Solve The System Of Equations:$\[ \begin{array}{c} 2c - 5z = 21 \\ -6c = 12 \end{array} \\]
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Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. 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{array}{c} 2c - 5z = 21 \\ -6c = 12 \end{array} \}
This system consists of two linear equations with two variables, c and z. Our goal is to find the values of c and z that satisfy both equations.
Step 1: Isolate One Variable
To solve the system of equations, we can start by isolating one variable in one of the equations. Let's isolate the variable c in the second equation.
{ -6c = 12 \}
To isolate c, we can divide both sides of the equation by -6.
{ c = -\frac{12}{6} \}
{ c = -2 \}
Step 2: Substitute the Value of c into the First Equation
Now that we have isolated the variable c, we can substitute its value into the first equation.
{ 2c - 5z = 21 \}
Substituting c = -2 into the first equation, we get:
{ 2(-2) - 5z = 21 \}
{ -4 - 5z = 21 \}
Step 3: Solve for z
To solve for z, we can add 4 to both sides of the equation.
{ -4 - 5z + 4 = 21 + 4 \}
{ -5z = 25 \}
To solve for z, we can divide both sides of the equation by -5.
{ z = -\frac{25}{5} \}
{ z = -5 \}
Conclusion
In this article, we have solved a system of two linear equations with two variables. We started by isolating one variable in one of the equations, and then substituted its value into the other equation. Finally, we solved for the remaining variable. The values of c and z that satisfy both equations are c = -2 and z = -5.
Example Use Cases
Solving systems of linear equations has many practical 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.
- Economics: Systems of linear equations are used to model economic systems, such as supply and demand curves.
- Computer Science: Systems of linear equations are used in computer graphics, machine learning, and data analysis.
Tips and Tricks
When solving systems of linear equations, it's essential to:
- Check your work: Verify that the solution satisfies both equations.
- Use substitution or elimination: Choose the method that works best for the given system.
- Simplify the equations: Simplify the equations before solving to make the solution process easier.
Conclusion
Solving systems of linear equations is a fundamental concept in mathematics that has numerous practical applications. By following the steps outlined in this article, you can solve systems of linear equations with ease. Remember to check your work, use substitution or elimination, and simplify the equations to make the solution process easier.
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Q: What is a system of linear equations?
A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.
Q: How do I know which method to use to solve a system of linear equations?
You can use either the substitution method or the elimination method to solve a system of linear equations. The substitution method involves isolating one variable in one of the equations and substituting its value into the other equation. The elimination method involves adding or subtracting the equations to eliminate one of the variables.
Q: What is the difference between the substitution method and the elimination method?
The substitution method involves isolating one variable in one of the equations and substituting its value into the other equation. The elimination method involves adding or subtracting the equations to eliminate one of the variables.
Q: How do I know if a system of linear equations has a unique solution, no solution, or infinitely many solutions?
To determine the number of solutions to a system of linear equations, you can use the following criteria:
- Unique solution: If the system has a unique solution, the equations are consistent and the number of equations is equal to the number of variables.
- No solution: If the system has no solution, the equations are inconsistent and the number of equations is equal to the number of variables.
- Infinitely many solutions: If the system has infinitely many solutions, the equations are consistent and the number of equations is less than the number of variables.
Q: How do I check if a system of linear equations has a unique solution, no solution, or infinitely many solutions?
To check if a system of linear equations has a unique solution, no solution, or infinitely many solutions, you can use the following criteria:
- Unique solution: If the system has a unique solution, the equations are consistent and the number of equations is equal to the number of variables.
- No solution: If the system has no solution, the equations are inconsistent and the number of equations is equal to the number of variables.
- Infinitely many solutions: If the system has infinitely many solutions, the equations are consistent and the number of equations is less than the number of variables.
Q: What is the importance of solving systems of linear equations?
Solving systems of linear equations is a fundamental concept in mathematics that has numerous practical applications in various fields, including physics, engineering, economics, and computer science.
Q: How do I apply the concepts of solving systems of linear equations in real-world problems?
To apply the concepts of solving systems of linear equations in real-world problems, you can use the following steps:
- Model the problem: Model the problem using a system of linear equations.
- Solve the system: Solve the system of linear equations using the substitution method or the elimination method.
- Interpret the results: Interpret the results of the solution to the system of linear equations.
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 the consistency of the equations: Make sure to check the consistency of the equations before solving the system.
- Not using the correct method: Choose the correct method to solve the system, either the substitution method or the elimination method.
- Not simplifying the equations: Simplify the equations before solving to make the solution process easier.
Q: How do I determine the number of solutions to a system of linear equations?
To determine the number of solutions to a system of linear equations, you can use the following criteria:
- Unique solution: If the system has a unique solution, the equations are consistent and the number of equations is equal to the number of variables.
- No solution: If the system has no solution, the equations are inconsistent and the number of equations is equal to the number of variables.
- Infinitely many solutions: If the system has infinitely many solutions, the equations are consistent and the number of equations is less than the number of variables.