Solving Equations Using Properties A Comprehensive Guide
In the realm of mathematics, solving equations is a fundamental skill. This article will delve into how properties can be used to rewrite equations while maintaining their solutions. Specifically, we'll address the equation 2.3p - 10.1 = 6.5p - 4 - 0.01p and identify which among the given options have the same solution. Understanding the properties of equality is crucial for manipulating equations and finding accurate solutions. This involves applying operations to both sides of the equation to isolate the variable. The addition, subtraction, multiplication, and division properties of equality allow us to add, subtract, multiply, or divide both sides of an equation by the same number without changing the solution. Our focus will be on leveraging these properties to transform the original equation into equivalent forms, thereby making it easier to identify equations with identical solutions. The core of this process involves combining like terms, simplifying expressions, and rearranging the equation to isolate the variable p. By carefully applying these techniques, we can determine which of the provided options maintain the same solution set as the original equation. This exploration will not only enhance your equation-solving skills but also deepen your understanding of the underlying principles that govern algebraic manipulations. Let's embark on this journey to master the art of rewriting equations while preserving their solutions.
Understanding the Properties of Equations
To effectively rewrite equations, it’s essential to grasp the properties that govern mathematical equality. These properties allow us to manipulate equations without altering their solutions. The addition property of equality states that adding the same quantity to both sides of an equation maintains the equality. For instance, if we have an equation a = b, adding c to both sides gives us a + c = b + c. Similarly, the subtraction property allows us to subtract the same quantity from both sides without affecting the solution. If a = b, then a - c = b - c. These two properties are fundamental in isolating variables by undoing addition or subtraction operations. The multiplication property of equality dictates that multiplying both sides of an equation by the same non-zero quantity preserves the equality. Thus, if a = b, then ac = bc, provided c is not zero. Likewise, the division property of equality allows us to divide both sides by the same non-zero quantity. If a = b, then a/c = b/c, where c cannot be zero. These properties are vital for scaling equations and eliminating coefficients. In addition to these, the distributive property is a cornerstone in simplifying equations. It allows us to expand expressions by multiplying a term across a sum or difference. For example, a(b + c) = ab + ac. This property is particularly useful when dealing with parentheses or complex expressions. Lastly, combining like terms on each side of the equation simplifies the equation before applying other properties. By understanding and applying these properties judiciously, we can rewrite equations into simpler, equivalent forms, making them easier to solve and analyze. This comprehensive understanding forms the backbone of algebraic manipulation and is crucial for solving a wide array of mathematical problems. The application of these properties ensures that the solution set of the equation remains unchanged throughout the transformation process.
Rewriting the Original Equation
Our primary task is to rewrite the given equation, 2.3p - 10.1 = 6.5p - 4 - 0.01p, using mathematical properties to identify equivalent equations. The first step involves simplifying the equation by combining like terms. On the right side of the equation, we have 6.5p and -0.01p, which are like terms since they both contain the variable p. Combining these gives us 6.5p - 0.01p = 6.49p. Thus, the equation becomes 2.3p - 10.1 = 6.49p - 4. This is a crucial simplification that makes the equation easier to analyze and compare with the given options. Next, we can manipulate the equation further by applying the addition and subtraction properties of equality. To consolidate the terms involving p on one side, we can subtract 2.3p from both sides of the equation. This gives us 2.3p - 10.1 - 2.3p = 6.49p - 4 - 2.3p, which simplifies to -10.1 = 4.19p - 4. This step eliminates p from the left side, bringing us closer to isolating the variable. To further isolate p, we can add 4 to both sides of the equation: -10.1 + 4 = 4.19p - 4 + 4. This simplifies to -6.1 = 4.19p. Now, to finally solve for p, we would divide both sides by 4.19. However, our current goal is to rewrite the equation into equivalent forms, so we'll hold off on this step for now. Another approach to rewriting the original equation involves eliminating the decimals to work with whole numbers. We can multiply both sides of the original equation 2.3p - 10.1 = 6.49p - 4 by 100 to remove the decimal points. This yields 100(2.3p - 10.1) = 100(6.49p - 4). Applying the distributive property on both sides, we get 230p - 1010 = 649p - 400. By strategically applying the properties of equality, we’ve successfully rewritten the original equation into various equivalent forms, which will help us compare it with the provided options.
Analyzing the Given Options
Now, let's analyze the given options to determine which equations have the same solution as the original equation, 2.3p - 10.1 = 6.5p - 4 - 0.01p. We've already simplified the original equation to 2.3p - 10.1 = 6.49p - 4 and further transformed it to 230p - 1010 = 649p - 400. We’ll compare each option to these equivalent forms to identify matching equations.
The first option is 2.3p - 10.1 = 6.4p - 4. Comparing this with our simplified equation 2.3p - 10.1 = 6.49p - 4, we can see that the right-hand sides are different. The first option has 6.4p, while our simplified equation has 6.49p. This difference, though seemingly small, means that the equations are not equivalent and will not have the same solution. Therefore, this option is not a match.
The second option is 2.3p - 10.1 = 6.49p - 4. This equation is identical to the simplified form of our original equation. Thus, it is equivalent and will have the same solution. This option is a definite match.
The third option is 230p - 1010 = 650p - 400 - p. We need to simplify this equation to see if it matches any of our equivalent forms. Combining the p terms on the right side, we get 650p - p = 649p. So, the equation becomes 230p - 1010 = 649p - 400. This equation is exactly the same as one of the equivalent forms we derived from the original equation by multiplying both sides by 100. Hence, this option has the same solution as the original equation.
The fourth option is 23p - 101 = 65p - 40 - p. To determine if this equation is equivalent, we can multiply the original equation 2.3p - 10.1 = 6.49p - 4 by 10. This gives us 23p - 101 = 64.9p - 40. However, the given option has 65p - 40 - p on the right side, which simplifies to 64p - 40. The coefficients of p are different (64.9 vs. 64), so this option does not have the same solution as the original equation.
The fifth option is 2.3p - 14.1 = 6.4p - 4. Comparing this to our simplified equation 2.3p - 10.1 = 6.49p - 4, we can see that both sides are different. The left side has -14.1 instead of -10.1, and the right side has 6.4p instead of 6.49p. Therefore, this option is not equivalent and does not share the same solution.
Conclusion
In summary, by carefully applying the properties of equality, we've rewritten the original equation 2.3p - 10.1 = 6.5p - 4 - 0.01p into various equivalent forms. Through a detailed analysis, we've identified that the equations 2.3p - 10.1 = 6.49p - 4 and 230p - 1010 = 650p - 400 - p have the same solution as the original equation. These equations are equivalent because they can be derived from the original equation through valid algebraic manipulations, such as combining like terms, multiplying by a constant, and applying the distributive property. Understanding and applying these properties is crucial for solving equations accurately and efficiently. This exercise highlights the importance of mastering fundamental algebraic techniques for success in more advanced mathematical problem-solving. The ability to rewrite equations while preserving their solutions is a powerful tool in mathematics, enabling us to tackle complex problems with confidence and precision.
