify equations and expressions. It involves rewriting an expression as a product of its factors. For example, given the polynomial:
\(7x^2 - 7x\)
We can factor out the greatest common factor (GCF). Here, the GCF is \(7x\), so we can rewrite it as:
\[7x(x - 1)\]
In the original solution, we factored the equation:
\(0 = 7x^2 - 7x\)
into:
\[0 = 7x(x - 1)\]
This shows that either:

• \(7x = 0\)
• \(x - 1 = 0\)

So, \(x = 0\) or \(x = 1\). Factoring helps in breaking down equations into simpler parts which are easier to solve.

In algebra, a statement can be true, false, or conditional. A true statement holds for all values of variables. A false statement does not hold for any value (sometimes excepting specific cases), and a conditional statement holds under certain conditions. For example, consider the statement:
\(3x + 4x = 7x^2\)
If we combine like terms, we get:
\(7x = 7x^2\)
Rearranging it gives:
\(0 = 7x(x - 1)\)
This shows that the equation only holds for \(x = 0\) or \(x = 1\). Thus, generally, the statement is false.
On the other hand, the statement:
\(3x + 4x = 7x\)
Combines to \(7x = 7x\), which holds true for all \(x\). Thus, it is a true statement. It's important to verify the conditions under which algebraic statements hold true.