Problem 65 - Algebra and Trigonometry

Question

Solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. $$ 10 x+3=8 x+3 $$

Step-by-Step Solution

Answer

The solution for the equation is \(x = 1.5\). The equation is a conditional equation.

Step 1: Simplify the equation

Start by isolating the variables on one side of the equation. Subtract \(8x\) from both sides to get \[10x - 8x = 8x - 8x + 3\], which simplifies to \(2x = 3\].

Step 2: Solve the equation

To isolate \(x\), divide each side of the equation by 2: \[2x / 2 = 3 / 2\], which simplifies to \(x = 1.5\].

Step 3: Determine the type of equation

Now that we have a solution, we can identify the type of equation. This equation is a conditional equation since it is not always true but true given the condition \(x = 1.5\)

Key Concepts

Solve Equations

Solving equations is about finding the value of the unknown variable that makes the equation true. Here, the unknown is represented by a symbol, typically a letter such as \(x\). The process involves manipulating the equation to isolate the variable:

• Begin by organizing the terms. Move variable terms (those including \(x\)) to one side and constant terms (those without \(x\)) to the other side of the equation.
• Use addition or subtraction to eliminate terms on one side.
• Apply multiplication or division to solve for the variable.

In the original exercise, by subtracting \(8x\) from both sides, we simplified the equation to \(2x = 3\). This arrangement allowed us to clearly see what operations were needed next to solve for \(x\).

Conditional Equations

A conditional equation is true only for certain values of the variable. This means that the equation will not hold for every possible number substituted into \(x\). Instead, it will only be true under specific conditions:

• Example: The solution \(x = 1.5\) for the equation \(10x + 3 = 8x + 3\) is a condition where the equation is satisfied.
• Conditional equations usually result from solving equations that require finding particular solutions.

It's important to recognize these equations as they apply only to particular scenarios, where the condition is met, just like in our exercise.

Identity Equations

An identity equation is an equation that is true for every possible value of the variable involved. These appear in more straightforward situations where no specific solution set exists as all values satisfy the equation:

• For example, consider the equation \(x + x = 2x\) for any value of \(x\). This is an identity equation as substituting any value for \(x\) will always make the equation true.
• Symbols and expressions might cancel out or simplify to a universally true statement, indicating an identity.

In our step-by-step solution, we didn't have such a case. Rather, our equation had one specific solution (\(x = 1.5\)), qualifying it not as an identity.

Inconsistent Equations

An inconsistent equation has no solution, which means it is impossible to find a value for the variable that makes the equation true. This often happens when the equation simplifies to a statement that is always false:

• Example: An equation like \(x + 2 = x + 3\) simplifies to \(2 = 3\), which is impossible.
• Inconsistent equations typically emerge from errors or contradictions within the problem structure.

Understanding the lack of solutions, as seen in inconsistent equations, contrasts with our exercise's outcome, where we found a specific solution making it a conditional equation.