Example 1. Find The Solution Of 15 X + 12 Y + 30 Z = 24 15x + 12y + 30z = 24 15 X + 12 Y + 30 Z = 24 .

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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 finding the solution of a linear equation in three variables, specifically $15x + 12y + 30z = 24$. We will break down the solution process into manageable steps, using a combination of algebraic manipulations and logical reasoning.

Understanding the Equation

The given equation is a linear equation in three variables, $x$, $y$, and $z$. It can be written in the general form:

$$ax + by + cz = d$$

where $a$, $b$, $c$, and $d$ are constants. In this case, we have:

$$15x + 12y + 30z = 24$$

Our goal is to find the values of $x$, $y$, and $z$ that satisfy this equation.

Step 1: Isolate One Variable

To solve the equation, we can start by isolating one variable. Let's choose to isolate $x$. We can do this by subtracting $12y + 30z$ from both sides of the equation:

$$15x = 24 - 12y - 30z$$

Now, we can divide both sides by $15$ to get:

$$x = \frac{24 - 12y - 30z}{15}$$

Step 2: Substitute the Expression for $x$

Now that we have an expression for $x$, we can substitute it into the original equation. This will give us an equation in two variables, $y$ and $z$.

$$15\left(\frac{24 - 12y - 30z}{15}\right) + 12y + 30z = 24$$

Simplifying the equation, we get:

$$24 - 12y - 30z + 12y + 30z = 24$$

Step 3: Simplify the Equation

The equation simplifies to:

$$24 = 24$$

This is a true statement, which means that the equation is an identity. In other words, the equation is true for all values of $y$ and $z$.

Conclusion

In this article, we have shown how to solve a linear equation in three variables, specifically $15x + 12y + 30z = 24$. We broke down the solution process into manageable steps, using a combination of algebraic manipulations and logical reasoning. The final solution is an identity, which means that the equation is true for all values of $y$ and $z$.

Example Use Cases

Linear equations are used in a wide range of applications, including:

• Physics: Linear equations are used to describe the motion of objects under the influence of forces.
• 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.

Tips and Tricks

Here are some tips and tricks for solving linear equations:

• Use algebraic manipulations: Linear equations can be solved using a combination of algebraic manipulations, such as addition, subtraction, multiplication, and division.
• Use logical reasoning: Linear equations can be solved using logical reasoning, such as identifying patterns and relationships between variables.
• Check your work: Always check your work to ensure that the solution is correct.

Further Reading

If you want to learn more about linear equations, here are some additional resources:

• Textbooks: There are many textbooks available on linear equations, including "Linear Algebra and Its Applications" by Gilbert Strang and "Introduction to Linear Algebra" by Gilbert Strang.
• Online resources: There are many online resources available on linear equations, including Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
• Practice problems: Practice problems are a great way to reinforce your understanding of linear equations. You can find practice problems online or in textbooks.
  

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 answer some of the most frequently asked questions about 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 is an equation that can be written in the form:

$$ax + by + cz = d$$

where $a$, $b$, $c$, and $d$ are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use a combination of algebraic manipulations and logical reasoning. Here are the steps:

1. Isolate one variable: Choose one variable and isolate it on one side of the equation.
2. Substitute the expression: Substitute the expression for the isolated variable into the original equation.
3. Simplify the equation: Simplify the equation to get a true statement.

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

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. In other words, a linear equation can be written in the form:

$$ax + by + cz = d$$

while a quadratic equation can be written in the form:

$$ax^2 + bx + c = d$$

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 to ensure that the solution is correct.

Q: How do I check my work?

A: To check your work, you can plug the solution back into the original equation and see if it's true. If it's true, then the solution is correct.

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

A: Here are some common mistakes to avoid when solving linear equations:

• Not isolating one variable: Make sure to isolate one variable on one side of the equation.
• Not substituting the expression: Make sure to substitute the expression for the isolated variable into the original equation.
• Not simplifying the equation: Make sure to simplify the equation to get a true statement.

Q: Can I use linear equations to model real-world problems?

A: Yes, linear equations can be used to model real-world problems. For example, you can use linear equations to model the motion of an object under the influence of forces, or to design and optimize systems.

Q: What are some examples of real-world applications of linear equations?

A: Here are some examples of real-world applications of linear equations:

• Physics: Linear equations are used to describe the motion of objects under the influence of forces.
• 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: Here are some ways to practice solving linear equations:

• Practice problems: Practice problems are a great way to reinforce your understanding of linear equations. You can find practice problems online or in textbooks.
• Online resources: There are many online resources available on linear equations, including Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
• Textbooks: There are many textbooks available on linear equations, including "Linear Algebra and Its Applications" by Gilbert Strang and "Introduction to Linear Algebra" by Gilbert Strang.

Conclusion

In this article, we have answered some of the most frequently asked questions about linear equations. We hope that this article has been helpful in clarifying any confusion you may have had about linear equations. If you have any further questions, please don't hesitate to ask.