Problem 19 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. \(3(5-x)=4(2 x+1)\)

Step-by-Step Solution

Answer

The solution to the equation is x=1

Step 1: Distribute and open up the parentheses

First, we'll distribute on both sides of the equation: \(3(5-x)=4(2 x+1)\) which becomes \(15 - 3x = 8x + 4\)

Step 2: Group like terms

We shall bring terms containing 'x' to one side and constant terms to the opposite side. We then get \(15 - 4 = 8x + 3x\), which simplifies to \(11 = 11x\).

Step 3: Solve for x

Finally solve for 'x' by dividing both sides by 11, \(x = 11 / 11\), therefore \(x = 1\).

Step 4: Verify the solution

Substitute \(x=1\) in the original equation and see if it balances out. Thus, \(3(5-1) = 4(2 \cdot 1 + 1)\), that equates to \(12 = 12\) which is true, hence, confirming the solution is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term inside a set of parentheses.
Take the equation from the original exercise:
\(3(5-x) = 4(2x+1)\)
Applying the distributive property, we multiply 3 by each term inside the first set of parentheses, resulting in \(15 - 3x\), and 4 by each term inside the second set, producing \(8x + 4\).
Understanding the distributive property is crucial because it simplifies expressions and makes solving equations more manageable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In our example, \(15 - 3x\) and \(8x + 4\) are algebraic expressions created from applying the distributive property.
A key aspect of dealing with these expressions is knowing how to manipulate them by combining like terms, which are terms that have the same variable raised to the same power, and isolating variables to solve equations.

Equation Solving

Equation solving is the process of finding the value of the variable that makes the equation true.
After distributing and combining like terms, we obtained a simpler equation: \(11 = 11x\).
To solve for \(x\), we need to get the variable by itself on one side. By dividing both sides of the equation by 11, we found \(x = 1\). It's essential to perform the same operation on both sides of the equation to maintain balance and accurately determine the variable's value.

Verification of Solutions

Verifying the solution is a crucial step to ensure that our answer is correct. This involves substituting the value back into the original equation and checking if both sides equal the same number.
For the original equation \(3(5-x) = 4(2x+1)\), substituting \(x = 1\) yields \(3(5-1) = 4(2 \cdot 1 + 1)\), which simplifies to \(12 = 12\). This proves that our solution is accurate. Always verify to prevent and detect any mistakes made during the problem-solving process.