In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 10x + 3 = 8x + 3

Verified step by step guidance

1. Start by writing down the given equation: \$10x + 3 = 8x + 3$.
2. To isolate the variable terms on one side, subtract \$8x\( from both sides: \)10x + 3 - 8x = 8x + 3 - 8x\(, which simplifies to \)2x + 3 = 3$.
3. Next, subtract 3 from both sides to isolate the term with \(x\): \$2x + 3 - 3 = 3 - 3\(, which simplifies to \)2x = 0$.
4. Now, solve for \(x\) by dividing both sides by 2: \(\frac{2x}{2} = \frac{0}{2}\), which simplifies to \(x = 0\).
5. Finally, analyze the solution: since you found a specific value for \(x\), the equation is a conditional equation (true only for \(x = 0\)).

Key Concepts

Solving Linear Equations

Solving linear equations involves isolating the variable on one side of the equation using inverse operations such as addition, subtraction, multiplication, or division. The goal is to find the value of the variable that makes the equation true.

Types of Equations: Identity, Conditional, and Inconsistent

An identity is true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solution. Identifying the type depends on the solution set after solving the equation.

Properties of Equality

Properties of equality, such as the addition, subtraction, multiplication, and division properties, allow manipulation of equations without changing their solutions. These properties are essential for maintaining balance while solving equations.