Solve The System Of Equations:${ \begin{array}{l} 4a = 1 - 3b \ 4b = -1 - 6a \end{array} }$

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.

The System of Equations

The given system of equations is:

$$\begin{array}{l} 4a = 1 - 3b \\ 4b = -1 - 6a \end{array}$$

Step 1: Write Down the Equations

The first step in solving a system of linear equations is to write down the equations. In this case, we have two equations:

1. $4a = 1 - 3b$
2. $4b = -1 - 6a$

Step 2: Solve One Equation for One Variable

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method. We will solve one equation for one variable and then substitute that expression into the other equation.

Let's solve the first equation for $a$:

$$4a = 1 - 3b$$

$$a = \frac{1 - 3b}{4}$$

Now, substitute this expression for $a$ into the second equation:

$$4b = -1 - 6\left(\frac{1 - 3b}{4}\right)$$

Step 3: Simplify the Equation

Simplify the equation by multiplying both sides by 4:

$$16b = -4 - 6(1 - 3b)$$

$$16b = -4 - 6 + 18b$$

$$16b = -10 + 18b$$

Step 4: Isolate the Variable

Subtract $18b$ from both sides:

$$-2b = -10$$

Divide both sides by $-2$:

$$b = 5$$

Step 5: Find the Value of the Other Variable

Now that we have found the value of $b$, we can substitute it into one of the original equations to find the value of the other variable. Let's substitute $b = 5$ into the first equation:

$$4a = 1 - 3(5)$$

$$4a = 1 - 15$$

$$4a = -14$$

Divide both sides by 4:

$$a = -\frac{14}{4}$$

$$a = -\frac{7}{2}$$

Conclusion

In this article, we have solved a system of two linear equations with two variables using the elimination method. We have found the values of both variables, $a$ and $b$, that satisfy both equations in the system. The final answer is:

$a = -\frac{7}{2}$

$b = 5$

Example Use Cases

Solving systems of linear equations has many practical applications in various fields, including:

• Physics and Engineering: Solving systems of linear equations is essential in physics and engineering to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
• Computer Science: Solving systems of linear equations is used in computer science to solve problems in machine learning, data analysis, and computer graphics.
• Economics: Solving systems of linear equations is used in economics to model economic systems, such as supply and demand, and to make predictions about economic trends.

Tips and Tricks

• Use the elimination method: The elimination method is a powerful tool for solving systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables.
• Use substitution: Substitution is another method for solving systems of linear equations. It involves substituting one expression into another equation to eliminate one of the variables.
• Check your work: Always check your work by plugging the values of the variables back into the original equations to make sure they are true.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics that has many practical applications in various fields. In this article, we have solved a system of two linear equations with two variables using the elimination method. We have found the values of both variables, $a$ and $b$, that satisfy both equations in the system. The final answer is:

$a = -\frac{7}{2}$

$b = 5$

Introduction

In our previous article, we discussed how to solve a system of two linear equations with two variables using the elimination method. However, we understand that solving systems of linear equations can be a challenging task, and many students and professionals have questions about this topic. In this article, we will address some of the most frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. In other words, it is a collection of equations that are all true at the same time.

Q: How do I know if a system of linear equations has a solution?

A: To determine if a system of linear equations has a solution, you need to check if the equations are consistent. If the equations are consistent, then the system has a solution. If the equations are inconsistent, then the system does not have a solution.

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

A: A consistent system of linear equations is one that has a solution. In other words, the equations are true at the same time. An inconsistent system of linear equations is one that does not have a solution. In other words, the equations are contradictory.

Q: How do I solve a system of linear equations using the elimination method?

A: To solve a system of linear equations using the elimination method, you need to follow these steps:

1. Write down the equations.
2. Multiply both sides of one or both equations by a constant to make the coefficients of one of the variables the same.
3. Add or subtract the equations to eliminate one of the variables.
4. Solve for the remaining variable.
5. Substitute the value of the remaining variable back into one of the original equations to find the value of the other variable.

Q: What is the difference between the elimination method and the substitution method?

A: The elimination method involves adding or subtracting equations to eliminate one of the variables. The substitution method involves substituting one expression into another equation to eliminate one of the variables.

Q: How do I know if a system of linear equations has infinitely many solutions?

A: A system of linear equations has infinitely many solutions if the equations are dependent. In other words, one of the equations is a multiple of the other equation.

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

A: A dependent system of linear equations is one that has infinitely many solutions. An independent system of linear equations is one that has a unique solution.

Q: Can a system of linear equations have no solution?

A: Yes, a system of linear equations can have no solution. This occurs when the equations are inconsistent.

Q: How do I check if a system of linear equations is consistent or inconsistent?

A: To check if a system of linear equations is consistent or inconsistent, you need to follow these steps:

1. Write down the equations.
2. Check if the equations are true at the same time.
3. If the equations are true at the same time, then the system is consistent.
4. If the equations are not true at the same time, then the system is inconsistent.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics that has many practical applications in various fields. In this article, we have addressed some of the most frequently asked questions about solving systems of linear equations. We hope this article has provided a clear and concise guide to solving systems of linear equations.

Example Use Cases

Solving systems of linear equations has many practical applications in various fields, including:

• Physics and Engineering: Solving systems of linear equations is essential in physics and engineering to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
• Computer Science: Solving systems of linear equations is used in computer science to solve problems in machine learning, data analysis, and computer graphics.
• Economics: Solving systems of linear equations is used in economics to model economic systems, such as supply and demand, and to make predictions about economic trends.

Tips and Tricks

• Use the elimination method: The elimination method is a powerful tool for solving systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables.
• Use substitution: Substitution is another method for solving systems of linear equations. It involves substituting one expression into another equation to eliminate one of the variables.
• Check your work: Always check your work by plugging the values of the variables back into the original equations to make sure they are true.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics that has many practical applications in various fields. In this article, we have addressed some of the most frequently asked questions about solving systems of linear equations. We hope this article has provided a clear and concise guide to solving systems of linear equations.