91Ó°ÊÓ

Linear equations are equations of the first order. This means the variable (often denoted as x) is not raised to any power other than one. Solving linear equations involves finding the value of the variable that makes the equation true.
To solve a linear equation, follow these general steps:

• Simplify both sides of the equation, if necessary, by combining like terms and removing parentheses.
• Move all terms containing the variable to one side of the equation.
• Move all constant terms to the other side of the equation.
• Isolate the variable by dividing or multiplying both sides of the equation.

Remember to always perform the same operation on both sides to maintain the equality.

The distributive property is a useful tool in algebra that allows you to eliminate parentheses by distributing a factor outside the parentheses to each term inside the parentheses:
a(b + c) = ab + ac
For example, in the given exercise \(\frac{1}{3}(90x - 12)\), we distribute \(\frac{1}{3}\) over \((90x - 12)\):
\[\frac{1}{3} \times 90x - \frac{1}{3} \times 12 = 30x - 4\]
Similarly, on the other side, we distribute \(\frac{1}{2}\) over \((8x + 10)\):
\[\frac{1}{2} \times 8x + \frac{1}{2} \times 10 = 4x + 5\]
This helps in transforming the equation to one without parentheses, making it easier to solve.

Isolating the variable is crucial to solving an equation. The goal is to get the variable by itself on one side of the equation. Here's how we did it in the exercise:
After distributing and simplifying, we had:
\[30x - 4 = 4x + 5\]
Our next step is to move all terms with \x\ to one side. By subtracting \4x\ from both sides, we get:
\[26x - 4 = 5\]
Next, we move the constant term -4 to the other side by adding 4 to both sides:
\[26x = 9\]
Now, the variable term is isolated and we can easily solve for \x\ by dividing both sides by 26:
\[x = \frac{9}{26}\]
This process ensures all steps maintain the equation’s balance.

Dealing with fractions in equations might seem daunting, but it's manageable with the right approach. Here’s how:
In the given problem, we started with fractions:\ \frac{1}{3}(90x - 12) = \frac{1}{2}(8x + 10)\
Applying the distributive property helped us eliminate the parentheses and simplify:
\[\frac{1}{3}(90x - 12) = \frac{1}{2}(8x + 10)\]
distributes to:
\[30x - 4 = 4x + 5\]
When faced with equations containing fractions, it's often useful to clear the fractions by multiplying through by the denominators, if possible. This makes the equation easier to handle. In our solution, we distributed the fractions right away, simplifying our equation to one without fractions.
In equations, always ensure common denominators before combining fractions, and consider converting fractions to simpler forms if it aids in solving.