Problem 50 - Algebra 2

Question

Solve each equation. Check your solution. $$ 4 x-9=23 $$

Step-by-Step Solution

Answer

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

Step 1: Isolate the variable term

Start by isolating the term with the variable on one side of the equation. To do this, add 9 to both sides of the equation.\[ 4x - 9 + 9 = 23 + 9 \]Simplifying both sides, we have:\[ 4x = 32 \]

Step 2: Solve for the variable

Next, solve for \( x \) by dividing both sides of the equation by 4.\[ \frac{4x}{4} = \frac{32}{4} \]This simplifies to:\[ x = 8 \]

Step 3: Verify the solution

Verify the solution by substituting \( x = 8 \) back into the original equation.\[ 4(8) - 9 = 23 \]Calculate to check accuracy:\[ 32 - 9 = 23 \]Since both sides are equal, the solution \( x = 8 \) is correct.

Key Concepts

Solving Equations

When tackling a problem involving linear equations, the goal is to find the value of the unknown variable that makes the equation true. In simple terms, we need to get the variable all by itself on one side of the equation. This process is known as solving equations. Let's explore how to solve the given equation: \[4x - 9 = 23\] To start solving, our aim is to isolate the term with the variable, which in this case is \(4x\). We do this by performing algebraic operations that simplify the equation, gradually "undoing" other operations to solve for \(x\). Here's what you typically do:

• Identify the term with the variable and look at other numbers (constants) on the same side of the equation.
• Use inverse operations (like using addition to "undo" subtraction) to eliminate these constants. This makes the term with the variable stand out brightly!

In this exercise, adding \(9\) to both sides keeps the balance intact and simplifies the equation, leading us closer to the solution.

Algebraic Manipulation

Algebraic manipulation is the art of rearranging an equation to highlight the information we need, like the value of a variable. It's like solving a puzzle where each move brings you closer to the finished picture. In our example: 1. After isolating \(4x\), we have \[4x = 32\] 2. The next step is to "release" \(x\) from its multiplication bond by dividing both sides by \(4\). This is crucial to finally solve for \(x\): \[\frac{4x}{4} = \frac{32}{4}\] 3. As a result, \(x\) simplifies beautifully to \(8\), giving us the solution. Why is this manipulation important? It allows us to reformat the equation neatly and decipher the answer, just like peeling away layers to reveal the core idea.

Checking Solutions

Checking solutions is an essential last step in solving equations, much like proofreading an essay before submission. It's our chance to catch any possible errors and confirm that our solution indeed satisfies the original equation. For our solution:
We substitute \(x = 8\) back into the original equation:
\[4(8) - 9 = 23\] Calculating gives:
\[32 - 9 = 23\] And it works! 23 equals 23, confirming that \(x = 8\) balances the whole equation perfectly. Why is this step crucial?

• It serves as a double-check for mistakes in prior steps.
• Assures us that our solution is reliable and accurate.

So remember, verifying the solution isn't just a routine step—it's vital for ensuring correctness in algebraic solutions.