Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Match each equation in Column I with its solution \((s)\) in Column \(I I\). A. \(\pm 2 i\) B. \(\pm 2 \sqrt{2}\) C. \(\pm i \sqrt{2}\) D. 2 E. \(\pm \sqrt{2} \quad\) F. \(-2\) G. \(\pm 2\) H. \(\pm 2 i \sqrt{2}\) $$x-2=0$$

Step-by-Step Solution

Answer

The solution \(2\) from Column II matches the equation \(x-2=0\) from Column I.

Step 1: Understand the Given Equation

The equation given in Column I is \(x - 2 = 0\). To solve for \(x\), we need to isolate \(x\) on one side of the equation.

Step 2: Solve the Equation

Add 2 to both sides of the equation to isolate \(x\):\[x - 2 + 2 = 0 + 2\]This simplifies to \(x = 2\).

Step 3: Match the Solution with Column II

Compare the solution \(x = 2\) with the options in Column II. Look for the solution that matches \(x = 2\). In Column II, option D corresponds to \(2\).

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements where two expressions are set equal to each other. They often involve variables, which are symbols that represent unknown values. Solving these equations means finding the value of the variable that makes the equation true. Usually, algebraic equations can be simple, like linear equations, or complex, involving quadratic or polynomial forms.
The main challenge when dealing with algebraic equations is ensuring you correctly manipulate the expressions to maintain equality. This requires understanding various algebraic rules and operations, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same value. It is essential to keep the equation balanced to find the correct solution.
Algebra is a foundational skill in mathematics, and mastering it opens doors to solving more advanced mathematical problems. Being comfortable with operations and principles of equations can make problem-solving more intuitive and less daunting.

Isolating Variables

To solve an equation, one of the primary goals is to isolate the variable. This means arranging the equation so that the variable stands alone on one side of the equation, with its coefficient being 1. This process makes it easier to identify the value of the variable.
For the equation \(x - 2 = 0\), isolating the variable \(x\) involves reversing the operation performed on \(x\). Here, 2 is subtracted from \(x\), so to isolate \(x\), we add 2 to both sides. This gives us:

• Adding 2: \(x - 2 + 2 = 0 + 2\)
• Resulting equation: \(x = 2\)

This process is straightforward with linear equations like this one but can involve more steps when dealing with complex equations. Still, the principle remains the same: apply the reverse operation to what is currently applied to the variable.
Isolating variables is a key skill in algebra, allowing students to systematically solve equations and find unknown values efficiently.

Matching Solutions

Once you've solved an equation and determined the variable's value, the next step in many exercises is to match the solution with a set of options. This step checks your understanding of the solution and offers a way to verify your work. In this exercise, option D in Column II, which corresponds to \(2\), matches the solution of the equation \(x = 2\).
Matching the solution involves looking for the exact value or equivalent expression among provided choices. It's crucial to be precise and consult each option thoroughly. Misreading or mistaking a solution can lead to incorrect matching, so attention to detail is key.
This step can act as a self-check mechanism in math exercises. Not only does it confirm your calculated solution, but it also reinforces the accuracy of your equation-solving process. Practicing this frequently helps build confidence in solving and verifying equations correctly.