91Ó°ÊÓ

In algebra, combining like terms is one of the foundational steps in solving equations. It involves merging terms that contain the same variable raised to the same power. This helps to simplify the equation and reduce the number of terms, making it easier to solve. For example, in the equation given: \(\frac{1}{4} x - 6 = -\frac{3}{4} x + 14\)...we first want to gather all terms containing 'x' on one side. To achieve this, we added \(\frac{3}{4}x\) to both sides:\(\frac{1}{4} x + \frac{3}{4} x - 6 = 14\).
The next step is to simplify the x terms. By adding the fractions, \(\frac{1}{4} + \frac{3}{4}\) becomes \(1\).
So, the equation now looks like:\(x - 6 = 14\).
Remembering to combine like terms is essential, as it sets the stage for isolating the variable and ultimately solving the equation.

Isolating the variable is another critical step when solving linear equations. This process involves manipulating the equation to get the variable on one side by itself. We can achieve this through operations like addition, subtraction, multiplication, or division. After combining like terms in our example, we had: \(x - 6 = 14\).
To isolate 'x', we need to get rid of -6. We do this by adding 6 to both sides of the equation:\(x - 6 + 6 = 14 + 6\).
This simplifies to:\(x = 20\).
At this point, 'x' is isolated, and we see that 'x' equals 20. Isolating the variable is crucial because it lets us directly identify the solution to the equation.

A linear equation is an equation between two variables that produces a straight line when graphed. They generally take the form, \(ax + b = c\), where 'a', 'b', and 'c' are constants. The primary goal when solving linear equations is to determine the value of the unknown variable that makes the equation true. Let's revisit our solved linear equation from the exercise:\(\frac{1}{4} x - 6 = - \frac{3}{4} x + 14\).
After combining like terms and isolating the variable, we found that:\(x = 20\).
Linear equations are straightforward once you learn the steps for solving them, such as combining like terms and isolating the variable. They form the basis for more complex algebraic concepts, making them an essential part of mathematics.