Problem 12 - Prealgebra

Question

Solve each equation. Check your solution. $$5 x-4=11$$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Set Up the Equation

We start with the given equation: \(5x - 4 = 11\). Our goal is to solve for \(x\).

Step 2: Add 4 to Both Sides

To isolate the term with \(x\), add 4 to both sides of the equation: \(5x - 4 + 4 = 11 + 4\). This simplifies to \(5x = 15\).

Step 3: Divide Both Sides by 5

Now, divide both sides by 5 to solve for \(x\): \(\frac{5x}{5} = \frac{15}{5}\). This simplifies to \(x = 3\).

Step 4: Check the Solution

Substitute \(x = 3\) back into the original equation to verify: \(5(3) - 4 = 15 - 4 = 11\). Since both sides of the equation are equal, \(x = 3\) is the correct solution.

Key Concepts

solving equations

Solving linear equations is like being a detective looking for the unknown number. Our goal is to find what variable, in this case \( x \), makes the equation true. The equation given is \( 5x - 4 = 11 \). The key to solving equations is to isolate the variable on one side. This means getting \( x \) by itself.

• Focus on balance: Think of an equation as a balanced scale. Whatever you do to one side must be done to the other. This keeps the equation intact.
• Undo operations: Consider reverse operations to simplify. Here, subtraction is undone by addition.

By systematically applying these principles, we move closer to discovering the value of \( x \). In our equation, we added 4 to both sides to clear the \(-4\), simplifying our problem and keeping our equation trustworthy.

algebraic manipulation

Algebraic manipulation is the art of rearranging and solving equations, step by step. Once you start getting comfortable with it, you'll find it becomes second nature.
After adding 4 to both sides of the equation, we got to \( 5x = 15 \). Remember, every part of this process brings us closer to our answer.

• Simplifying expressions: Simplification makes equations easier to work with. Here, adding 4 to each side resulted in a simpler expression \( 5x = 15 \).
• Misdirection avoidance: Keep your goal in view: isolating \( x \). It helps in avoiding unnecessary steps or errors.

Finally, dividing both sides by 5 was our last step in manipulation. We were left with \( x = 3 \), our solution! It's like carefully peeling back layers to reveal the core of the problem.

checking solutions

Checking solutions is an essential step in verifying the accuracy of an equation's solution. This ensures that all your preceding steps were correct.
After finding \( x = 3 \), we need to test if this value satisfies the original equation \( 5x - 4 = 11 \).

• Substitution: Replace \( x \) in the original equation with 3 to see if both sides equal: \( 5(3) - 4 = 11 \).
• Verification: Simplifying the left side, you get \( 15 - 4 = 11 \), confirming the equation holds true.

Since both sides of the equation match, \( x = 3 \) was indeed the correct solution. Always verify your solutions—it’s the last piece of the puzzle ensuring everything fits perfectly together.