Solving Algebraic Equations And Expressions A Step By Step Guide
Solving algebraic equations is a fundamental skill in mathematics. This section will walk you through the step-by-step process of solving the equation 4(x-7) - x - 4 = 6. We will cover the necessary algebraic manipulations to isolate the variable x and find its value. This involves applying the distributive property, combining like terms, and performing inverse operations to maintain the equality. Understanding these steps is crucial for tackling more complex equations in algebra and beyond.
To effectively solve the equation, let's break down the process into manageable steps. First, we address the distributive property, which is essential for removing parentheses and simplifying the equation. Next, we'll focus on combining like terms to streamline the expression further. This involves identifying terms with the same variable and constant terms, then adding or subtracting them accordingly. Once the equation is simplified, we will employ inverse operations to isolate the variable x. This typically involves adding or subtracting constants from both sides of the equation, followed by dividing or multiplying to solve for x. Each step is crucial for maintaining the balance of the equation and arriving at the correct solution. Through this detailed explanation, you'll gain a solid understanding of how to solve similar equations systematically.
Let's start with the given equation: 4(x-7) - x - 4 = 6. The first step is to apply the distributive property to the term 4(x-7). This means multiplying 4 by both x and -7. This gives us 4x - 28. Now, substitute this back into the equation: 4x - 28 - x - 4 = 6. The next step involves combining like terms. We have two terms with x, namely 4x and -x, which combine to 3x. We also have two constant terms, -28 and -4, which combine to -32. So, the equation simplifies to 3x - 32 = 6. Now, our goal is to isolate the variable x. To do this, we first add 32 to both sides of the equation to eliminate the constant term on the left side: 3x - 32 + 32 = 6 + 32, which simplifies to 3x = 38. Finally, we divide both sides by 3 to solve for x: 3x / 3 = 38 / 3, which gives us x = 38 / 3. Therefore, the solution to the equation 4(x-7) - x - 4 = 6 is x = 38 / 3. This thorough step-by-step approach not only solves the equation but also reinforces the fundamental algebraic principles involved, making it easier to tackle similar problems in the future.
Solving the Equation 4(x-7)^2 - (x-7) = 7
Solving quadratic equations often requires a different set of techniques compared to linear equations. In this section, we will address the equation 4(x-7)^2 - (x-7) = 7, which is a quadratic equation in disguise. To solve this, we'll employ a substitution method to simplify the equation, making it easier to factor or solve using the quadratic formula. Understanding this approach is essential for handling more complex quadratic equations and recognizing patterns that simplify the solving process.
To solve this equation efficiently, the substitution method is an excellent strategy. By substituting a simpler variable for a complex term, we can transform the equation into a more manageable form. Once we solve for the substituted variable, we can then back-substitute to find the value of x. This technique is particularly useful when dealing with expressions that repeat within the equation, as it allows us to reduce the complexity and focus on the underlying structure. Additionally, we will discuss factoring as a method for solving quadratic equations, which involves breaking down the quadratic expression into its factors. Factoring is a powerful technique that, when applicable, can lead to straightforward solutions. Through this comprehensive explanation, you'll gain proficiency in using substitution and factoring to solve a variety of quadratic equations.
Let's consider the equation: 4(x-7)^2 - (x-7) = 7. To make this equation easier to handle, we will use substitution. Let y = (x-7). Substituting y into the equation, we get: 4y^2 - y = 7. Now, we can rewrite the equation in the standard quadratic form by subtracting 7 from both sides: 4y^2 - y - 7 = 0. Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to 4 * -7 = -28 and add up to -1. These numbers are -7 and 4. So, we can rewrite the middle term: 4y^2 - 7y + 4y - 7 = 0. Now, we factor by grouping: y(4y - 7) + 1(4y - 7) = 0. This gives us (4y - 7)(y + 1) = 0. Setting each factor equal to zero, we have: 4y - 7 = 0 or y + 1 = 0. Solving for y, we get y = 7/4 or y = -1. Now, we back-substitute x - 7 for y: x - 7 = 7/4 or x - 7 = -1. Solving for x, we get: x = 7 + 7/4 = 35/4 or x = 7 - 1 = 6. Therefore, the solutions to the equation 4(x-7)^2 - (x-7) = 7 are x = 35/4 and x = 6. This step-by-step solution demonstrates the power of substitution and factoring in simplifying and solving quadratic equations, providing a valuable tool for tackling more complex algebraic problems.
Simplifying and Evaluating the Expression (0, 7+9) ÷ 7
Simplifying numerical expressions is a foundational skill in arithmetic. This section focuses on evaluating the expression (0, 7+9) ÷ 7. We will walk through the order of operations, which is crucial for obtaining the correct result. Understanding how to simplify and evaluate expressions is essential for success in mathematics, as it forms the basis for more advanced concepts.
When evaluating numerical expressions, adhering to the order of operations is paramount. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) serves as a helpful guide to ensure calculations are performed in the correct sequence. First, we address any operations within parentheses, then exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (from left to right). This systematic approach guarantees that the expression is simplified accurately. In this particular example, we will focus on the operations within the parentheses and then perform the division. Through this detailed explanation, you'll reinforce your understanding of the order of operations and gain confidence in simplifying various numerical expressions.
Let's evaluate the expression: (0, 7+9) ÷ 7. Following the order of operations (PEMDAS), we first address the operation within the parentheses. Inside the parentheses, we have 7 + 9, which equals 16. So, the expression becomes (0, 16) ÷ 7. However, the notation (0, 16) is somewhat ambiguous. If it represents a coordinate point, the expression doesn't make sense as a simple arithmetic operation. If it's intended as 0.16, we proceed as follows: 0.16 ÷ 7. To perform this division, we can convert the decimal to a fraction: 16/100 ÷ 7. Dividing by 7 is the same as multiplying by 1/7, so we have (16/100) * (1/7) = 16/700. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4: (16 ÷ 4) / (700 ÷ 4) = 4/175. Therefore, if (0, 16) is interpreted as 0.16, the result of the expression (0, 7+9) ÷ 7 is 4/175. However, without clear context, the original notation introduces ambiguity, underscoring the importance of precise notation in mathematics. This step-by-step evaluation highlights the need for careful interpretation and the correct application of the order of operations to arrive at an accurate solution.
Analyzing the Expression (ax^2 - 4x(3)θ - 7)
Analyzing algebraic expressions involves understanding their structure and components. In this section, we will examine the expression (ax^2 - 4x(3)θ - 7). This analysis will cover the terms, coefficients, variables, and constants present in the expression. Recognizing these elements is crucial for manipulating and simplifying algebraic expressions effectively.
To comprehend an algebraic expression thoroughly, it is essential to identify its individual components. These components include variables, which are symbols representing unknown values; coefficients, which are the numerical factors multiplying the variables; constants, which are fixed numerical values; and terms, which are the individual parts of the expression separated by addition or subtraction. In the expression (ax^2 - 4x(3)θ - 7), we will dissect each of these components to understand how they contribute to the overall structure of the expression. Additionally, we will discuss the importance of simplifying expressions by combining like terms and rearranging terms to make the expression more readable and manageable. Through this detailed analysis, you'll gain a deeper appreciation for the anatomy of algebraic expressions and improve your ability to work with them effectively.
Let's analyze the expression: (ax^2 - 4x(3)θ - 7). First, let’s identify the terms in the expression. A term is a single number or variable, or numbers and variables multiplied together. In this expression, we have three terms: ax^2, -4x(3)θ, and -7. Now, let’s look at the variables. The variables are the symbols that represent unknown values. In this expression, we have x and θ (theta) as variables. Next, we identify the coefficients. A coefficient is the numerical factor of a term. In the term ax^2, the coefficient is a. In the term -4x(3)θ, we can simplify the coefficient by multiplying -4 and 3, which gives us -12. So, the coefficient is -12. The term becomes -12xθ. Finally, let’s identify the constants. A constant is a fixed numerical value. In this expression, we have one constant: -7. Rewriting the expression with the simplified coefficient, we have: ax^2 - 12xθ - 7. This expression consists of a quadratic term ax^2, a mixed term -12xθ, and a constant term -7. This analysis provides a clear understanding of the components of the algebraic expression, which is crucial for further manipulation or simplification. By breaking down the expression into its constituent parts, we can better understand its structure and behavior, paving the way for more advanced algebraic operations.
