Failing to distribute the negative sign correctly

Oct 13, 2025 by windailys.com 34 views

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific linear equation, which involves variables and constants. We will break down the solution into manageable steps, making it easy to understand and follow.

The Equation

The given equation is:

-5(1 + 5r) + 3 = 123

Our goal is to isolate the variable $r$ and find its value.

Step 1: Distribute the Negative Sign

The first step is to distribute the negative sign to the terms inside the parentheses:

-5(1 + 5r) = -5 - 25r

So, the equation becomes:

-5 - 25r + 3 = 123

Step 2: Combine Like Terms

Next, we combine the like terms on the left-hand side of the equation:

-2 - 25r = 123

Step 3: Add 2 to Both Sides

To isolate the term with the variable, we add 2 to both sides of the equation:

-25r = 125

Step 4: Divide Both Sides by -25

Finally, we divide both sides of the equation by -25 to solve for $r$ :

r = -\frac{125}{25}

Simplifying the Fraction

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 25:

r = -5

Conclusion

In this article, we solved a linear equation step by step, using basic algebraic operations. We started with a complex equation and broke it down into manageable parts, making it easy to understand and follow. By distributing the negative sign, combining like terms, adding 2 to both sides, and dividing both sides by -25, we were able to isolate the variable $r$ and find its value.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS).
Distributing the negative sign can be a challenging step, but it's crucial to get it right.
Combining like terms can help simplify the equation and make it easier to solve.
Adding or subtracting the same value to both sides of the equation is a common technique used to isolate the variable.

Real-World Applications

Linear equations have numerous real-world applications, including:

Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and distance.
Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and structural analysis.
Economics: Linear equations are used to model economic systems, including supply and demand, cost-benefit analysis, and optimization problems.

Common Mistakes

Failing to distribute the negative sign correctly.
Not combining like terms.
Adding or subtracting the wrong value to both sides of the equation.
Not simplifying the fraction correctly.

Practice Problems

Try solving the following linear equations:

$2x + 5 = 11$
$-3x - 2 = 7$
$x + 2 = 9$

Conclusion

Introduction

In our previous article, we explored the step-by-step process of solving linear equations. However, we understand that sometimes, it's not just about following a set of rules, but also about understanding the underlying concepts and addressing common questions and concerns. In this article, we'll tackle some of the most frequently asked questions about solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it's an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation has a highest power of 1, while a quadratic equation has a highest power of 2. In other words, a linear equation can be written in the form ax + b = c, while a quadratic equation can be written in the form ax^2 + bx + c = 0.

Q: How do I know if an equation is linear or quadratic?

A: To determine if an equation is linear or quadratic, look for the highest power of the variable(s). If it's 1, it's a linear equation. If it's 2, it's a quadratic equation.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictates the order in which mathematical operations should be performed. It stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I distribute a negative sign?

A: To distribute a negative sign, multiply the negative sign by each term inside the parentheses. For example, if you have -2(3x + 4), you would multiply the negative sign by each term inside the parentheses, resulting in -6x - 8.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation with one variable, while a system of linear equations is a set of two or more linear equations with the same variable(s).

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you can use the following methods:

Substitution method: Substitute the expression for one variable from one equation into the other equation.
Elimination method: Add or subtract the equations to eliminate one variable.
Graphing method: Graph the equations on a coordinate plane and find the point of intersection.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

Failing to distribute the negative sign correctly.
Not combining like terms.
Adding or subtracting the wrong value to both sides of the equation.
Not simplifying the fraction correctly.

Conclusion

Solving linear equations is a fundamental skill that requires practice and patience. By understanding the underlying concepts and addressing common questions and concerns, you can become proficient in solving linear equations and apply them to real-world problems. Remember to follow the order of operations, distribute the negative sign correctly, combine like terms, and simplify the fraction correctly. With practice, you'll become a master of solving linear equations!