Solving Equations 3) ((⅚x-1/3)-½)×2½=1½ And 4) ⅗×((2x+1¾)-7½)=2 A Step-by-Step Guide
In this article, we will explore the process of solving two algebraic equations: 3) ((⅚x-1/3)-½)×2½=1½ and 4) ⅗×((2x+1¾)-7½)=2. These equations involve fractions and require a systematic approach to isolate the variable 'x'. Solving equations like these is a fundamental skill in mathematics, with applications in various fields such as physics, engineering, and economics. We will break down each step, providing clear explanations to help you understand the underlying principles. Understanding the methods to solve such equations not only aids in academic pursuits but also enhances problem-solving capabilities in real-world situations. Mastering these techniques ensures that you can tackle complex mathematical problems with confidence and precision. We'll cover everything from simplifying expressions to isolating the variable, making this guide an invaluable resource for anyone looking to improve their algebra skills. Each step will be thoroughly explained, making it easier for both beginners and those looking to brush up on their math skills to follow along. So, let’s dive in and unravel these equations together!
Equation 3: ((⅚x - 1/3) - ½) × 2½ = 1½
To solve the equation ((⅚x - 1/3) - ½) × 2½ = 1½, we'll follow a step-by-step approach to isolate 'x'. Solving complex equations often involves multiple steps, but by breaking it down, it becomes much more manageable. Our initial task is to simplify the equation and remove any mixed numbers and fractions that might complicate the process. This involves converting mixed numbers into improper fractions and then using the order of operations to simplify expressions within parentheses. By carefully following each step, we can methodically isolate the variable 'x' and find its value. Understanding the sequence of operations and how they affect the equation is crucial for solving not just this, but any algebraic equation. Mastering this process builds a strong foundation for more advanced mathematical concepts. Let's start by addressing the mixed numbers and fractions to make the equation easier to work with. This systematic approach is vital for achieving accuracy and confidence in solving algebraic problems.
Step 1: Convert Mixed Numbers to Improper Fractions
The first step in simplifying this equation is to convert the mixed numbers into improper fractions. We have 2½ and 1½. Converting 2½, we multiply the whole number (2) by the denominator (2) and add the numerator (1), giving us (2 * 2) + 1 = 5. So, 2½ becomes 5/2. Similarly, converting 1½, we multiply the whole number (1) by the denominator (2) and add the numerator (1), resulting in (1 * 2) + 1 = 3. Therefore, 1½ becomes 3/2. By converting these mixed numbers to improper fractions, we eliminate one potential source of confusion and make the equation easier to manipulate. This transformation is a crucial first step in solving the equation and sets the stage for subsequent simplifications. Improper fractions are generally easier to work with in equations because they allow for simpler arithmetic operations, such as multiplication and division. Making this conversion streamlines the process and helps avoid common errors.
Our equation now looks like this: ((⅚x - 1/3) - ½) × 5/2 = 3/2
Step 2: Divide Both Sides by 5/2
To further isolate the expression containing 'x', we need to undo the multiplication by 5/2. We accomplish this by dividing both sides of the equation by 5/2. Dividing by a fraction is the same as multiplying by its reciprocal, so we will multiply both sides by 2/5. This step is essential to simplifying the equation and bringing us closer to isolating the variable 'x'. When we perform the same operation on both sides of the equation, we maintain the equality, which is a fundamental principle in solving algebraic equations. This technique is widely used and is critical for simplifying complex equations. By applying this step, we effectively remove the fraction multiplying the expression in parentheses, thus making the equation easier to solve. This meticulous approach ensures we are progressing correctly towards our final solution. Dividing by 5/2, or equivalently multiplying by 2/5, is a pivotal step in simplifying the given equation.
((⅚x - 1/3) - ½) × 5/2 × 2/5 = 3/2 × 2/5
This simplifies to:
(⅚x - 1/3) - ½ = 3/5
Step 3: Combine Constants
Next, we need to combine the constant terms on the left side of the equation. We have -1/3 and -½. To combine these, we need a common denominator, which is 6. So, -1/3 becomes -2/6 and -½ becomes -3/6. Adding these gives us -2/6 - 3/6 = -5/6. This process of finding a common denominator is crucial when adding or subtracting fractions and ensures that we can accurately combine these terms. By combining these constants, we simplify the equation further and move closer to isolating 'x'. This step is not only important for this specific equation but is a general technique used in many algebraic problems. By simplifying the equation, we make it more manageable and easier to solve. Combining constants is a key step in the overall strategy of solving for a variable.
Our equation now looks like this:
⅚x - 5/6 = 3/5
Step 4: Add 5/6 to Both Sides
To isolate the term with 'x', we need to add 5/6 to both sides of the equation. This will cancel out the -5/6 on the left side, leaving us with just the term containing 'x'. Adding the same value to both sides maintains the balance of the equation, a fundamental principle in algebra. This step brings us closer to isolating the variable and simplifies the equation further. Adding 5/6 to both sides helps to consolidate the constant terms on one side, which is a common strategy in solving algebraic equations. This process allows us to work towards a solution by gradually isolating the unknown variable. This step is essential for progressing towards the final answer and streamlining the solution process.
⅚x - 5/6 + 5/6 = 3/5 + 5/6
This simplifies to:
⅚x = 3/5 + 5/6
Step 5: Find a Common Denominator and Add Fractions
To add 3/5 and 5/6, we need a common denominator. The least common multiple of 5 and 6 is 30. So, we convert 3/5 to 18/30 and 5/6 to 25/30. Now we can add the fractions: 18/30 + 25/30 = 43/30. This process of finding a common denominator is crucial when adding or subtracting fractions, ensuring that we can accurately combine the terms. Adding these fractions simplifies the right side of the equation and moves us closer to isolating 'x'. This step is not only important for this specific equation but is a general technique used in many algebraic problems involving fractions. By simplifying the equation, we make it more manageable and easier to solve. Working with a common denominator allows for straightforward addition and subtraction, a key skill in algebra.
Our equation now looks like this:
⅚x = 43/30
Step 6: Multiply Both Sides by 6/5
To isolate 'x', we need to multiply both sides of the equation by the reciprocal of 5/6, which is 6/5. Multiplying by the reciprocal effectively cancels out the fraction multiplying 'x', leaving 'x' by itself. This step is essential to finding the value of 'x' and completing the solution. When we perform the same operation on both sides of the equation, we maintain the equality, a fundamental principle in solving algebraic equations. This technique is widely used and is critical for simplifying complex equations. By applying this step, we isolate the variable, thus enabling us to determine its value. This meticulous approach ensures we are progressing correctly towards our final solution.
⅚x × 6/5 = 43/30 × 6/5
This simplifies to:
x = 43/25
Therefore, the solution to the equation ((⅚x - 1/3) - ½) × 2½ = 1½ is x = 43/25.
Equation 4: ⅗ × ((2x + 1¾) - 7½) = 2
Now let's tackle the second equation: ⅗ × ((2x + 1¾) - 7½) = 2. Solving this equation also requires a systematic approach, similar to the previous one. We'll start by converting any mixed numbers into improper fractions, which will make the equation easier to manipulate. Then, we'll work to isolate the variable 'x' by performing inverse operations in the correct order. This involves dividing, adding, subtracting, and potentially multiplying to gradually simplify the equation. The key to successfully solving this equation is to maintain balance by performing the same operations on both sides. This ensures that the equality remains valid throughout the process. Each step we take brings us closer to determining the value of 'x'. This process is not just about finding the answer; it's about understanding the logic and principles of algebra. With a clear and methodical approach, we can confidently find the solution to this equation. Let’s begin by converting the mixed numbers to improper fractions.
Step 1: Convert Mixed Numbers to Improper Fractions
As with the previous equation, we begin by converting mixed numbers into improper fractions. We have 1¾ and 7½. Converting 1¾, we multiply the whole number (1) by the denominator (4) and add the numerator (3), giving us (1 * 4) + 3 = 7. So, 1¾ becomes 7/4. For 7½, we multiply the whole number (7) by the denominator (2) and add the numerator (1), resulting in (7 * 2) + 1 = 15. Therefore, 7½ becomes 15/2. Converting to improper fractions simplifies the equation, making it easier to work with by removing the complexity of mixed numbers. This step is crucial for making the subsequent calculations more straightforward and efficient. Improper fractions allow us to perform arithmetic operations more directly, which is essential for solving equations accurately. By converting mixed numbers to improper fractions, we set the stage for the next steps in the solution process.
Our equation now looks like this:
⅗ × ((2x + 7/4) - 15/2) = 2
Step 2: Divide Both Sides by ⅗
To isolate the expression in parentheses, we need to divide both sides of the equation by ⅗. Dividing by a fraction is the same as multiplying by its reciprocal, so we will multiply both sides by 5/3. This operation will help us to peel away the outer layers of the equation and bring us closer to the variable 'x'. Maintaining balance in the equation is paramount, which is why we perform the same operation on both sides. This ensures that the equation remains valid and that we are on the correct path to finding the solution. This step is a key element in the simplification process and helps to streamline the subsequent calculations. By dividing by ⅗ (or multiplying by 5/3), we effectively remove this factor from the left side, making the equation more manageable.
⅗ × ((2x + 7/4) - 15/2) × 5/3 = 2 × 5/3
This simplifies to:
(2x + 7/4) - 15/2 = 10/3
Step 3: Combine Constants Inside Parentheses
Now we need to combine the constants inside the parentheses: 7/4 and -15/2. To do this, we need a common denominator, which is 4. So, -15/2 becomes -30/4. Combining these fractions gives us 7/4 - 30/4 = -23/4. This step simplifies the expression inside the parentheses, making the equation easier to handle. Combining constants is a crucial step in solving algebraic equations, as it reduces the complexity and allows us to isolate the variable more effectively. By finding a common denominator and performing the subtraction, we consolidate these terms into a single fraction, which is an essential technique in algebraic manipulation. This simplification is a necessary step towards isolating 'x' and finding its value.
Our equation now looks like this:
2x - 23/4 = 10/3
Step 4: Add 23/4 to Both Sides
To further isolate the term with 'x', we need to add 23/4 to both sides of the equation. This will cancel out the -23/4 on the left side, leaving us with just the term containing 'x'. Adding the same value to both sides is a fundamental principle in algebra, ensuring that the equation remains balanced. This step moves us closer to solving for 'x' by eliminating the constant term on the left side. Adding 23/4 to both sides helps to consolidate all the constant terms on the right side, simplifying the equation further. This operation is crucial for the subsequent steps in solving the equation and isolating the variable.
2x - 23/4 + 23/4 = 10/3 + 23/4
This simplifies to:
2x = 10/3 + 23/4
Step 5: Find a Common Denominator and Add Fractions
To add 10/3 and 23/4, we need a common denominator. The least common multiple of 3 and 4 is 12. So, we convert 10/3 to 40/12 and 23/4 to 69/12. Now we can add the fractions: 40/12 + 69/12 = 109/12. This process of finding a common denominator is crucial when adding or subtracting fractions, ensuring that we can accurately combine the terms. Adding these fractions simplifies the right side of the equation and moves us closer to isolating 'x'. This step is not only important for this specific equation but is a general technique used in many algebraic problems involving fractions. By simplifying the equation, we make it more manageable and easier to solve.
Our equation now looks like this:
2x = 109/12
Step 6: Divide Both Sides by 2
To isolate 'x', we need to divide both sides of the equation by 2. Dividing by 2 is the same as multiplying by ½. This step will finally give us the value of 'x'. This is the final step in isolating the variable, and it completes the solution process. Dividing both sides by 2 ensures that we maintain the balance of the equation, a key principle in algebra. This step directly reveals the value of 'x', which is our ultimate goal in solving the equation. By performing this division, we arrive at the solution, which represents the value of the unknown variable.
2x × ½ = 109/12 × ½
This simplifies to:
x = 109/24
Therefore, the solution to the equation ⅗ × ((2x + 1¾) - 7½) = 2 is x = 109/24.
In this article, we have successfully solved two algebraic equations involving fractions and mixed numbers: 3) ((⅚x - 1/3) - ½) × 2½ = 1½ and 4) ⅗ × ((2x + 1¾) - 7½) = 2. We found that the solution for the first equation is x = 43/25, and for the second equation, it is x = 109/24. The process of solving these equations involved converting mixed numbers to improper fractions, simplifying expressions, and isolating the variable 'x' using inverse operations. The key steps included finding common denominators, adding and subtracting fractions, and multiplying or dividing both sides of the equation to maintain balance. These techniques are fundamental in algebra and can be applied to solve a wide variety of equations. Mastering these steps enhances problem-solving skills and builds a strong foundation for more advanced mathematical concepts. Each step in the process was carefully explained to ensure clarity and understanding. The systematic approach we used can be applied to solve similar equations, making it a valuable tool for anyone learning algebra. By breaking down complex problems into smaller, manageable steps, we can confidently find solutions and gain a deeper understanding of mathematical principles. We hope this guide has been helpful in clarifying the process of solving equations with fractions and mixed numbers.
