Solving Equations Explained How Step 1 Isolates Variables In 3x + 18 = 54
When tackling algebraic equations, the primary goal is to isolate the variable. This means manipulating the equation through a series of steps until the variable stands alone on one side, revealing its value. Let's dissect the provided equation, 3x + 18 = 54, to understand the logic behind each step and how it contributes to the final solution. We will delve into the significance of Step 1 and illuminate the underlying mathematical principles.
Original Equation
The original equation, 3x + 18 = 54, represents a balance. The left-hand side (3x + 18) is equal in value to the right-hand side (54). Our mission is to find the value of 'x' that maintains this equilibrium. This equation involves a variable term (3x) and a constant term (18) on one side, and a constant term (54) on the other. To find the value of x, we must isolate it by reversing the operations that are being performed on it. The order of operations dictates that we typically perform multiplication before addition. Therefore, in reverse, we will address the addition first, followed by the multiplication.
Step 1 The Subtraction Property of Equality
Step 1 showcases a fundamental principle in algebra: the Subtraction Property of Equality. This principle states that if you subtract the same value from both sides of an equation, the equation remains balanced. In our case, subtracting 18 from both sides is the key to simplifying the equation. This operation is explicitly shown as:
3x + 18 - 18 = 54 - 18
Why Subtract 18?
The strategic choice of subtracting 18 is crucial. The goal is to eliminate the constant term on the left side of the equation, bringing us closer to isolating the variable term 3x. By subtracting 18, we effectively cancel out the +18 on the left side:
3x + 18 - 18 simplifies to 3x
On the right side, we perform the subtraction:
54 - 18 = 36
This leads us to the simplified equation:
3x = 36
The Significance of Step 1
Step 1 is a pivotal step because it significantly simplifies the equation. Before Step 1, we had a variable term alongside a constant term on one side. After Step 1, we are left with only the variable term (3x) on one side and a constant (36) on the other. The equation 3x = 36 is much easier to solve than the original equation 3x + 18 = 54. Step 1 has essentially cleared the path for the next step in isolating x. The application of the Subtraction Property of Equality is a standard technique in solving linear equations, and mastering this step is crucial for success in algebra.
Step 2 The Division Property of Equality
The next step in solving the equation 3x = 36 involves isolating x completely. We achieve this by applying the Division Property of Equality. This property states that if you divide both sides of an equation by the same non-zero value, the equation remains balanced. In this case, we divide both sides by 3, the coefficient of x:
(3x) / 3 = 36 / 3
Isolating x
Dividing 3x by 3 cancels out the coefficient, leaving x alone:
(3x) / 3 simplifies to x
On the right side, we perform the division:
36 / 3 = 12
This gives us the solution:
x = 12
Verification
To ensure the correctness of our solution, we can substitute x = 12 back into the original equation:
3(12) + 18 = 54
36 + 18 = 54
54 = 54
The equation holds true, confirming that x = 12 is indeed the solution.
Conclusion
Solving equations involves a systematic approach of isolating the variable by performing inverse operations. Step 1, employing the Subtraction Property of Equality, is a critical step in simplifying the equation by removing the constant term. Step 2 then completes the isolation of the variable using the Division Property of Equality. By understanding and applying these principles, we can confidently solve a wide range of algebraic equations. Remember, the key is to maintain balance by performing the same operations on both sides of the equation, ensuring that the equality remains valid throughout the solution process. Each step is a deliberate move towards isolating the variable and revealing its value. Understanding the underlying properties of equality empowers you to manipulate equations effectively and arrive at accurate solutions. This step-by-step approach not only solves the immediate problem but also builds a strong foundation for tackling more complex algebraic challenges. The methodical application of these principles is the hallmark of successful problem-solving in mathematics and related fields. The journey to mastering algebra is paved with understanding these fundamental steps and their logical underpinnings. With practice and a clear grasp of these concepts, you can confidently navigate the world of equations and unlock their solutions.
