Solve The Following Equation For $x$. Express Your Answer In The Simplest Form. 5 ( − 2 X − 3 ) = 9 X + 21 5(-2x - 3) = 9x + 21 5 ( − 2 X − 3 ) = 9 X + 21 The Equation Has:- No Solution- Infinitely Many Solutions- One Solution
Introduction
Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation, $5(-2x - 3) = 9x + 21$, and explore the different possibilities for the solution.
Understanding the Equation
Before we dive into solving the equation, let's break it down and understand its components. The equation is a linear equation, which means it is in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants. In this case, the equation is $5(-2x - 3) = 9x + 21$.
Step 1: Distribute the Coefficient
To solve the equation, we need to start by distributing the coefficient $5$ to the terms inside the parentheses. This will give us $-10x - 15 = 9x + 21$.
Step 2: Add or Subtract the Same Value
Next, we need to get all the terms with $x$ on one side of the equation. To do this, we can add $10x$ to both sides of the equation, which will give us $-15 = 19x + 21$.
Step 3: Subtract the Same Value
Now, we need to get rid of the constant term on the right-hand side of the equation. To do this, we can subtract $21$ from both sides of the equation, which will give us $-36 = 19x$.
Step 4: Solve for x
Finally, we can solve for $x$ by dividing both sides of the equation by $19$. This will give us $x = -\frac{36}{19}$.
Conclusion
In this article, we have solved the linear equation $5(-2x - 3) = 9x + 21$ and found that the solution is $x = -\frac{36}{19}$. This means that the equation has one solution, which is a unique value of $x$ that satisfies the equation.
The Possibilities for the Solution
But what if the equation had no solution or infinitely many solutions? Let's explore these possibilities in more detail.
No Solution
If the equation had no solution, it would mean that there is no value of $x$ that satisfies the equation. This can happen if the equation is inconsistent, meaning that it is impossible to find a value of $x$ that makes both sides of the equation true.
Infinitely Many Solutions
If the equation had infinitely many solutions, it would mean that there are an infinite number of values of $x$ that satisfy the equation. This can happen if the equation is an identity, meaning that it is always true for any value of $x$.
How to Determine the Possibility of the Solution
So, how can we determine whether the equation has one solution, no solution, or infinitely many solutions? Here are some tips:
- Check for consistency: If the equation is inconsistent, it will have no solution.
- Check for identity: If the equation is an identity, it will have infinitely many solutions.
- Solve the equation: If you can solve the equation, it will have one solution.
Real-World Applications of Linear Equations
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 future trends.
Conclusion
In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, you can solve linear equations and determine whether they have one solution, no solution, or infinitely many solutions. Remember to check for consistency and identity, and to solve the equation to determine the possibility of the solution. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.
Final Thoughts
Introduction
In our previous article, we explored the concept of solving linear equations and provided a step-by-step guide on how to solve a specific linear equation. In this article, we will answer some 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 (usually x) is 1. It is in the form of ax + b = c, where a, b, and c are constants.
Q: How do I know if an equation is linear?
A: To determine if an equation is linear, look for the highest power of the variable. If it is 1, then the equation is linear. If it is greater than 1, then the equation is not linear.
Q: What are the steps to solve a linear equation?
A: The steps to solve a linear equation are:
- Distribute the coefficient to the terms inside the parentheses.
- Add or subtract the same value to both sides of the equation.
- Solve for x by dividing both sides of the equation by the coefficient.
Q: What if the equation has no solution?
A: If the equation has no solution, it means that there is no value of x that satisfies the equation. This can happen if the equation is inconsistent, meaning that it is impossible to find a value of x that makes both sides of the equation true.
Q: What if the equation has infinitely many solutions?
A: If the equation has infinitely many solutions, it means that there are an infinite number of values of x that satisfy the equation. This can happen if the equation is an identity, meaning that it is always true for any value of x.
Q: How do I determine if an equation is consistent or inconsistent?
A: To determine if an equation is consistent or inconsistent, check if the equation is true for any value of x. If it is true for any value of x, then the equation is consistent. If it is not true for any value of x, then the equation is inconsistent.
Q: How do I determine if an equation is an identity?
A: To determine if an equation is an identity, check if the equation is true for any value of x. If it is true for any value of x, then the equation is an identity.
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 future trends.
Q: How can I practice solving linear equations?
A: You can practice solving linear equations by working on problems and exercises. You can also use online resources, such as math websites and apps, to practice solving linear equations.
Conclusion
In conclusion, solving linear equations is a crucial skill for students to master. By understanding the concept of linear equations and following the steps outlined in this article, you can solve linear equations and determine whether they have one solution, no solution, or infinitely many solutions. Remember to check for consistency and identity, and to solve the equation to determine the possibility of the solution. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.
Final Thoughts
Solving linear equations is a fundamental concept in mathematics, and it has many real-world applications. By understanding the concept of linear equations and practicing solving them, you can become proficient in solving linear equations and apply them to real-world problems. Remember to check for consistency and identity, and to solve the equation to determine the possibility of the solution. With practice and patience, you can master the art of solving linear equations.