Solving For The Value Of An Algebraic Expression Given An Equation

by ADMIN 67 views

In the realm of mathematics, specifically algebra, solving expressions given certain conditions is a fundamental skill. This article delves into a problem where we are given an equation, 3a - 7b = -3, and are tasked with finding the value of the expression 2(2a + b - 1) + 5(a - 4b) - 3b. This problem showcases the importance of algebraic manipulation, simplification, and substitution. We will break down the problem step-by-step, making it easy to understand for anyone, regardless of their mathematical background. Understanding these concepts is crucial for success in algebra and related fields.

The ability to manipulate algebraic expressions is not just an academic exercise. It has real-world applications in various fields, including engineering, computer science, economics, and finance. By mastering these skills, you gain a powerful tool for problem-solving and decision-making in diverse contexts. This article aims to provide you with a comprehensive understanding of how to approach and solve such problems, equipping you with the knowledge and confidence to tackle similar challenges in the future.

Algebraic expressions are the building blocks of mathematical equations and models. They consist of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. The key to solving algebraic problems lies in understanding how to manipulate these expressions to isolate variables, simplify equations, and find solutions. This article will guide you through the process of simplifying and evaluating the given expression, demonstrating the practical application of algebraic principles. Through clear explanations and step-by-step solutions, we will empower you to confidently navigate the world of algebraic problem-solving.

The problem at hand involves finding the numerical value of a complex algebraic expression, given a specific condition. We are given that 3a - 7b = -3. The expression we need to evaluate is 2(2a + b - 1) + 5(a - 4b) - 3b. This requires us to first simplify the expression using the distributive property and combining like terms. Subsequently, we will use the given equation to substitute and find a numerical value. This type of problem emphasizes the core principles of algebra, including simplification, substitution, and the order of operations. Mastering these techniques is essential for solving more complex mathematical problems.

1. Simplify the Expression

The first step towards solving the problem is to simplify the given expression. This involves applying the distributive property and combining like terms. The distributive property states that a(b + c) = ab + ac. We will apply this property to the expression 2(2a + b - 1) + 5(a - 4b) - 3b.

Applying the Distributive Property:

  • Distribute the 2 in the first term: 2 * (2a + b - 1) = 4a + 2b - 2
  • Distribute the 5 in the second term: 5 * (a - 4b) = 5a - 20b

Now, we can rewrite the entire expression as:

4a + 2b - 2 + 5a - 20b - 3b

Combining Like Terms:

Next, we combine the like terms, which are the terms with the same variable. In this case, we have terms with 'a' and terms with 'b'.

  • Combine the 'a' terms: 4a + 5a = 9a
  • Combine the 'b' terms: 2b - 20b - 3b = -21b

So, the simplified expression is:

9a - 21b - 2

2. Utilize the Given Equation

We are given the equation 3a - 7b = -3. Notice that the coefficients in this equation are related to the coefficients in our simplified expression, 9a - 21b - 2. Specifically, the coefficients 9 and -21 are multiples of 3 and -7, respectively. This suggests that we can manipulate the given equation to help us evaluate the expression.

Multiplying the Given Equation:

Multiply both sides of the equation 3a - 7b = -3 by 3. This gives us:

3 * (3a - 7b) = 3 * (-3)

9a - 21b = -9

Now, we have an equation that contains the terms 9a and -21b, which are also present in our simplified expression.

3. Substitute and Evaluate

We now have the simplified expression 9a - 21b - 2 and the equation 9a - 21b = -9. We can substitute the value of 9a - 21b from the equation into the expression.

Substitution:

Replace 9a - 21b in the expression with -9:

9a - 21b - 2 = -9 - 2

Evaluation:

Finally, we perform the subtraction:

-9 - 2 = -11

Therefore, the value of the expression 2(2a + b - 1) + 5(a - 4b) - 3b is -11.

Another way to approach this problem is to solve the given equation for one variable in terms of the other and then substitute that into the expression. Let's solve the equation 3a - 7b = -3 for 'a'.

1. Solve for 'a'

Add 7b to both sides of the equation:

3a = 7b - 3

Divide both sides by 3:

a = (7b - 3) / 3

2. Substitute into the Expression

Substitute this expression for 'a' into the simplified expression 9a - 21b - 2:

9 * ((7b - 3) / 3) - 21b - 2

Simplify:

3 * (7b - 3) - 21b - 2

21b - 9 - 21b - 2

3. Evaluate

The terms 21b and -21b cancel out, leaving us with:

-9 - 2 = -11

This confirms our previous result that the value of the expression is -11.

In this article, we successfully found the value of the algebraic expression 2(2a + b - 1) + 5(a - 4b) - 3b, given the condition 3a - 7b = -3. We explored two different methods to solve this problem. The first method involved simplifying the expression, manipulating the given equation, and then substituting to find the value. The second method involved solving the given equation for one variable and then substituting that into the expression.

Both methods yielded the same result, -11, demonstrating the consistency and versatility of algebraic techniques. This problem highlights the importance of understanding algebraic manipulation, simplification, and substitution. These are fundamental skills that are essential for solving a wide range of mathematical problems.

By mastering these techniques, you can confidently tackle more complex algebraic challenges and apply these skills in various fields that rely on mathematical problem-solving. The ability to simplify expressions and solve equations is a valuable asset in both academic and professional settings. We encourage you to practice similar problems to further solidify your understanding and enhance your problem-solving abilities.

Algebraic expressions, simplification, substitution, equation solving, mathematics, problem-solving, distributive property, combining like terms, evaluation, algebra.