Problem 55 - Algebra 2

Question

Solve each equation. Check your solution. \(3 x+6=22\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{16}{3}\).

Step 1: Isolate the Variable Term

The given equation is \(3x + 6 = 22\). Our first task is to isolate the term with the variable (\(3x\)) on one side. To do this, we subtract 6 from both sides of the equation. This gives us:\[3x + 6 - 6 = 22 - 6\]\[3x = 16\]

Step 2: Solve for the Variable

Now that we have \(3x = 16\), we need to solve for \(x\). We do this by dividing both sides by 3:\[x = \frac{16}{3}\]

Step 3: Check the Solution

To ensure that \(x = \frac{16}{3}\) is correct, substitute \(x\) back into the original equation and verify that both sides are equal:\[3 \left( \frac{16}{3} \right) + 6 = 22\]This simplifies to:\[16 + 6 = 22\]\[22 = 22\]The sides are equal, so the solution is confirmed.

Key Concepts

Isolating the Variable

When solving equations, one of the most important steps is isolating the variable. This means we try to get the variable, usually represented as a letter like \(x\), on one side of the equation by itself. We do this so that we can figure out what value the variable holds. In the given equation, \(3x + 6 = 22\), our task is to separate \(3x\) from other numbers. Here’s a simple way to think about it:

• Look at the equation and notice the number added to \(3x\), which is 6.
• To remove the 6, do the opposite operation. Since it is adding, you subtract 6 from both sides.
• This gives us \(3x = 16\).

This step is crucial because it makes the equation simpler and helps us focus on finding the value of \(x\). Notice how we do the same operation on both sides to keep the balance of the equation.

Checking Solutions

Once you think you've solved an equation, it's always a smart move to check your solution. Checking solutions ensures that your answer is correct and confirms that the equation is balanced. Here’s how we check our solution for \(x = \frac{16}{3}\):

• Substitute the value back into the original equation. Replace \(x\) with \(\frac{16}{3}\) in \(3x + 6 = 22\).
• This changes the equation to \(3 \times \frac{16}{3} + 6 = 22\).
• Simplify both sides: \(16 + 6 = 22\).
• See that both sides equal 22, which means our solution \(x = \frac{16}{3}\) is correct.

Always re-substitute your answer back into the original equation. If both sides are not equal, revisit your steps and see where you might have gone wrong. This way, you'll catch any mistakes and ensure you're on the right track.

Linear Equations

Linear equations are one of the most basic forms of algebraic equations. They're called "linear" because when you graph them, they make a straight line. The general form of a linear equation is \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.In our example, the equation \(3x + 6 = 22\) is a linear equation. Here’s why they're important:

• They help us find unknown values that make the equation true.
• They're the foundation for more complicated math topics.
• They are used in real-life situations like budgeting or calculating distances.

Solving these equations typically involves simple arithmetic operations such as addition, subtraction, multiplication, and division, used strategically to isolate the variable. Understanding these concepts will make it easier to approach not just math problems but practical problems you may encounter in everyday life.