Use substitution or elimination

Oct 13, 2025 by pynkocean.com 102 views

=====================================================

Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. 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 to demonstrate the steps involved in solving such a system.

The System of Equations

The given system of equations is:

{ \begin{align*} 5a - 2z &= -5 \\ 2a &= 2 \end{align*} \}

This system consists of two linear equations with two variables, $a$ and $z$ . The first equation is $5a - 2z = -5$ , and the second equation is $2a = 2$ .

Step 1: Solve the Second Equation for $a$

The second equation is $2a = 2$ . To solve for $a$ , we can divide both sides of the equation by 2.

{ 2a = 2 \Rightarrow a = \frac{2}{2} = 1 \}

So, the value of $a$ is 1.

Step 2: Substitute the Value of $a$ into the First Equation

Now that we have the value of $a$ , we can substitute it into the first equation to solve for $z$ .

{ 5a - 2z = -5 \Rightarrow 5(1) - 2z = -5 \}

Simplifying the equation, we get:

{ 5 - 2z = -5 \}

Step 3: Solve for $z$

To solve for $z$ , we can isolate the variable by adding 2 to both sides of the equation.

{ 5 - 2z = -5 \Rightarrow -2z = -10 \}

Dividing both sides of the equation by -2, we get:

{ -2z = -10 \Rightarrow z = \frac{-10}{-2} = 5 \}

So, the value of $z$ is 5.

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 to demonstrate the steps involved in solving such a system. We first solved the second equation for $a$ , then substituted the value of $a$ into the first equation to solve for $z$ . The final solution is $a = 1$ and $z = 5$ .

Example Use Cases

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

Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects under the influence of forces.
Computer Science: Linear equations are used in computer graphics, game development, and machine learning.
Economics: Systems of linear equations are used to model economic systems, such as supply and demand curves.

Tips and Tricks

When solving systems of linear equations, it's essential to:

Check for consistency: Ensure that the system of equations is consistent, meaning that the equations are not contradictory.
Use substitution or elimination: Choose the method that is most convenient for the given system of equations.
Check for extraneous solutions: Verify that the solutions obtained are not extraneous, meaning that they do not satisfy the original equations.

Final Thoughts

Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields. By following the steps outlined in this article, you can solve systems of linear equations with ease. Remember to check for consistency, use substitution or elimination, and check for extraneous solutions to ensure that you obtain the correct solutions.

====================================================================

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it can be a bit challenging for some students. 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 system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is a linear equation, meaning that it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

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

A system of linear equations has a solution if and only if the equations are consistent. Consistent equations are equations that have at least one solution in common. If the equations are inconsistent, then they have no solution in common.

Q: What are the two main methods for solving systems of linear equations?

The two main methods for solving systems of linear equations are the substitution method and the elimination method.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one of the equations is already solved for one variable.

Elimination Method

The elimination method involves adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of the variables are the same in both equations.

Q: How do I choose between the substitution method and the elimination method?

You should choose the method that is most convenient for the given system of equations. If one of the equations is already solved for one variable, then the substitution method may be the best choice. If the coefficients of the variables are the same in both equations, then the elimination method may be the best choice.

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

A dependent system of linear equations is a system that has infinitely many solutions. This occurs when the equations are equivalent, meaning that one equation can be obtained from the other by multiplying both sides by a constant.

An independent system of linear equations is a system that has a unique solution. This occurs when the equations are not equivalent, meaning that one equation cannot be obtained from the other by multiplying both sides by a constant.

Q: How do I determine if a system of linear equations is dependent or independent?

You can determine if a system of linear equations is dependent or independent by checking if the equations are equivalent. If the equations are equivalent, then the system is dependent. If the equations are not equivalent, then the system is independent.

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

Some common mistakes to avoid when solving systems of linear equations include:

Not checking for consistency: Make sure that the system of equations is consistent before solving it.
Not choosing the correct method: Choose the method that is most convenient for the given system of equations.
Not checking for extraneous solutions: Verify that the solutions obtained are not extraneous, meaning that they do not satisfy the original equations.