Problem 22 - Elementary Algebra

Question

For the following problems, solve each conditional equation. If the equation is not conditional, identify it as an identity or a contradiction. $$ 5 x+6=-9 $$

Step-by-Step Solution

Answer

The equation 5x + 6 = -9 is a conditional equation, and the solution for x is -3.

Step 1: Define the equation

The given equation is: $$ 5x + 6 = -9 $$

Step 2: Isolate the variable (x)

In order to isolate x, we need to subtract 6 from both sides of the equation: $$ 5x + 6 - 6 = -9 - 6 \\ 5x = -15 $$

Step 3: Solve for x

Divide both sides of the equation by 5 to find the value of x: $$ \frac{5x}{5} = \frac{-15}{5} \\ x = -3 $$

Step 4: Identify the type of equation

Since we were able to find a value for x that satisfies the equation, it is a conditional equation. The solution to the given conditional equation is: $$ x = -3 $$

Key Concepts

Conditional Equations

A conditional equation is a type of equation that is true only under certain conditions. This implies that the equation holds true only when specific values are assigned to the variables involved. In our example, the equation \(5x + 6 = -9\) is conditional because the solution \(x = -3\) makes the statement true, while other values of \(x\) will not. Understanding when an equation is conditional is crucial because it allows us to determine the circumstances or values that allow the equation to be balanced. Conditional equations differ from other types of equations like identities or contradictions, which are always true or never true respectively. In summary, identifying whether an equation is conditional helps in understanding the scope and limitation of the variables involved.

Isolate the Variable

Isolating the variable is a fundamental step in solving equations. The goal is to rearrange the equation so that the variable you are solving for is alone on one side of the equation. For the equation \(5x + 6 = -9\), isolating the variable \(x\) involved two key operations:

• Subtracting 6 from both sides to eliminate the constant term on the side with the variable, which simplifies the equation to \(5x = -15\).
• Dividing both sides by 5 to have \(x\) by itself, resulting in \(x = -3\).

Isolating the variable simplifies the problem and makes it easier to see the value that makes the equation true. Mastering this skill is essential for successfully solving various types of equations.

Types of Equations

Understanding the different types of equations is an important part of mastering algebra. Here are the major types you will encounter:

• Conditional Equations: As discussed, these equations hold true only for specific values of the variable. Our equation \(5x + 6 = -9\) is conditional because only \(x = -3\) makes it true.
• Identity Equations: These equations are true for all possible values of the variables involved. An example is the equation \(2x = 2x\), which is true no matter what value \(x\) takes.
• Contradiction Equations: These equations have no solution because they are never true. For example, the equation \(x + 1 = x\) implies \(1 = 0\), which is impossible, making it a contradiction.

Classifying equations helps us understand how to approach solving them and what kinds of solutions to expect.