Problem 57 - Introductory and Intermediate Algebra for College Students 4th

Question

Contain small geometric figures that represent real numbers. Use the addition property of equality to isolate \(x\) on one side of the equation and the geometric figures on the other side. $$2 x+\Delta=3 x+\square$$

Step-by-Step Solution

Answer

The solution is \(x = \Delta - \square\).

Step 1: Rearrange the equation to isolate x

The first step is to move the \(3x\) term from the right side of the equation to the left side. This will leave only \(\square\) on the right side. This can be done by subtracting \(3x\) from both sides, yielding the equation \((2x - 3x) + \Delta = \square\) or \(-x + \Delta = \square\).

Step 2: Rearrange the equation to leave x alone

The second step involves moving the \(\Delta\) term from the left side of the equation to the right side. This can be done by subtracting \(\Delta\) from both sides of the equation, yielding the equation \(-x = \square - \Delta\).

Step 3: Make x positive

Now, we have \(-x\) on the left side, but we want to isolate \(x\). We can make \(x\) positive by multiplying the whole equation by -1. The equation becomes \(x = -\square + \Delta\).

Step 4: Simplify the equation

The equation \(x = -\square + \Delta\) can be simplified to \(x = \Delta - \square\), making it clear that \(x\) is simply the difference between the values represented by \(\Delta\) and \(\square\).

Key Concepts

Isolating Variables

In mathematics, isolating the variable is a key technique used to solve equations, especially when solving for unknowns like \(x\). The goal is to have the variable appear by itself on one side of the equation. Think of it like balancing a scale - whatever you do to one side, you must do to the other. This keeps the equation in balance. For example, to move \(3x\) from one side of the equation to the other, you subtract \(3x\) from both sides. This ensures the equation remains balanced.

Steps for isolating variables:

• First, decide which side you want your variable on.
• Use inverse operations to move other terms to the opposite side.
• Perform the same operation on both sides of the equation to keep it balanced.
• Ensure your variable is positive by multiplying or dividing as necessary.

These steps not only help solve equations but are also foundational for algebra.

Linear Equations

Linear equations are equations that graph as straight lines, and have a form exemplified by \(y = mx + b\). Often, they involve finding one single solution or determining a relationship between variables. In a linear equation, each term is either a constant or the product of a constant and a single variable.

Linear equations are straightforward because:

• They typically have simple solutions, like the equation \(x = 3\).
• They follow predictable rules compared to more complex equations with variables raised to higher powers.

Being familiar with linear equations prepares you for tackling a wide range of algebraic challenges. Recognizing linear equations involves knowing the structure and the manipulations needed to isolate a variable, as shown in our original problem.

Simplifying Equations

Simplifying equations is the process of rewriting them to make them easier to work with or to understand. It involves combining like terms, arranging the equation in a simpler form, and eliminating unnecessary complexities. This process can make solving equations much easier, especially when preparing for the final step of finding a solution.

Considerations for simplifying:

• Identify and combine like terms effectively.
• Rearrange the equation to have a clear and direct expression.
• Convert any negative expressions or awkward terms into a standard form where possible.

In our example, once isolating the variable was achieved, the final expression \(x = \Delta - \square\) was simplified by recognizing that combining terms could give us a clearer insight into the solution. Clear equations are not just tidier; they also help in avoiding calculation errors.