Problem 53 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Solve the equation. $$ 6 x+5=35 $$

Step-by-Step Solution

Answer

The solution to the equation \(6x + 5 = 35\) is \(x = 5\)

Step 1: Identify the Equation

The given equation to be solved is \(6x + 5 = 35\)

Step 2: Simplify the Equation

Begin isolating x by subtracting 5 from both sides of the equation to get: \(6x = 35 - 5\)

Step 3: Solve the Modified Equation

Now we simplify the right side of the equation to get: \(6x = 30\)

Step 4: Isolate the variable

To solve for x, divide both sides of the equation by 6: \(x = \frac{30}{6}\)

Step 5: Simplify the answer

By dividing, we find \(x = 5\)

Key Concepts

Isolating Variable

To solve any equation, isolating the variable is key. This means getting the variable alone on one side of the equation. When you focus on isolating the variable, you're in essence breaking down the problem into simpler parts.

In the equation \(6x + 5 = 35\), you start by removing any numbers that are added or subtracted around the variable. This is typically done by performing the opposite arithmetic operation. In this example, subtract 5 from both sides to eliminate it from around \(x\):

- Subtract 5 from both sides: \(6x + 5 - 5 = 35 - 5\)
- Resulting in: \(6x = 30\)

Now, \(x\) is closer to being isolated, as it only remains multiplied by a number.

Simplifying Equations

Simplifying equations means making the equation as basic as possible without changing its value. This step can often involve removing fractions, reducing coefficients, or combining like terms.

In this specific problem, after isolating the variable through subtraction, we're left with \(6x = 30\). Here, the equation is already quite simplified, but to further simplify, we need to handle multiplication.

Simplification mostly involves:

• Performing arithmetic operations like addition, subtraction, multiplication, or division
• Reducing equations to their simplest form by solving logical chunks, one step at a time

This ensures clarity and accuracy before solving for the unknown variable.

Arithmetic Operations

Arithmetic operations are crucial for manipulating and solving equations. They include addition, subtraction, multiplication, and division. Applying these operations correctly ensures we move through each solving stage without error.

For the given equation \(6x + 5 = 35\), we used these operations in sequence:

• Subtraction: \(35 - 5 = 30\) to remove constants
• Division: Knowing \(6x = 30\), divide by 6
• Result: \(x = \frac{30}{6}\)

Using division helps us express \(x\) as a single number. Performing these straightforward arithmetic actions methodically moves the equation closer to the solution.

Linear Equations

A linear equation is an equation of the first degree, meaning it contains no exponents, which makes it straightforward to solve.

Linear equations such as \(6x + 5 = 35\) present a direct relationship between two values: \(x\) and constant numbers. The goal is to find the exact number \(x\) represents.

Characteristics of linear equations include:

• Having variables raised to the power of one (linear)
• A direct and clear path to isolating the variable
• The solution offers a specific number resulting from rearranging and simplifying the terms

This clear structure makes linear equations relatively simple to understand and solve, offering consistent methods applicable across diverse problems.