Solve The System Of Equations:${ \begin{cases} 2x + 5y = 14 \ 4x + 2y = -4 \end{cases} }$

Mar 7, 2025 by ADMIN 91 views

**Solving Systems of Linear Equations: A Step-by-Step Guide**

Introduction

Systems of 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 system of two linear equations with two variables. We will use the given system of equations as an example and provide a step-by-step guide on how to solve it.

What are Systems of Linear Equations?

A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is a statement that two expressions are equal, and the variables are the unknown values that we need to find. In this article, we will focus on solving a system of two linear equations with two variables.

The Given System of Equations

The given system of equations is:

{ \begin{cases} 2x + 5y = 14 \\ 4x + 2y = -4 \end{cases} \}

Step 1: Write Down the System of Equations

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

$2x + 5y = 14$
$4x + 2y = -4$

Step 2: Solve One Equation for One Variable

The next step is to solve one equation for one variable. We can choose either equation and solve it for either variable. Let's solve the first equation for $x$ :

$2x + 5y = 14$

Subtract $5y$ from both sides:

$2x = 14 - 5y$

Divide both sides by 2:

$x = \frac{14 - 5y}{2}$

Step 3: Substitute the Expression into the Other Equation

Now that we have an expression for $x$ , we can substitute it into the other equation. Let's substitute the expression into the second equation:

$4x + 2y = -4$

Substitute $x = \frac{14 - 5y}{2}$ into the equation:

$4\left(\frac{14 - 5y}{2}\right) + 2y = -4$

Simplify the equation:

$2(14 - 5y) + 2y = -4$

Expand and simplify:

$28 - 10y + 2y = -4$

Combine like terms:

$28 - 8y = -4$

Step 4: Solve for the Other Variable

Now that we have a single equation with one variable, we can solve for that variable. Let's solve for $y$ :

$28 - 8y = -4$

Subtract 28 from both sides:

$-8y = -32$

Divide both sides by -8:

$y = \frac{-32}{-8}$

Simplify:

$y = 4$

Step 5: Find the Value of the Other Variable

Now that we have the value of $y$ , we can find the value of the other variable. Let's find the value of $x$ :

$x = \frac{14 - 5y}{2}$

Substitute $y = 4$ into the equation:

$x = \frac{14 - 5(4)}{2}$

Simplify:

$x = \frac{14 - 20}{2}$

Combine like terms:

$x = \frac{-6}{2}$

Simplify:

$x = -3$

Conclusion

In this article, we solved a system of two linear equations with two variables. We used the given system of equations as an example and provided a step-by-step guide on how to solve it. We wrote down the system of equations, solved one equation for one variable, substituted the expression into the other equation, solved for the other variable, and found the value of the other variable. The final solution is $x = -3$ and $y = 4$ .

Tips and Tricks

When solving a system of linear equations, it's essential to write down the system of equations clearly and accurately.
When solving one equation for one variable, make sure to simplify the expression as much as possible.
When substituting the expression into the other equation, make sure to simplify the equation as much as possible.
When solving for the other variable, make sure to simplify the equation as much as possible.
When finding the value of the other variable, make sure to substitute the value of the other variable into the equation.

Common Mistakes

Not writing down the system of equations clearly and accurately.
Not simplifying the expression when solving one equation for one variable.
Not simplifying the equation when substituting the expression into the other equation.
Not simplifying the equation when solving for the other variable.
Not substituting the value of the other variable into the equation when finding the value of the other variable.

Real-World Applications

Systems of linear equations have many real-world applications, including:

Physics: Systems of linear equations are used to describe the motion of objects in physics.
Engineering: Systems of linear equations are used to design and optimize systems in engineering.
Economics: Systems of linear equations are used to model economic systems and make predictions about economic trends.
Computer Science: Systems of linear equations are used in computer science to solve problems in machine learning and data analysis.

Conclusion

Introduction

In our previous article, we provided a step-by-step guide on how to solve a system of two linear equations with two variables. In this article, we will answer 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 two or more variables. Each equation is a statement that two expressions are equal, and the variables are the unknown values that we need to find.

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, we need to check if the two equations are consistent. If the two equations are consistent, then the system has a solution. If the two equations are inconsistent, then the system does not have a solution.

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

A: A consistent system of linear equations is one that has a solution. An inconsistent system of linear equations is one that does not have a solution.

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

A: To solve a system of linear equations with three variables, we need to use the method of substitution or elimination. We can solve one equation for one variable and substitute it into the other equations, or we can add or subtract the equations to eliminate one variable.

Q: What is the method of substitution?

A: The method of substitution is a technique used to solve a system of linear equations by substituting one equation into the other equations.

Q: What is the method of elimination?

A: The method of elimination is a technique used to solve a system of linear equations by adding or subtracting the equations to eliminate one variable.

Q: How do I use the method of substitution to solve a system of linear equations?

A: To use the method of substitution to solve a system of linear equations, we need to solve one equation for one variable and substitute it into the other equations.

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

A: To use the method of elimination to solve a system of linear equations, we need to add or subtract the equations to eliminate one variable.

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

A: A linear equation is an equation that can be written in the form ax + by = c, where a, b, and c are constants. A nonlinear equation is an equation that cannot be written in the form ax + by = c.

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

A: To solve a system of nonlinear equations, we need to use numerical methods or algebraic methods. Numerical methods involve using a computer to approximate the solution, while algebraic methods involve using algebraic techniques to solve the system.

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

A: A homogeneous system of linear equations is one that has the form ax + by = 0, where a and b are constants. A nonhomogeneous system of linear equations is one that has the form ax + by = c, where c is a constant.

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

A: To solve a homogeneous system of linear equations, we need to find the solution to the system. If the system has a nontrivial solution, then the solution is a non-zero vector. If the system has only the trivial solution, then the solution is the zero vector.

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

A: To solve a nonhomogeneous system of linear equations, we need to find the solution to the system. If the system has a nontrivial solution, then the solution is a non-zero vector. If the system has only the trivial solution, then the solution is the zero vector.

Conclusion

In this article, we answered some of the most frequently asked questions about solving systems of linear equations. We covered topics such as the definition of a system of linear equations, the method of substitution, the method of elimination, and the difference between a linear equation and a nonlinear equation. We also covered topics such as homogeneous and nonhomogeneous systems of linear equations and how to solve them.