Problem 4 - Elementary Algebra

Question

Write "identity," "contradiction," or "conditional." If you can, find the solution by making an educated guess based on your knowledge of arithmetic. $$ \frac{x}{4}=9 $$

Step-by-Step Solution

Answer

Question: Identify the type of the given equation and find its solution using arithmetic: $\frac{x}{4}=9$. Answer: The given equation is a simple linear equation, and the solution for x, using arithmetic, is $x = 36$.

Step 1: Identify the type of equation

The given equation is a linear equation, as it involves one variable (x) and no powers higher than 1.

Step 2: Analyze the equation

The given equation is: $$ \frac{x}{4}=9 $$ Here, we need to find the value of 'x' that satisfies the equation.

Step 3: Solve for x

To solve for x, we need to isolate x on one side of the equation. In this case, we can multiply both sides of the equation by 4 to eliminate the denominator: $$ \frac{x}{4} \times 4 = 9 \times 4 $$ On the left side, the '4' in the denominator cancels out with the '4' multiplication. We get: $$ x = 36 $$

Step 4: Answer

The equation type is a simple linear equation, and the solution for x, using arithmetic, is: $$ x = 36 $$

Key Concepts

Equation Solving

Solving an equation is like solving a puzzle. In the world of mathematics, equations express relationships between numbers and unknowns, often represented by letters like \(x\). For example, the equation \(\frac{x}{4} = 9\) tells us that when \(x\) is divided by 4, the result is 9. To find the value of \(x\), we need to undo the division by 4. This is done by performing the opposite operation, which is multiplication. So, we multiply both sides of the equation by 4. This helps isolate \(x\) on one side, making it easier to find its value.

In equation solving, it's crucial to keep both sides balanced. This means whatever you do to one side, you must also do to the other. It's like a seesaw that needs to remain level. By isolating \(x\), we discover that \(x = 36\). The joy of equation solving comes from the satisfaction of finding a value that makes the original equation true.

Arithmetic Reasoning

Arithmetic reasoning involves understanding and applying simple mathematical operations to solve problems. It's the logical process that allows us to work with numbers and symbols efficiently. When we look at the equation \(\frac{x}{4} = 9\), arithmetic reasoning helps us identify the necessary steps to find \(x\).

We observe that \(x\) is divided by 4, and we need the "undo" or opposite operation to solve it. Multiplying by 4 reverses the division, bringing us back to the original number. This logical process is essential in problem-solving and is a big part of everyday decision-making.

• Identify the operation performed on \(x\): division by 4.
• Decide on the reverse operation: multiplication.
• Apply the operation consistently to both sides of the equation.

Arithmetic reasoning helps us navigate through numbers, making complex problems simpler and more approachable.

Identity, Contradiction, Conditional

In mathematics, equations can be categorized as an identity, contradiction, or conditional. An identity is an equation that's true for all possible values of the variable, like \(x + 0 = x\). A contradiction, on the other hand, has no solution; it's like saying \(x + 1 = x\), which is never true.

Conditional equations are the most common type, only true for specific values of the variable. Our equation \(\frac{x}{4} = 9\) is a conditional equation because it holds true only when \(x = 36\).

Recognizing if an equation is identity, contradiction, or conditional helps us understand what we're dealing with. For conditional equations, the goal is to find that particular solution making the equation true. This categorization is crucial in understanding the nature of solutions and planning the strategy to solve different types of equations effectively.