91Ó°ÊÓ

Linear equations are foundational to algebra. They represent a relationship where one variable, usually denoted as \( x \), is multiplied by a coefficient and set equal to a constant. A general form is \(-ax = b\), which reads as a constant \( a \) multiplied by the variable equals another constant \( b \). In our problem, \(-14x = -84\), where \( a = 14 \) and \( b = 84 \). This type of equation will graph as a straight line if plotted, hence the name "linear."
The core aspect of linear equations is their predictability and simplicity. They have a single solution if \( a eq 0 \). If you're solving these equations, the goal is to pinpoint that unique solution using algebraic methods.

When solving linear equations like \(-14x = -84\), it's crucial to isolate the variable to one side of the equation. Isolation means untangling \( x \) from other numbers or coefficients.
Here's a simple step-by-step:

• Start by identifying the operation surrounding the variable, which in this case involves multiplication by \(-14\).
• To isolate \( x \), perform the inverse operation. Here, divide both sides of the equation by \(-14\). This gives \( x = \frac{-84}{-14} \).

Through this division, the negative signs cancel each other out, yielding a positive result, and hence we get \( x = 6 \). Remember, whatever operation you do on one side, do it equally to the other to maintain balance.

After figuring out \( x \), our next step is verification. Verification ensures our solution is mathematically sound:

• Substitute the value obtained (\( x = 6 \)) back into the original equation.
• Replace \( x \) with 6: \(-14(6)\).
• Simplify the operation: \(-84\).

By confirming \(-84 = -84\), both sides of the equation agree. Verification not only checks it's mathematically correct, it also boosts confidence in problem-solving prowess. Remember, always verify to avoid simple errors.