Problem 1 - Precalculus

Question

Determine whether the given value for the variable is a root of the equation. $$12 x-8=112 ; x=10$$

Step-by-Step Solution

Answer

Yes, \(x = 10\) is a root of the equation.

Step 1: Substitute the given value into the equation

Start by substituting the given value \(x = 10\) into the equation \(12x - 8 = 112\). This gives us the left-hand side expression of the equation as \(12(10) - 8\).

Step 2: Calculate the left-hand side expression

Calculate \(12 \times 10 - 8\). First, multiply \(12\) by \(10\) to get \(120\). Then, subtract \(8\) from \(120\) to get \(112\). So, the left-hand side becomes \(112\).

Step 3: Compare both sides of the equation

Now, compare the calculated left-hand side (\(112\)) with the right-hand side of the original equation (\(112\)). Since both sides are equal, \(x = 10\) satisfies the equation.

Key Concepts

Substitution Method

To determine if a particular value is a root of an equation, mathematicians often employ the substitution method. This is a powerful tool for testing whether a specific value satisfies the equation or not. Here's how it works in simple terms:

1. **Identify the Variable:** Start by identifying the variable in the equation. In our exercise, this variable is \(x\).
2. **Substitute the Value:** Replace the variable with the given value. In our example, we substitute \(x = 10\) into the equation \(12x - 8 = 112\).
3. **Simplify the Equation:** Perform arithmetic operations to simplify the equation after substitution. By substituting \(10\) for \(x\), the equation becomes \(12(10) - 8\).

By simplifying, you get \(120 - 8 = 112\).

Using the substitution method helps verify if plugging in a certain value yields true results on both sides of the equation. It's like testing if a key fits into a lock!

Basic Algebra

Basic algebra involves performing various operations to manipulate and solve equations. It's the foundation that allows us to work with unknowns like \(x\) and solve problems efficiently. Let's break down these fundamental operations:

1. **Operations:** The four fundamental operations — addition, subtraction, multiplication, and division — are crucial in algebra. In our example:

• We multiply \(12\) by \(10\) resulting in \(120\).
• Then, we subtract \(8\) which simplifies the expression to \(112\).

2. **Expression Evaluation:** Evaluate expressions by following the correct order of operations, often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).Basic algebra helps in expressing relationships and solving for unknowns effortlessly to find meaningful solutions.

Mastering these skills allows you to manipulate equations and arrive at the correct answer, as demonstrated in solving \(12x - 8 = 112\) when \(x = 10\).

Solving Equations

Solving equations is all about finding the value of variables that make the equation true. It's a significant skill in mathematics, helping to reveal unknown elements. Let's explore this fundamental process:

1. **Isolating the Variable:** Begin by doing what's necessary to isolate the variable on one side of the equation. Although not obvious here since we check a value substitution, this is a general approach to solving equations.

2. **Verifying Solutions:** After substituting a conjectured value (like \(x = 10\)) and solving both sides, check if both sides are equal. If they are, like in our case, the given value is a solution.

3. **Equality Check:** Both sides of the equation must hold the same value after performing operations; thus, the equation ensures equality. Here, solving \(12x - 8 = 112\) with \(x = 10\) confirms that equality.Understanding these steps helps to navigate through different types of equations, thus enhancing math proficiency. This process is vital as it confirms the results' accuracy by directly computing and analyzing values.