Problem 64 - Introductory Algebra for College Students

Question

Solve each equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. $$2+3(2 x-7)=9-4(3 x+1)$$

Step-by-Step Solution

Answer

The solution to the equation \(2+3(2x-7)=9-4(3x+1)\) is \(x=4/3\)

Step 1: Apply distributive property

Simplify the equation \(2+3(2x-7)=9-4(3x+1)\) by distributing the multiplicand over the terms in the parentheses: \(2+6x-21=9-12x-4\). Simplify this further to get: \(6x-19=-12x+5\).

Step 2: Re-arrange the equation

Rearrange the equation by adding \(12x\) to both sides and also adding \(19\) to both sides to solve for \(x\). This will result in: \(18x=24\). This simplifies the equation and prepares it to solve for \(x\).

Step 3: Solve for x

To isolate \(x\), divide both sides of the equation by 18. Doing this gives: \(x=24/18\). Simplifying the fraction on the right gives \(x=4/3\). Hence, the solution of the equation is \(x=4/3\).

Step 4: Verify the solution

To verify the solution, substitute \(x=4/3\) into the original equation. For the left side, \(2+3(2*(4/3)-7)\) becomes \(2+3*(8/3-7) = 2-3\) which equals \(-1\). For the right side, \(9-4(3*(4/3)+1)\) becomes \(9-4(4+1) = 9-4*5 = 9-20\) which also equals \(-1\). Since the left side equals the right side, the solution is verified.

Key Concepts

Distributive Property

Understanding the distributive property is essential when dealing with algebraic expressions that involve parentheses. This property is a cornerstone in simplifying and solving equations since it allows us to eliminate the parentheses by distributing a multiplier to each term inside them.

For instance, if we have an expression like \( a(b + c) \), the distributive property tells us that we can 'distribute' the multiplier \( a \) to both \( b \) and \( c \). As a result, the expression becomes \( ab + ac \). In the exercise \( 2+3(2x-7)=9-4(3x+1) \), applying the distributive property involves multiplying \( 3 \) by both \( 2x \) and \( -7 \) on the left side, while \( -4 \) gets distributed to \( 3x \) and \( 1 \) on the right side. It simplifies the equation and sets the stage for further steps to find the value of \( x \).

Equation Solving Steps

Solving linear equations follows a methodical process aimed at isolating the variable to find its value. After applying the distributive property, the next steps typically include combining like terms, rearranging the equation to get the variable on one side, and performing operations that reverse the procedures used to build the equation.

In our exercise, after distributing and combining like terms, we get \( 6x - 19 = -12x + 5 \). The goal is to isolate \( x \), which we accomplish by rearranging the equation. This involves adding \( 12x \) to both sides to consolidate the \( x \) terms, and adding \( 19 \) to eliminate the constant on the left, yielding \( 18x = 24 \). Finally, we divide by \( 18 \) to solve for \( x \), resulting in \( x = 4/3 \).

The beauty of a stepped approach is that it breaks down the problem into manageable parts, allowing for careful manipulation of the equation and making the process of finding the solution less daunting.

Verifying Equation Solutions

Verifying a solution is just as important as finding it. This step confirms that the value obtained is correct and satisfies the original equation. Verification is done by substituting the solution back into the original equation and simplifying to check if both sides are equal.

In the exercise, after obtaining \( x = 4/3 \), we substitute it back into each side of the equation to verify if they are indeed the same. On the left side, we have: \( 2 + 3(2(4/3) - 7) \), which simplifies to \( -1 \) after conducting the operations. The right side \( 9 - 4(3(4/3) + 1) \) also simplifies to \( -1 \). Since both sides equal \( -1 \) after the substitution, the solution \( x = 4/3 \) is verified to be correct.

Verification provides the reassurance needed to confidently state that our solution is not only a mathematical possibility but an accurate solution to the problem presented.