Problem 30 - Elementary Algebra

Question

A student said that the solutions of \(3(x+8)=0\) are \(x=3\) and \(x=-8\). Explain what is wrong with this thinking.

Step-by-Step Solution

Answer

The equation \(3(x+8)=0\) has only one solution: \(x=-8\).

Step 1: Identify the given equation

The given equation is: \(3(x+8)=0\)

Step 2: Simplify the equation

To solve the equation, first divide both sides by 3 to simplify: \(x + 8 = 0\)

Step 3: Solve for x

Subtract 8 from both sides to isolate \(x\): \(x = -8\)

Step 4: Check the possible values

The student incorrectly stated that \(x=3\) is a solution. However, when substituting \(x=3\) into the equation, \(3(3+8)=33\), which does not equal zero. Therefore, \(x=3\) is not a valid solution.

Step 5: Verify the correct solution

To check the correct solution, substitute \(x=-8\) into the original equation: \(3(-8 + 8) = 0\), which is true. Therefore, \(x=-8\) is the correct and only solution.

Key Concepts

linear equations

Linear equations are mathematical statements that show the equality of two expressions with a variable in it. Typically, you will see them in the form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. In our problem, the linear equation given is \(3(x+8)=0\). This indicates that the expression \(3(x+8)\) should equal zero. Understanding linear equations is crucial because they form the foundation for more complex algebraic concepts.

solving equations

Solving equations is the process of finding the value of the unknown variable that makes the equation true. To solve the equation \(3(x+8)=0\), we take several steps. First, we simplify by dividing both sides by 3, leading to \(x+8=0\). This helps isolate the variable on one side. Next, we solve for \(x\) by subtracting 8 from both sides, giving us \(x=-8\). Solving equations is about balance; whatever operation you do to one side, you must do to the other. This method ensures the equation remains true as you work towards finding \(x\).

simplification

Simplification is a technique for reducing equations or expressions to their simplest form. In the given problem, \(3(x+8)=0\), simplification begins by dividing every term on both sides by the same non-zero number, which is 3 in this case. This results in \(x+8=0\). Simplification makes the equation easier to solve. By isolating terms and getting rid of common factors, you can see the relationship between the variable and the constants more clearly. Performing these simple arithmetic steps correctly is crucial for reaching the correct solution. Verifying the final answer by substituting it back into the original equation ensures the accuracy of the simplification and solving process.