Problem 65 - Beginning and Intermediate Algebra

Question

Decide whether the given number is a solution of the given equation. Is 5 a solution of \(3 x+30=9 x ?\)

Step-by-Step Solution

Answer

Yes, 5 is a solution because substituting it into the equation satisfies both sides.

Step 1: Substitute the Given Number into the Equation

Start by substituting the value 5 for the variable \( x \) in the given equation \( 3x + 30 = 9x \). This substitution will help us determine if 5 is a solution.Substitute to get: \[3(5) + 30 = 9(5)\]

Step 2: Calculate Both Sides of the Equation

Now, calculate the expressions on each side of the equation after substitution. Left side: \[3(5) + 30 = 15 + 30 = 45\]Right side: \[9(5) = 45\]

Step 3: Compare the Results

Compare the calculated values from both sides of the equation. The left side equals 45 and the right side equals 45. Since both sides are equal, the substitution satisfies the equation.

Key Concepts

Substitution Method

Understanding the substitution method can make solving algebraic equations much easier. It's a process where we replace the variable in an equation, often denoted as \(x\), with a specific value. In our example, we were checking if 5 is a solution to the equation \(3x + 30 = 9x \). By substituting \(5\) for \(x\), you change the equation to a numeric expression: \(3(5) + 30 = 9(5)\).
This kind of substitution transforms an equation into a simple arithmetic problem. It allows you to directly compute the left and right sides to see if they are equal, making it straightforward to determine if your chosen number is indeed a solution. The substitution method is valuable because it systematically verifies potential solutions without guesswork. Remember, it's like plugging in numbers to check if they work in an equation.

Algebraic Equations

Algebraic equations may seem complex at first, but they fundamentally express relationships between different variables. In our case, the equation \(3x + 30 = 9x\) is an algebraic equation that looks to find a balance or equality between two expressions.
When handling these kinds of equations, it's crucial to handle them using algebraic principles. This involves operations like addition, subtraction, multiplication, or division to manipulate the equation. In its simplest form, an algebraic equation tells us when two mathematical expressions hold the same value, often using an equal sign to create a balance.

• Purpose: Help determine unknown quantities.
• Structure: Made of variables, constants, and operations.
• Balance: Always work towards balancing both sides.

Understanding algebraic equations lays the groundwork for solving complex mathematical problems effectively.

Verifying Solutions

Verifying solutions is a crucial step in the problem-solving process. Once you've substituted a number into the equation, as we did with \(5\) in \(3x + 30 = 9x\), the next crucial task is to verify that this number satisfies the equation.
After performing the substitution, calculate the left and right sides of the equation. You arrive at \(45\) on both sides of our example equation. When both sides are equal, it confirms that your number is a solution. This process helps eliminate errors and strengthens your understanding of whether your assumed solution works.
Why Verify?

• Ensures accuracy in problem-solving.
• Builds confidence in your solutions.
• Identifies any errors made during the process.

Verifying not only confirms your answer but also reinforces your algebra skills, ensuring you understand every step in the equation-solving journey.