Solve The Following System Of Equations:${ \begin{cases} 5x + 3y = 28 \ 7x + 2y = -7 \end{cases} }$
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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. These equations are linear because they are in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. In this article, we will focus on solving a system of two linear equations with two variables.
The System of Equations
The system of equations we will be solving is:
{ \begin{cases} 5x + 3y = 28 \\ 7x + 2y = -7 \end{cases} \}
This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations simultaneously.
Method of Elimination
One of the most common methods for solving a system of linear equations is the method of elimination. This method involves adding or subtracting the equations to eliminate one of the variables. In this case, we can multiply the first equation by 2 and the second equation by 3 to make the coefficients of y in both equations equal.
Step 1: Multiply the First Equation by 2
Multiplying the first equation by 2 gives us:
{ 10x + 6y = 56 \}
Step 2: Multiply the Second Equation by 3
Multiplying the second equation by 3 gives us:
{ 21x + 6y = -21 \}
Step 3: Subtract the Second Equation from the First Equation
Now that we have the coefficients of y in both equations equal, we can subtract the second equation from the first equation to eliminate the y variable.
{ (10x + 6y) - (21x + 6y) = 56 - (-21) \}
Simplifying the equation gives us:
{ -11x = 77 \}
Step 4: Solve for x
Now that we have the equation -11x = 77, we can solve for x by dividing both sides of the equation by -11.
{ x = -\frac{77}{11} \}
Simplifying the equation gives us:
{ x = -7 \}
Step 5: Substitute x into One of the Original Equations
Now that we have the value of x, we can substitute it into one of the original equations to solve for y. Let's use the first equation:
{ 5x + 3y = 28 \}
Substituting x = -7 into the equation gives us:
{ 5(-7) + 3y = 28 \}
Simplifying the equation gives us:
{ -35 + 3y = 28 \}
Step 6: Solve for y
Now that we have the equation -35 + 3y = 28, we can solve for y by adding 35 to both sides of the equation and then dividing both sides by 3.
{ 3y = 28 + 35 \}
Simplifying the equation gives us:
{ 3y = 63 \}
Dividing both sides of the equation by 3 gives us:
{ y = \frac{63}{3} \}
Simplifying the equation gives us:
{ y = 21 \}
Conclusion
In this article, we solved a system of two linear equations with two variables using the method of elimination. We multiplied the first equation by 2 and the second equation by 3 to make the coefficients of y in both equations equal, then subtracted the second equation from the first equation to eliminate the y variable. We solved for x by dividing both sides of the equation by -11, then substituted x into one of the original equations to solve for y. The values of x and y that satisfy both equations simultaneously are x = -7 and y = 21.
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 gravity or the flow of fluids through pipes.
- Computer Science: Systems of linear equations are used in computer graphics to perform tasks such as 3D modeling and animation.
- Economics: Systems of linear equations are used to model economic systems and make predictions about the behavior of markets.
Tips and Tricks
When solving systems of linear equations, it's essential to:
- Check your work: Make sure to check your solutions by plugging them back into the original equations.
- Use the correct method: Choose the method that best suits the problem, such as the method of elimination or substitution.
- Simplify your equations: Simplify your equations as much as possible to make them easier to solve.
Conclusion
Solving systems of linear equations is a fundamental skill in mathematics that has many practical applications in various fields. By following the steps outlined in this article, you can solve systems of linear equations using the method of elimination. Remember to check your work, use the correct method, and simplify your equations to make them easier to solve.
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Q: What is a system of linear equations?
A: 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. These equations are linear because they are in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: What are the different methods for solving systems of linear equations?
A: There are several methods for solving systems of linear equations, including:
- Method of Elimination: This method involves adding or subtracting the equations to eliminate one of the variables.
- Method of Substitution: This method involves solving one equation for one variable and then substituting that expression into the other equation.
- Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
Q: How do I choose the correct method for solving a system of linear equations?
A: The choice of method depends on the specific problem and the variables involved. If the coefficients of the variables are easy to work with, the method of elimination may be the best choice. If the coefficients are more complex, the method of substitution may be more suitable.
Q: What are some common mistakes to avoid when solving systems of linear equations?
A: Some common mistakes to avoid when solving systems of linear equations include:
- Not checking your work: Make sure to check your solutions by plugging them back into the original equations.
- Not using the correct method: Choose the method that best suits the problem.
- Not simplifying your equations: Simplify your equations as much as possible to make them easier to solve.
Q: How do I know if a system of linear equations has a unique solution, no solution, or infinitely many solutions?
A: To determine the number of solutions, follow these steps:
- Check if the equations are consistent: If the equations are consistent, then the system has either a unique solution or infinitely many solutions.
- Check if the equations are inconsistent: If the equations are inconsistent, then the system has no solution.
- Check if the equations are dependent: If the equations are dependent, then the system has infinitely many solutions.
Q: What are some real-world applications of solving systems of linear equations?
A: 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 gravity or the flow of fluids through pipes.
- Computer Science: Systems of linear equations are used in computer graphics to perform tasks such as 3D modeling and animation.
- Economics: Systems of linear equations are used to model economic systems and make predictions about the behavior of markets.
Q: How can I practice solving systems of linear equations?
A: There are many ways to practice solving systems of linear equations, including:
- Working through example problems: Practice solving systems of linear equations using different methods and techniques.
- Using online resources: There are many online resources available that provide practice problems and exercises for solving systems of linear equations.
- Taking online courses or tutorials: Consider taking online courses or tutorials to learn more about solving systems of linear equations.
Q: What are some common types of systems of linear equations?
A: Some common types of systems of linear equations include:
- Homogeneous systems: These systems have the form Ax = 0, where A is a matrix and x is a vector.
- Nonhomogeneous systems: These systems have the form Ax = b, where A is a matrix, x is a vector, and b is a vector.
- Systems with multiple variables: These systems have more than two variables and require the use of matrices and other advanced techniques to solve.
Q: How can I use technology to solve systems of linear equations?
A: There are many ways to use technology to solve systems of linear equations, including:
- Using graphing calculators: Graphing calculators can be used to graph the equations and find the point of intersection.
- Using computer algebra systems: Computer algebra systems, such as Mathematica or Maple, can be used to solve systems of linear equations and perform other advanced mathematical operations.
- Using online tools: There are many online tools available that can be used to solve systems of linear equations, including online graphing calculators and computer algebra systems.