Problem 96 - College Algebra

Question

In your group, describe the best procedure for solving the following equation: $$0.47 x+\frac{19}{4}=-0.2+\frac{2}{5} x$$ Use this procedure to actually solve the equation. Then compare procedures with other groups working on this problem. Which group devised the most streamlined method?

Step-by-Step Solution

Answer

The solution to the equation is \(x = -67.14\).

Step 1: Standardize the Equation

First, arrange the equation in a standard form. To do this, move all the term containing \(x\) to one side of the equation, and constants to the other side, such that: \(0.47x - \frac{2}{5}x = -0.2 - \frac{19}{4}\)

Step 2: Combine Like Terms

Secondly, combine like terms on each side of the equation to simplify. This results in: \(0.07x = -4.7\)

Step 3: Isolate the Variable

Thirdly, isolate the variable by dividing both sides of the equation by 0.07. This leads to: \(x = -4.7 / 0.07\)

Step 4: Calculate the Solution

Finally, calculate the value of \(x\), which comes out to be \(x = -67.14\)

Key Concepts

Combining Like Terms

When solving linear equations, one of the first steps is often to simplify the equation by 'combining like terms.' This process requires you to identify terms that have the same variable raised to the same power and combine them into a single term. To do this effectively, you add or subtract the coefficients (the numbers in front of the variables) and keep the variable unchanged.

Consider the linear equation from the exercise: \(0.47x + \frac{19}{4} = -0.2 + \frac{2}{5}x\). Here, the terms containing \(x\) are on both sides of the equation. Combining like terms involves moving all these terms to one side to easily work with them. This method often simplifies the equation significantly and reveals a clearer path towards the solution.

For instance, in the solution given, the equation is rearranged to bring the \(x\)-terms on one side and the constants on the other, resulting in \(0.47x - \frac{2}{5}x = -0.2 - \frac{19}{4}\). Then by converting the coefficients to a common denominator, or by using decimal conversion, the like terms with \(x\) are combined to get \(0.07x\). Note how this step has drastically simplified the equation, bringing you one step closer to isolating the variable and solving for \(x\).

Isolating the Variable

After combining like terms, the next goal is to 'isolate the variable' in order to solve the equation. Isolating the variable means manipulating the equation so that the variable you are solving for is by itself on one side of the equal sign, and everything else is on the other side. This is typically done by performing mathematically allowable operations that cancel out the other terms from the side of the variable.

In the provided equation \(0.07x = -4.7\), the next step is to get \(x\) alone. To isolate \(x\), we divide both sides by the coefficient of \(x\), which is 0.07. By performing this operation, we ensure that we're left with \(x\) by itself on one side of the equation. The resulting equation is \(x = -4.7 / 0.07\), which then can be calculated to find the specific value of \(x\).

This example illustrates the importance of isolating the variable as one of the final steps in solving linear equations. It's a methodical process that requires careful execution of basic arithmetic operations and is crucial for finding the precise answer.

Standard Form of Equation

In the context of linear equations, the 'standard form' is a way of writing equations that makes them easy to analyze and solve. The standard form of a linear equation in one variable \(x\) is expressed as \(Ax + B = C\), where \(A\), \(B\), and \(C\) are constants, and \(A\) is not zero.

Starting with the standard form can make solving the equation more straightforward. For example, in our exercise, we begin by reorganizing the equation to get terms with variables on one side and constants on the other. By doing so, we're setting up the equation in a standard form that's closer to \(Ax + B = C\), which is a preferred starting point for the combination of like terms, and ultimately, isolating the variable.

The standard form plays a crucial role in simplifying and solving equations, as it provides a clear structure for handling the equation. It assures a systematic approach in which each term is accounted for properly without overlooking any portion of the equation. It's the first step in a methodical process that leads to a clear and concise solution.