91Ó°ÊÓ

Algebraic equations form the foundation of algebra and are equations where the two sides are equal when a certain value or set of values are substituted for the variables. They can range from simple, like the one given in the exercise, to much more complex. In the provided exercise, \frac{1}{4}x + 0.5x = 1.5\, we are guided to find the value of \(x\) that makes the equation true. This process is commonly referred to as 'solving for \(x\)'.

Understanding the principles of balancing equations and performing operations equally on both sides is critical. The objective is to rearrange the equation to isolate the variable on one side, thereby finding its value. We start by combining like terms and follow a series of algebraic steps to simplify and solve the equation.

Combining like terms is a crucial step in simplifying algebraic equations. Like terms are terms that have the same variables raised to the same powers. In other words, they're terms that are 'alike' in the algebraic sense. In the equation from the exercise, \(\frac{1}{4}x\) and \(0.5x\) are like terms because they both have the same variable, \(x\), to the first power.

To combine like terms, add or subtract their coefficients. In the solution provided, \(0.5x\) was converted to \(\frac{2}{4}x\) for convenience, making it easier to add to \(\frac{1}{4}x\). This resulted in \(\frac{3}{4}x\), simplifying the left-hand side of the equation. Remember, without combining like terms correctly, simplifying the equation further wouldn't be possible.

After finding a potential solution to an equation, it's essential to check that the solution is valid. This step confirms whether the value obtained for the variable, when substituted back into the original equation, makes the equation true. It's like retracing your steps to ensure no errors were made along the way.

For our example, the solution indicates that \(x=2\) might be the value that satisfies the original algebraic equation. The checking process involves substituting \(2\) for every instance of \(x\) in the equation and verifying if both sides are indeed equal. As shown in the steps, substituting \(2\) yields \(1.5\), which matches the right-hand side of the equation. This successful check reassures us that the solution \(x=2\) is correct for the given problem.