In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 5x + 7 = 2x + 7

Verified step by step guidance

1. Start by isolating the variable term on one side of the equation. Subtract 2x from both sides of the equation: $5x\; +\; 7\; -\; 2x\; =\; 2x\; +\; 7\; -\; 2x$.
2. Simplify the equation by combining like terms: $(5x\; -\; 2x)\; +\; 7\; =\; 7$, which simplifies to $3x\; +\; 7\; =\; 7$.
3. Next, isolate the term with the variable by subtracting 7 from both sides: $3x\; +\; 7\; -\; 7\; =\; 7\; -\; 7$.
4. Simplify the equation further: $3x\; =\; 0$.
5. Finally, solve for $x$ by dividing both sides of the equation by 3: $x\; =\; \backslash frac\{0\}\{3\}$. After solving, determine whether the equation is an identity, a conditional equation, or an inconsistent equation based on the solution.

Key Concepts

Types of Equations

In algebra, equations can be classified into three main types: identities, conditional equations, and inconsistent equations. An identity is true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solutions. Understanding these classifications helps in determining the nature of the solution set for any given equation.

Solving Linear Equations

Solving linear equations involves isolating the variable to find its value. This typically includes combining like terms, using inverse operations, and simplifying both sides of the equation. For the equation 5x + 7 = 2x + 7, one would rearrange the terms to isolate x, which is essential for determining the type of equation it represents.

Checking Solutions

After solving an equation, it is crucial to check the solution by substituting it back into the original equation. This verification process confirms whether the solution is valid and helps identify the type of equation. For instance, if the left-hand side equals the right-hand side after substitution, it indicates whether the equation is an identity or a conditional equation.