Problem 12 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$x+4-3=9$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the equation

The given equation is \(x+4-3=9\). The first step is to simplify the equation. Combine like terms on the left-hand side of the equation, which results in \(x+1=9\).

Step 2: Isolate the variable

Isolate the variable \(x\) on one side of the equation. To do this, subtract 1 from both sides of the equation, which gives you \(x=9-1\).

Step 3: Solve for the variable

Solve for \(x\) by simplifying the right-hand side of the equation: \(x=9-1\), which simplifies to \(x=8\).

Key Concepts

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. An equation can be thought of as a balance scale where both sides need to be equal. The goal is to perform operations on both sides to maintain this balance until the variable is isolated.

• Start by carefully examining the equation to understand what operations are needed.
• Make sure each step keeps the equation balanced by doing the same operation to both sides.

In our exercise, we begin with the equation \(x+4-3=9\). First, we'll simplify and then isolate \(x\) using smart algebraic steps to find \(x = 8\). This process is key for accurate solutions.

Algebraic Manipulation

Algebraic manipulation is all about adjusting an equation to simplify it or to isolate the variable. It involves using inverse operations to "undo" what has been done to the variable so that you can solve for it.
To manipulate the equation \(x+4-3=9\), we start by simplifying the left-hand side:

• We combine 4 and -3 to get 1, leading to \(x+1=9\).
• Next, we subtract 1 from both sides to simplify further, arriving at \(x=8\).

Algebraic manipulation requires practice, but understanding the inverse relationships of operations can make it much easier. Remember, each step you take is getting you closer to solving the equation.

Combining Like Terms

Combining like terms is an essential skill in the simplification process of solving equations. Like terms are terms that contain the same variable raised to the same power or terms that are just constants.
In the equation \(x+4-3=9\), the terms \(+4\) and \(-3\) are like because they are both constants. Here's how you handle them:

• Add or subtract the coefficients (numbers in front of the variable or the constants).
• In this case, \(4-3\) gives \(1\).

This simplification step is crucial because it reduces the equation to its simplest form. Then you can focus on isolating the variable, making the solution process more straightforward.