Problem 35 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(3(2 x+1)-2(x-2)=5\)

Step-by-Step Solution

Answer

The solution is \( x = -\frac{1}{2} \). It is neither an identity nor a contradiction.

Step 1: Distribute the coefficients

Expand the equation by distributing the coefficients. For the term with 3: 3(2x + 1) = 6x + 3 For the term with -2: -2(x - 2) = -2x + 4 Now the equation becomes: 6x + 3 - 2x + 4 = 5

Step 2: Combine like terms

Combine the like terms on the left side of the equation. 6x - 2x + 3 + 4 = 4x + 7 Now the equation is: 4x + 7 = 5

Step 3: Isolate the variable term

Subtract 7 from both sides of the equation to isolate the variable term. 4x + 7 - 7 = 5 - 7 4x = -2

Step 4: Solve for x

Divide both sides by 4 to solve for x. \( x = \frac{-2}{4} \) \( x = -\frac{1}{2} \)

Step 5: Check the solution

Substitute \(x = -\frac{1}{2} \) back into the original equation to verify the solution is correct. Original equation: 3(2x + 1) - 2(x - 2) = 5 Substitute \(x = -\frac{1}{2} \): 3(2(-\frac{1}{2}) + 1) - 2(-\frac{1}{2} - 2) = 5 3(-1 + 1) - 2(-\frac{5}{2}) = 5 3(0) + 5 = 5 5 = 5 Since both sides of the equation are equal, the solution \( x = -\frac{1}{2} \) checks out.

Key Concepts

Distributive Property

When solving linear equations, the distributive property is a key tool to simplify expressions.
It helps us to