Problem 67 - Algebra A Combined Function

Question

Decide whether the given number is a solution of the given equation. $$ 3 x-6=9 ; 5 $$

Step-by-Step Solution

Answer

Yes, 5 is a solution to the equation since it satisfies \(3x - 6 = 9\).

Step 1: Understand the Problem

We are tasked with determining if the number 5 is a solution to the equation \(3x - 6 = 9\). To do this, we will substitute \(x = 5\) into the equation and check if both sides of the equation are equal.

Step 2: Substitute the Given Number

Substitute \(x = 5\) into the left-hand side of the equation \(3x - 6\):\[3(5) - 6\]

Step 3: Perform the Arithmetic

Calculate the expression:\[3 \times 5 = 15\]Then, subtract 6 from 15:\[15 - 6 = 9\]

Step 4: Compare Both Sides of the Equation

Now, compare the result from the left-hand side, which is 9, with the right-hand side of the equation, which is also 9. Since both sides are equal, \(x = 5\) satisfies the equation.

Key Concepts

Substitution Method

The substitution method is a technique used to determine if a particular value satisfies an equation. In this context, we need to find out if a guessed or given number is a solution to the equation. The process involves replacing the variable with the number you are testing. This step is crucial because it transforms the problem from one of algebra to one of simple arithmetic.

Key steps in the substitution method include:

• Identify the variable in the equation you need to substitute.
• Substitute the identified variable with the given number.
• Simplify the equation and see if both sides are equal.

By following these steps, you can easily check if the substitution number makes both sides of the equation the same. If they are, the number is a solution. If not, it's not a solution.

Linear Equations

Linear equations are mathematical expressions that graph as straight lines when plotted on a coordinate plane. They often appear in the form of ax + b = c, where 'x' represents the variable, and 'a', 'b', and 'c' are constants. Linear equations have extremely diverse applications, from simple calculations to complex real-world problem-solving.

The given equation, 3x - 6 = 9, is a classic example of a linear equation. The equation involves a single variable raised to the first power. To solve it, you need to isolate the variable, often using techniques like addition, subtraction, multiplication, or division.

Remember that a solution to a linear equation is the value of 'x' that will balance the equation, making both sides equal. Understanding linear equations is fundamental because they form the base for more advanced algebra topics. Grasping how to manipulate these equations prepares you for tackling a variety of algebraic equations.

Arithmetic Operations

Arithmetic operations are fundamental mathematical operations including addition, subtraction, multiplication, and division. These operations are the building blocks for solving equations. In our context, the arithmetic ensures we correctly simplify expressions after substitution.

Consider the expression from the exercise: 3(5) - 6. Here, using arithmetic operations involves:

• Multiplication: Start by multiplying 3 by 5, which gives you 15.
• Subtraction: Next, subtract 6 from 15 to yield 9.

Each step may seem simple, but performing these operations correctly is vital to finding the right solution. They help you methodically break down and simplify expressions, reaching a point where you can easily compare the result with the expected outcome. Mastering arithmetic operations breeds accuracy and confidence in solving linear equations.