Problem 79 - Introductory and Intermediate Algebra for College Students 4th

Question

Contain small figures ( \(\square, \triangle\) and \(\$$ ) that represent nonzero real numbers. Use properties of equality to isolate \)x\( on one side of the equation and the small figures on the other side. \)\frac{x}{\square}+\Delta=\$$

Step-by-Step Solution

Answer

The solution for \(x\) in terms of \(\square\), $\$$, and \(\Delta\) is \(x = \square (\$ - \Delta)\)

Step 1: Identify the given equation

The given equation is \( \frac{x}{\square} + \Delta = \$ \). The objective is to isolate \(x\) on one side of equation which can be done by subtracting \(\Delta\) from both sides of the equation.

Step 2: Subtract \(\Delta\) from both sides of the equation

Subtracting \(\Delta\) from both sides will isolate the \(x\) term on one side and provide an equation that looks like this: \( \frac{x}{\square} = \$ - \Delta \)

Step 3: Solve for x

To completely isolate \(x\), multiply both sides of the equation by \(\square\). This gives the final result: \(x = \square (\$ - \Delta) \).

Key Concepts

Properties of Equality

Understanding the properties of equality is essential when solving algebraic equations. These properties, including the addition, subtraction, multiplication, and division properties of equality, allow you to manipulate equations while maintaining the equality between the two sides. For instance, if you have an equation such as \(\frac{x}{\square} + \Delta = \$\), you can subtract \(\Delta\) from both sides due to the subtraction property of equality. This means that what you do to one side of an equation, you must do to the other to maintain balance. In our textbook problem, subtracting \(\Delta\) from each side keeps the equation balanced and simplifies it to \(\frac{x}{\square} = \$ - \Delta\).

It's important to do the same operation to both sides because the underlying principle of equality is that two values that are equal remain equal as long as the same operation is applied to both.

Isolating Variables

Isolating the variable, often the goal in algebra, involves rearranging the equation so that the variable of interest is alone on one side of the equals sign. This process makes it easier to identify the value of the variable. To isolate a variable, you may need to use several properties of equality such as addition or subtraction (to eliminate terms), multiplication or division (to get rid of coefficients or, as in our exercise, denominators), or even exponentiation and radicals (to address powers and roots).

In the example, \(\frac{x}{\square} = \$ - \Delta\), to isolate \(x\), you need to eliminate the division by \(\square\). This can be done by multiplying both sides of the equation by \(\square\), another use of the properties of equality, specifically the multiplication property. This yields \(x = \square(\$ - \Delta)\), effectively isolating \(x\) on one side of the equation. This step is crucial for solving for \(x\) and finding its value in terms of the other known, or in this case, represented figures.

Real Numbers

The real numbers are all the numbers that can be found on the number line. This includes both rational numbers (such as \(2\), \(3/4\), and \(0.5\)) and irrational numbers (like \(\pi\) and \(\sqrt{2}\)). In the context of our exercise, the small figures ( \(\square, \triangle\) and \(\$\) ) denote different nonzero real numbers. It's important to note that in algebra, we work with the assumption that the unknowns and the coefficients are real numbers unless specified otherwise.

In algebraic equations, we can add, subtract, multiply, and divide real numbers to help solve for a particular unknown, like \(x\) in our textbook exercise. It's critical to understand that dividing by zero is undefined in the real numbers, which is why the exercise specifies that the figures represent nonzero real numbers - to prevent any potential division by zero as we isolate \(x\). This ensures that each manipulation of the equation, in compliance with the properties of equality, results in an equation with meaningful, real number solutions.