Solving Linear Equations And Expressions A Comprehensive Guide
This article provides a comprehensive guide to solving linear equations and expressions, covering various techniques and examples. We will explore how to find the values of unknowns in equations, simplify expressions, and form linear equations from given information. This guide is designed to help students and anyone interested in improving their mathematical skills.
i) Solving Equations with Fractions
In this section, we will tackle equations involving fractions and demonstrate how to find the value of an unknown variable. Understanding how to manipulate fractions in equations is a crucial skill in algebra. Let's delve into the first problem:
Solving Equations Involving Fractions: A Detailed Approach
When dealing with equations involving fractions, the key is to manipulate the equation to isolate the variable we're trying to find. This often involves cross-multiplication, simplification, and basic arithmetic operations. Let's examine the given equation: 5/6 = (5-2)/54. Our objective is to determine the value that satisfies this equation. The method to solve this equation involves understanding the properties of proportions and equivalent fractions.
First, simplify the equation: 5/6 = 3/54. Next, reduce the fraction 3/54 to its simplest form by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This simplifies the fraction to 1/18. Now, the equation looks like this: 5/6 = 1/18. At first glance, it seems there might be an error, as 5/6 is not equal to 1/18. Let's re-evaluate the original equation, 5/6 = (5-2)/54, and proceed to solve for the unknown value correctly. The initial simplification was correct, leading to 5/6 = 3/54. The next step involves cross-multiplication to eliminate the fractions. Cross-multiplication means multiplying the numerator of the first fraction by the denominator of the second fraction and setting it equal to the product of the denominator of the first fraction and the numerator of the second fraction. Thus, we have 5 * 54 = 6 * 3, which simplifies to 270 = 18. However, this result is incorrect and indicates a misunderstanding of the original equation or an error in the transcription. If the equation was intended to be 5/x = (5-2)/54, where we need to find the value of x, the approach would be different. Let’s correct our approach by assuming the equation is meant to find a missing value. To clarify, let’s consider a slightly different interpretation where the question might be aiming to find a value related to the fractions, rather than directly solving for a variable within the fraction itself. If the original intent was to illustrate a property of proportions or equivalent fractions, the example falls short due to the inequality. Therefore, for instructional purposes, it’s essential to present a solvable and coherent problem.
Revised Approach for Clarity
Let’s consider a similar problem to illustrate the correct methodology for solving proportions. Suppose we have the equation 5/6 = x/54, where we want to find the value of x that makes the two fractions equivalent. In this scenario, we apply cross-multiplication correctly: 5 * 54 = 6 * x, which simplifies to 270 = 6x. To find x, we divide both sides of the equation by 6: x = 270 / 6, which gives x = 45. This revised approach demonstrates a clear and correct method for solving proportions, which is a fundamental concept in algebra. The key takeaway is to ensure that the problem is well-defined and that the steps taken lead to a logical solution. When dealing with fractions and equations, accuracy in both the problem statement and the solution process is paramount.
ii) Solving for 'k' in a Linear Equation
Understanding Linear Equations
Linear equations are fundamental in algebra and involve finding the value of an unknown variable. The equation given is k/5 = 15/61. To find the value of k, we need to isolate k on one side of the equation. This can be achieved by multiplying both sides of the equation by 5. This operation will cancel out the denominator on the left side, leaving k by itself. The step-by-step process is as follows:
Detailed Solution for Finding 'k'
Given the equation k/5 = 15/61, we want to isolate k. To do this, we multiply both sides of the equation by 5. This gives us: (k/5) * 5 = (15/61) * 5. On the left side, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with k. On the right side, we multiply 15/61 by 5, which results in 75/61. Therefore, the equation becomes: k = 75/61. This fraction is already in its simplest form, as 75 and 61 have no common factors other than 1. Thus, the value of k is 75/61. To express this as a mixed number, we divide 75 by 61. The quotient is 1, and the remainder is 14. So, k can also be written as 1 14/61. This means k is slightly greater than 1. Understanding how to solve for variables in linear equations is essential for more complex algebraic problems. By isolating the variable, we can find its value and solve the equation.
Q2. Evaluating Arithmetic Expressions
Arithmetic expressions involve performing operations such as addition, subtraction, multiplication, and division. In this section, we will evaluate two expressions that combine fractions and integers. These types of problems help reinforce the order of operations and the rules for handling fractions. Let's proceed with the evaluation.
(i) Evaluating 7x(1/2 + 3/4)
Step-by-Step Evaluation
The expression to evaluate is 7 * (1/2 + 3/4). According to the order of operations (PEMDAS/BODMAS), we first need to perform the operation inside the parentheses. This involves adding the two fractions, 1/2 and 3/4. To add fractions, they must have a common denominator. The least common denominator (LCD) for 2 and 4 is 4. We convert 1/2 to an equivalent fraction with a denominator of 4 by multiplying both the numerator and the denominator by 2: 1/2 = (1 * 2)/(2 * 2) = 2/4. Now we can add the fractions: 2/4 + 3/4. When adding fractions with the same denominator, we add the numerators and keep the denominator: 2/4 + 3/4 = (2 + 3)/4 = 5/4. So, the sum inside the parentheses is 5/4. Now the expression looks like this: 7 * (5/4). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same: 7 * (5/4) = (7 * 5)/4 = 35/4. The result is 35/4, which is an improper fraction (the numerator is greater than the denominator). We can convert this to a mixed number by dividing 35 by 4. The quotient is 8, and the remainder is 3. So, the mixed number is 8 3/4. Therefore, the value of the expression 7 * (1/2 + 3/4) is 35/4 or 8 3/4.
Key Takeaways
This evaluation highlights the importance of following the order of operations and understanding how to manipulate fractions. The steps include finding a common denominator, adding fractions, and multiplying a whole number by a fraction. Converting improper fractions to mixed numbers provides a clearer understanding of the quantity.
(ii) Evaluating 7(3/4 + 9/22)
Detailed Solution Process
Next, we evaluate the expression 7 * (3/4 + 9/22). As before, we start by simplifying the expression inside the parentheses. We need to add the fractions 3/4 and 9/22. To do this, we need to find the least common denominator (LCD) for 4 and 22. The prime factorization of 4 is 2^2, and the prime factorization of 22 is 2 * 11. The LCD is the product of the highest powers of all prime factors present in the denominators, which is 2^2 * 11 = 4 * 11 = 44. Now we convert both fractions to equivalent fractions with a denominator of 44. For 3/4, we multiply both the numerator and the denominator by 11: 3/4 = (3 * 11)/(4 * 11) = 33/44. For 9/22, we multiply both the numerator and the denominator by 2: 9/22 = (9 * 2)/(22 * 2) = 18/44. Now we can add the fractions: 33/44 + 18/44 = (33 + 18)/44 = 51/44. So, the sum inside the parentheses is 51/44. Now the expression looks like this: 7 * (51/44). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same: 7 * (51/44) = (7 * 51)/44 = 357/44. The result is 357/44, which is an improper fraction. We can convert this to a mixed number by dividing 357 by 44. The quotient is 8, and the remainder is 5. So, the mixed number is 8 5/44. Therefore, the value of the expression 7 * (3/4 + 9/22) is 357/44 or 8 5/44.
Key Insights
This evaluation further illustrates the importance of finding the least common denominator when adding fractions and converting improper fractions to mixed numbers. By breaking down the problem into smaller steps, we can systematically evaluate complex arithmetic expressions.
Q3. Simplifying Algebraic Expressions with Fractions
This question involves simplifying an algebraic expression that includes fractions and subtraction. Understanding how to combine like terms and work with negative signs is crucial in algebra. Let's explore the steps to simplify the expression (p+q) - (p-q), given that p = 3/10 and q = -4/9.
Step-by-Step Simplification
Given p = 3/10 and q = -4/9, we need to find the value of (p + q) - (p - q). First, let’s substitute the values of p and q into the expression: (3/10 + (-4/9)) - (3/10 - (-4/9)). Next, we simplify the terms inside the parentheses. For the first parenthesis (3/10 + (-4/9)), we need to add the fractions. To do this, we need to find the least common denominator (LCD) for 10 and 9. The prime factorization of 10 is 2 * 5, and the prime factorization of 9 is 3^2. The LCD is 2 * 5 * 3^2 = 90. Now we convert both fractions to equivalent fractions with a denominator of 90. For 3/10, we multiply both the numerator and the denominator by 9: 3/10 = (3 * 9)/(10 * 9) = 27/90. For -4/9, we multiply both the numerator and the denominator by 10: -4/9 = (-4 * 10)/(9 * 10) = -40/90. Now we can add the fractions: 27/90 + (-40/90) = (27 - 40)/90 = -13/90. So, the first parenthesis simplifies to -13/90. For the second parenthesis (3/10 - (-4/9)), we need to subtract the fractions. Subtracting a negative number is the same as adding the positive number, so we have 3/10 + 4/9. We already found the LCD for 10 and 9 is 90. We convert both fractions to equivalent fractions with a denominator of 90. For 3/10, we multiply both the numerator and the denominator by 9: 3/10 = (3 * 9)/(10 * 9) = 27/90. For 4/9, we multiply both the numerator and the denominator by 10: 4/9 = (4 * 10)/(9 * 10) = 40/90. Now we can add the fractions: 27/90 + 40/90 = (27 + 40)/90 = 67/90. So, the second parenthesis simplifies to 67/90. Now the expression looks like this: (-13/90) - (67/90). To subtract these fractions, we subtract the numerators and keep the denominator the same: (-13/90) - (67/90) = (-13 - 67)/90 = -80/90. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 10: -80/90 = (-80 ÷ 10)/(90 ÷ 10) = -8/9. Therefore, the value of the expression (p + q) - (p - q) is -8/9.
Key Concepts
This simplification demonstrates the importance of finding common denominators, adding and subtracting fractions, and simplifying the final result. The process also highlights the significance of handling negative signs correctly and following the order of operations.
Q4. Forming Simple Linear Equations
Forming linear equations from given information is a fundamental skill in algebra. It involves translating words and relationships into mathematical expressions. In this section, we will form simple linear equations based on the provided statements.
(i) Forming the Equation: 4m = 7m
Translating Words into Equations
In this case, the given equation is 4m = 7m. While this appears to be an equation already, it presents an interesting scenario for solving. Typically, forming an equation involves creating a relationship between variables and constants. Here, we are given an equality that seems to imply a specific condition for the variable m.
Solving the Given Equation
The equation 4m = 7m can be solved to find the value of m that satisfies it. To solve this, we can subtract 4m from both sides of the equation. This gives us: 4m - 4m = 7m - 4m, which simplifies to 0 = 3m. To isolate m, we divide both sides by 3: 0/3 = (3m)/3, which simplifies to 0 = m. Thus, the only solution to this equation is m = 0. This means that the equation 4m = 7m is only true when m is equal to 0.
Understanding the Implications
This exercise highlights the importance of understanding the solutions to equations. In this case, the equation has a unique solution, m = 0. This type of problem helps in developing analytical skills and reinforces the concept that not all equations have multiple solutions or a range of solutions.
In summary, this comprehensive guide has covered solving linear equations, evaluating arithmetic expressions, simplifying algebraic expressions with fractions, and forming simple linear equations. By mastering these concepts, you will build a strong foundation in algebra and enhance your problem-solving skills.
