Problem 11 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$4+x=8$$

Step-by-Step Solution

Answer

The solution to the equation \(4 + x = 8\) is \(x = 4\)

Step 1: Understand the problem

We have a simple linear equation in the form \(a + x = b\), where \(a\) is a constant, \(x\) is the variable and \(b\) is another constant.

Step 2: Solve for the variable

By the additive property of equality, we can subtract the constant value on the left side of the equation (4) from both sides. This gives us \(4 + x - 4 = 8 - 4\).

Step 3: Simplify

After subtraction, the equation simplifies to \(x = 4\).

Key Concepts

Additive Property of Equality

The additive property of equality is a fundamental concept used in solving equations. It simply states that if you add or subtract the same number to both sides of an equation, the two sides remain equal. This property is crucial because it allows us to isolate variables and solve equations.

In practical terms, if you start with the equation:

• \(4 + x = 8\)

You can subtract 4 from both sides to keep the equation balanced:

• \(4 + x - 4 = 8 - 4\)
• This simplifies to \(x = 4\)

By applying this property, we've effectively moved the constant from one side of the equation, allowing us to solve for \(x\). Remember, whatever operation you perform on one side, you must perform on the other.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. Linear equations, like the one we're working on, often require simple operations to isolate the variable.

Let's break it down:

• Identify the linear equation: \(a + x = b\)
• Use the additive property of equality to isolate \(x\)
• Perform any necessary arithmetic operations

In our example, the equation \(4 + x = 8\) starts with the goal of isolating \(x\). By subtracting 4 from both sides, we balance the equation, leaving \(x\) alone.

Once the variable is isolated, the equation \(x = 4\) reveals the solution. Solving linear equations is about following these logical steps to rearrange the terms until the variable stands by itself.

Simplifying Equations

Simplifying equations is all about making an equation easier to solve by using mathematical operations to condense it into a solvable form.

Here's what it involves:

• Gather like terms on both sides
• Reduce any complex expressions
• Perform the opposite operations necessary to isolate the variable

In our specific example, after subtracting 4 from both sides, we simplified the equation:

• \(4 + x - 4 = 8 - 4\)

This step leads us directly to the simplified form \(x = 4\). By simplifying, we make equations manageable, ensuring that our solutions are clear and concise.

The idea is to maintain the balance of the equation while transforming it into a solvable format. Each simplification step should bring you closer to uncovering the unknown variable's value.