Paula Used These Steps To Solve An Equation:Step 1: \[$-4(x+8)-2x=25\$\] Step 2: \[$-4x-32-2x=25\$\] Step 3: \[$-6x-32=25\$\] Step 4: \[$-6x=57\$\] Step 5: \[$x=-9 \frac{1}{2}\$\]Between Which Two Steps Did
Introduction
Solving linear equations is a fundamental concept in mathematics that involves isolating the variable to find its value. In this article, we will walk through the steps taken by Paula to solve a linear equation. We will analyze the equation and identify the key steps involved in solving it.
The Equation
The equation given to Paula is:
This equation involves a linear expression inside parentheses, which needs to be simplified before solving for the variable x.
Step 1: Simplifying the Equation
The first step in solving the equation is to simplify the expression inside the parentheses.
Using the distributive property, we can rewrite the equation as:
This simplification helps to combine like terms and makes it easier to solve for x.
Step 2: Combining Like Terms
In the previous step, we simplified the expression inside the parentheses. Now, we can combine like terms to further simplify the equation.
Combining the like terms -4x and -2x, we get:
This step helps to reduce the complexity of the equation and makes it easier to solve for x.
Step 3: Isolating the Variable
Now that we have simplified the equation, we can isolate the variable x by moving the constant term to the other side of the equation.
Adding 32 to both sides of the equation, we get:
This step helps to isolate the variable x and makes it easier to solve for its value.
Step 4: Solving for x
In the previous step, we isolated the variable x. Now, we can solve for its value by dividing both sides of the equation by the coefficient of x.
Dividing both sides of the equation by -6, we get:
This step provides the solution to the equation and gives us the value of the variable x.
Conclusion
In this article, we walked through the steps taken by Paula to solve a linear equation. We analyzed the equation and identified the key steps involved in solving it. By simplifying the expression inside the parentheses, combining like terms, isolating the variable, and solving for x, we were able to find the value of the variable x. This step-by-step guide provides a clear understanding of how to solve linear equations and can be applied to a variety of mathematical problems.
Between Which Two Steps Did the Solution Occur?
To determine between which two steps the solution occurred, we need to analyze the equation and identify the key steps involved in solving it.
The solution occurred between Step 3: Isolating the Variable and Step 4: Solving for x.
In Step 3, we isolated the variable x by moving the constant term to the other side of the equation. This step helped to reduce the complexity of the equation and made it easier to solve for x.
In Step 4, we solved for the value of x by dividing both sides of the equation by the coefficient of x. This step provided the solution to the equation and gave us the value of the variable x.
Therefore, the solution occurred between Step 3 and Step 4, as these two steps involved isolating the variable and solving for its value.
Key Takeaways
- Simplifying the expression inside the parentheses is an essential step in solving linear equations.
- Combining like terms helps to reduce the complexity of the equation and makes it easier to solve for x.
- Isolating the variable by moving the constant term to the other side of the equation is a crucial step in solving linear equations.
- Solving for the value of x by dividing both sides of the equation by the coefficient of x provides the solution to the equation.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants.
Q: What are the steps involved in solving a linear equation?
A: The steps involved in solving a linear equation are:
- Simplifying the expression inside the parentheses
- Combining like terms
- Isolating the variable by moving the constant term to the other side of the equation
- Solving for the value of x by dividing both sides of the equation by the coefficient of x
Q: Why is it important to simplify the expression inside the parentheses?
A: Simplifying the expression inside the parentheses helps to combine like terms and makes it easier to solve for x. It also helps to reduce the complexity of the equation.
Q: What is the difference between combining like terms and simplifying the expression inside the parentheses?
A: Combining like terms involves adding or subtracting terms that have the same variable and exponent. Simplifying the expression inside the parentheses involves rewriting the expression using the distributive property.
Q: How do I isolate the variable in a linear equation?
A: To isolate the variable in a linear equation, you need to move the constant term to the other side of the equation. This can be done by adding or subtracting the same value to both sides of the equation.
Q: What is the coefficient of x in a linear equation?
A: The coefficient of x in a linear equation is the number that is multiplied by the variable x. It can be a positive or negative number.
Q: How do I solve for the value of x in a linear equation?
A: To solve for the value of x in a linear equation, you need to divide both sides of the equation by the coefficient of x. This will give you the value of x.
Q: What is the solution to a linear equation?
A: The solution to a linear equation is the value of the variable x that makes the equation true.
Q: Can I use a calculator to solve a linear equation?
A: Yes, you can use a calculator to solve a linear equation. However, it's always a good idea to check your work by plugging the solution back into the original equation.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not simplifying the expression inside the parentheses
- Not combining like terms
- Not isolating the variable
- Not solving for the value of x
- Not checking your work
Q: How can I practice solving linear equations?
A: You can practice solving linear equations by working through examples and exercises in a textbook or online resource. You can also try solving linear equations on your own using a calculator or by hand.
Q: What are some real-world applications of linear equations?
A: Linear equations have many real-world applications, including:
- Physics: Linear equations are used to describe the motion of objects under constant acceleration.
- Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: Linear equations are used to model economic systems and make predictions about economic trends.
- Computer Science: Linear equations are used in computer graphics and game development to create realistic simulations.
By following these steps and avoiding common mistakes, you can become proficient in solving linear equations and apply them to a variety of real-world problems.