Solving Linear Equations: A Step-by-Step Guide To 7x - 9 = 28 + 4(x - 1)
In the realm of mathematics, linear equations form the bedrock of countless concepts and applications. Mastering the art of solving linear equations is not merely an academic exercise; it's a fundamental skill that empowers us to unravel real-world problems across various disciplines. This comprehensive guide delves into the intricacies of solving the linear equation 7x - 9 = 28 + 4(x - 1), providing a step-by-step approach coupled with insightful explanations to solidify your understanding.
Unveiling the Equation: 7x - 9 = 28 + 4(x - 1)
The equation at hand, 7x - 9 = 28 + 4(x - 1), presents a classic example of a linear equation in one variable. Our mission is to isolate the variable 'x' on one side of the equation, thereby determining its value. To achieve this, we shall embark on a journey of simplification and algebraic manipulation, employing the time-honored principles of equality.
Step 1: Distribution – Expanding the Realm
The first step in our algebraic odyssey involves simplifying the right-hand side of the equation. We encounter a term, 4(x - 1), which beckons us to employ the distributive property. This property dictates that we multiply the term outside the parentheses, 4, by each term within the parentheses, namely 'x' and '-1'.
Applying the distributive property, we transform 4(x - 1) into 4x - 4. Our equation now stands as:
7x - 9 = 28 + 4x - 4
Step 2: Combining Like Terms – Unifying the Elements
Our next endeavor entails combining like terms on each side of the equation. Like terms are those that share the same variable and exponent, or are simply constants. On the right-hand side, we observe the constants 28 and -4, which we can combine to yield 24. The equation now metamorphoses into:
7x - 9 = 4x + 24
Step 3: Isolating the Variable – The Quest for 'x'
The heart of our quest lies in isolating the variable 'x' on one side of the equation. To achieve this, we must strategically maneuver terms across the equality sign. Let us embark on the journey of grouping 'x' terms on the left-hand side and constants on the right-hand side.
To banish the '4x' term from the right-hand side, we subtract '4x' from both sides of the equation. This maintains the sacred balance of equality. Our equation now transforms into:
7x - 4x - 9 = 4x - 4x + 24
Simplifying, we obtain:
3x - 9 = 24
Step 4: Further Isolation – Liberating 'x'
Our pursuit of isolating 'x' continues. To liberate 'x' from the clutches of '-9', we add '9' to both sides of the equation, preserving the equilibrium.
3x - 9 + 9 = 24 + 9
This simplifies to:
3x = 33
Step 5: The Grand Finale – Unveiling the Value of 'x'
The final act in our algebraic drama involves unveiling the value of 'x'. To achieve this, we divide both sides of the equation by the coefficient of 'x', which is '3'.
3x / 3 = 33 / 3
This yields the triumphant solution:
x = 11
Verifying the Solution: Ensuring Accuracy
In the realm of mathematics, verification is paramount. To ensure the accuracy of our solution, we substitute the value of 'x', which is 11, back into the original equation:
7(11) - 9 = 28 + 4(11 - 1)
Simplifying the left-hand side:
77 - 9 = 68
Simplifying the right-hand side:
28 + 4(10) = 28 + 40 = 68
Behold! The left-hand side equals the right-hand side, thus validating our solution, x = 11.
Mastering Linear Equations: A Journey of Skill and Precision
Solving linear equations is not merely a mechanical process; it's an art that demands skill, precision, and a deep understanding of mathematical principles. By diligently practicing and mastering the techniques outlined in this guide, you shall fortify your algebraic prowess and unlock the doors to a myriad of mathematical adventures.
Delving Deeper: Advanced Techniques and Applications
Tackling Complex Linear Equations
As you delve deeper into the realm of linear equations, you may encounter equations adorned with fractions, decimals, or even nested parentheses. Fear not, for these challenges can be conquered with the aid of strategic techniques.
- Fractions: When confronted with fractions, the key is to eliminate them by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
- Decimals: Decimals can be tamed by multiplying both sides of the equation by a power of 10 that transforms the decimals into integers.
- Nested Parentheses: Nested parentheses demand a systematic approach. Start by simplifying the innermost parentheses and work your way outwards, applying the distributive property as needed.
Applications of Linear Equations: Real-World Scenarios
Linear equations are not confined to the ivory towers of academia; they permeate the fabric of our daily lives. From calculating grocery bills to determining travel distances, linear equations provide a framework for solving real-world problems.
- Word Problems: Word problems often present scenarios that can be translated into linear equations. By carefully identifying the unknowns and relationships, you can construct equations that pave the path to solutions.
- Financial Calculations: Linear equations are indispensable in financial calculations, such as determining loan payments, calculating interest, and analyzing investments.
- Scientific Applications: In the sciences, linear equations play a pivotal role in modeling physical phenomena, analyzing data, and making predictions.
Graphing Linear Equations: Visualizing the Relationship
Linear equations can be visualized through the elegant medium of graphs. The graph of a linear equation is a straight line, whose slope and intercepts reveal valuable information about the equation's behavior.
- Slope-Intercept Form: The slope-intercept form of a linear equation, y = mx + b, provides a clear view of the slope (m) and y-intercept (b) of the line.
- Graphing Techniques: Plotting points, using the slope and y-intercept, and employing graphing calculators are just a few of the techniques available for graphing linear equations.
Systems of Linear Equations: Unraveling Interconnectedness
Often, we encounter scenarios involving multiple linear equations that are intertwined. These systems of linear equations demand specialized techniques for their solution.
- Methods of Solution: Substitution, elimination, and matrix methods are among the arsenal of techniques employed to solve systems of linear equations.
- Applications: Systems of linear equations find applications in diverse fields, such as economics, engineering, and computer science.
Conclusion: Embracing the Power of Linear Equations
Linear equations, seemingly simple in their form, wield immense power in the world of mathematics and beyond. By mastering the art of solving linear equations, you equip yourself with a fundamental tool that unlocks countless doors to problem-solving, critical thinking, and a deeper understanding of the world around you. So, embrace the challenge, hone your skills, and embark on the rewarding journey of mastering linear equations.
The original question was: Solve . This is a clear and concise mathematical equation. No changes are needed to make it more understandable.
Solving Linear Equations A Step by Step Guide to 7x - 9 = 28 + 4(x - 1)
