Problem 35 - Algebra 1

Question

Solve the equation. $$x+4-3=6 \cdot 5$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 29\).

Step 1: Simplify the equation

Simplify the left side of the equation by performing the operation \(4-3\) to get \(x + 1\) and multiply the numbers on the right side of the equation \(6 \cdot 5\) to get \(30\), so the equation becomes \(x + 1 = 30\).

Step 2: Isolate x

To isolate \(x\), subtract 1 from both sides of the equation. This gives the simplified equation \(x = 30 - 1\).

Step 3: Solve for x

Finally, subtract to solve for \(x\). This gives \(x = 29\).

Key Concepts

Equation Simplification

When solving linear equations, it's often necessary to simplify the equation for ease of calculation. Simplification involves reducing the equation by performing basic arithmetic operations and rearranging terms into simpler forms. Initially, the equation given is \(x + 4 - 3 = 6 \cdot 5\). We can start by simplifying each side of the equation.

• Simplifying the Left Side: Combine like terms. Perform the operation \(4 - 3\) which equals 1, thus simplifying the left side to \(x + 1\).
• Simplifying the Right Side: Execute the multiplication \(6 \cdot 5\) to get 30, simplifying the right side to 30.

Hence, the simplified equation now reads \(x + 1 = 30\). This step brings the equation to a much simpler form, making it easier to handle. Simplifying equations is crucial because it can reduce potential errors and make further calculations more manageable.

Isolating Variables

The goal of isolating a variable in an equation is to get the variable by itself on one side of the equation. This process helps set the stage for solving the equation effectively. Here, our task is to isolate \(x\) in the equation \(x + 1 = 30\).

• Move Other Terms: To begin isolating \(x\), we subtract 1 from both sides of the equation. This operation effectively eliminates the constant from the left-hand side, leaving \(x\) by itself. So, \(x + 1 - 1 = 30 - 1\).
• Simplify: This subtraction simplifies the left side to \(x\) and the right side to 29, resulting in the equation \(x = 29\).

By isolating the variable, we are a step closer to finding its value, which can further be interpreted if needed. This process highlights the importance of working through each side of the equation systematically.

Arithmetic Operations

Arithmetic operations are timeless mathematical procedures that are key to solving equations. They include addition, subtraction, multiplication, and division. These operations are the tools we use to manipulate and simplify equations.

• Importance in Equations: Each operation serves to transform the equation into a form that is easier to interpret and solve. For example, the subtraction of 1 from both sides of \(x + 1 = 30\) is an arithmetic operation that isolates \(x\).
• Performing Operations: The multiplication \(6 \cdot 5\) we performed at the start is another such operation. It allows us to change expressions like \(6 \cdot 5\) into a numerical value (here, 30), simplifying our steps.

Understanding and correctly performing arithmetic operations is critical for ensuring the accuracy of your solution. One must be careful; missing a step or performing an operation incorrectly can lead to the wrong answer.