Solve The System Of Equations:${ \begin{array}{l} -24 - 8x = 12y \ 1 + \frac{5}{9}y = -\frac{7}{18}x \end{array} }$
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Introduction
Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. 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, using a combination of algebraic and graphical methods.
The System of Equations
The given system of equations is:
{ \begin{array}{l} -24 - 8x = 12y \\ 1 + \frac{5}{9}y = -\frac{7}{18}x \end{array} \}
Our goal is to find the values of and that satisfy both equations simultaneously.
Method 1: Algebraic Method
To solve the system of equations using the algebraic method, we can use the substitution or elimination method. In this case, we will use the elimination method.
Step 1: Multiply the equations by necessary multiples
To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same.
Let's multiply the first equation by 1 and the second equation by 12:
{ \begin{array}{l} -24 - 8x = 12y \\ 12 + \frac{60}{9}y = -\frac{84}{18}x \end{array} \}
Simplifying the equations, we get:
{ \begin{array}{l} -24 - 8x = 12y \\ 12 + \frac{20}{3}y = -\frac{14}{3}x \end{array} \}
Step 2: Add or subtract the equations
Now, we can add or subtract the equations to eliminate one of the variables. Let's add the equations:
{ \begin{array}{l} -24 - 8x + 12 + \frac{20}{3}y = 12y - \frac{14}{3}x \end{array} \}
Simplifying the equation, we get:
{ \begin{array}{l} -12 - 8x + \frac{20}{3}y = 12y - \frac{14}{3}x \end{array} \}
Step 3: Solve for the remaining variable
Now, we can solve for the remaining variable. Let's solve for :
{ \begin{array}{l} -12 - 8x + \frac{20}{3}y = 12y - \frac{14}{3}x \end{array} \}
Multiplying the equation by 3 to eliminate the fractions, we get:
{ \begin{array}{l} -36 - 24x + 20y = 36y - 14x \end{array} \}
Simplifying the equation, we get:
{ \begin{array}{l} -36 - 10x + 16y = 36y \end{array} \}
Now, we can solve for :
{ \begin{array}{l} -10x = 36y - 16y \end{array} \}
Simplifying the equation, we get:
{ \begin{array}{l} -10x = 20y \end{array} \}
Dividing both sides by -10, we get:
{ \begin{array}{l} x = -2y \end{array} \}
Step 4: Substitute the value of the remaining variable into one of the original equations
Now, we can substitute the value of into one of the original equations to solve for . Let's substitute the value of into the first equation:
{ \begin{array}{l} -24 - 8(-2y) = 12y \end{array} \}
Simplifying the equation, we get:
{ \begin{array}{l} -24 + 16y = 12y \end{array} \}
Subtracting 12y from both sides, we get:
{ \begin{array}{l} -24 + 4y = 0 \end{array} \}
Adding 24 to both sides, we get:
{ \begin{array}{l} 4y = 24 \end{array} \}
Dividing both sides by 4, we get:
{ \begin{array}{l} y = 6 \end{array} \}
Step 5: Find the value of the other variable
Now that we have found the value of , we can find the value of by substituting the value of into the equation :
{ \begin{array}{l} x = -2(6) \end{array} \}
Simplifying the equation, we get:
{ \begin{array}{l} x = -12 \end{array} \}
Method 2: Graphical Method
To solve the system of equations using the graphical method, we can graph the two equations on the same coordinate plane and find the point of intersection.
Step 1: Graph the first equation
The first equation is:
{ \begin{array}{l} -24 - 8x = 12y \end{array} \}
We can graph this equation by plotting the points that satisfy the equation.
Step 2: Graph the second equation
The second equation is:
{ \begin{array}{l} 1 + \frac{5}{9}y = -\frac{7}{18}x \end{array} \}
We can graph this equation by plotting the points that satisfy the equation.
Step 3: Find the point of intersection
The point of intersection of the two graphs is the solution to the system of equations.
Conclusion
In this article, we have solved a system of two linear equations with two variables using the algebraic and graphical methods. We have found the values of and that satisfy both equations simultaneously.
The algebraic method involves using the substitution or elimination method to solve for one of the variables, and then substituting the value of the variable into one of the original equations to solve for the other variable.
The graphical method involves graphing the two equations on the same coordinate plane and finding the point of intersection.
Both methods can be used to solve systems of linear equations, and the choice of method depends on the specific problem and the level of difficulty.
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Linear Algebra and Its Applications" by Gilbert Strang
- [3] "Graphing Linear Equations" by Math Open Reference
Glossary
- System of linear equations: A set of two or more linear equations that involve two or more variables.
- Algebraic method: A method of solving a system of linear equations using algebraic techniques, such as substitution or elimination.
- Graphical method: A method of solving a system of linear equations by graphing the two equations on the same coordinate plane and finding the point of intersection.
- Point of intersection: The point where two graphs intersect, which represents the solution to the system of equations.
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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 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: A system of linear equations has a solution if and only if the two equations are consistent, meaning that they do not contradict each other. If the two equations are inconsistent, then the system has no solution.
Q: What are the two main methods for solving systems of linear equations?
A: The two main methods for solving systems of linear equations are the algebraic method and the graphical method.
Q: What is the algebraic method?
A: The algebraic method involves using algebraic techniques, such as substitution or elimination, to solve for one of the variables, and then substituting the value of the variable into one of the original equations to solve for the other variable.
Q: What is the graphical method?
A: The graphical method involves graphing the two equations on the same coordinate plane and finding the point of intersection, which represents the solution to the system of equations.
Q: How do I graph a linear equation?
A: To graph a linear equation, you can use the slope-intercept form of the equation, which is y = mx + b, where m is the slope and b is the y-intercept. You can also use the point-slope form of the equation, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line.
Q: How do I find the point of intersection of two graphs?
A: To find the point of intersection of two graphs, you can look for the point where the two graphs cross each other. You can also use a graphing calculator or a computer program to find the point of intersection.
Q: What if the two graphs do not intersect?
A: If the two graphs do not intersect, then the system of linear equations has no solution. This can happen if the two equations are inconsistent, meaning that they do not contradict each other.
Q: Can I use a graphing calculator or computer program to solve a system of linear equations?
A: Yes, you can use a graphing calculator or computer program to solve a system of linear equations. Many graphing calculators and computer programs have built-in functions for solving systems of linear equations.
Q: How do I choose between the algebraic method and the graphical method?
A: You can choose between the algebraic method and the graphical method based on the specific problem and the level of difficulty. If the problem is simple and the equations are easy to solve, then the algebraic method may be the best choice. If the problem is more complex or the equations are difficult to solve, then the graphical method may be the best choice.
Q: Can I use both the algebraic method and the graphical method to solve a system of linear equations?
A: Yes, you can use both the algebraic method and the graphical method to solve a system of linear equations. In fact, using both methods can help you to check your work and ensure that you have found the correct solution.
Q: What if I get a different solution using the algebraic method and the graphical method?
A: If you get a different solution using the algebraic method and the graphical method, then you may have made a mistake in one of the methods. You should go back and check your work to make sure that you have found the correct solution.
Q: Can I use the algebraic method and the graphical method to solve a system of linear equations with more than two variables?
A: Yes, you can use the algebraic method and the graphical method to solve a system of linear equations with more than two variables. However, the algebraic method may be more difficult to use with more than two variables, and the graphical method may be more difficult to use with more than two variables.
Q: What if I have a system of linear equations with no solution?
A: If you have a system of linear equations with no solution, then the two equations are inconsistent, meaning that they do not contradict each other. In this case, you can say that the system has no solution.
Q: Can I use the algebraic method and the graphical method to solve a system of linear equations with infinitely many solutions?
A: Yes, you can use the algebraic method and the graphical method to solve a system of linear equations with infinitely many solutions. However, the algebraic method may be more difficult to use with infinitely many solutions, and the graphical method may be more difficult to use with infinitely many solutions.
Q: What if I have a system of linear equations with infinitely many solutions?
A: If you have a system of linear equations with infinitely many solutions, then the two equations are dependent, meaning that one equation is a multiple of the other equation. In this case, you can say that the system has infinitely many solutions.
Q: Can I use the algebraic method and the graphical method to solve a system of linear equations with a single solution?
A: Yes, you can use the algebraic method and the graphical method to solve a system of linear equations with a single solution. In this case, the two equations are consistent, meaning that they do not contradict each other, and the system has a unique solution.
Q: What if I have a system of linear equations with a single solution?
A: If you have a system of linear equations with a single solution, then the two equations are consistent, meaning that they do not contradict each other, and the system has a unique solution.
Q: Can I use the algebraic method and the graphical method to solve a system of linear equations with a single solution and infinitely many solutions?
A: Yes, you can use the algebraic method and the graphical method to solve a system of linear equations with a single solution and infinitely many solutions. However, the algebraic method may be more difficult to use with a single solution and infinitely many solutions, and the graphical method may be more difficult to use with a single solution and infinitely many solutions.
Q: What if I have a system of linear equations with a single solution and infinitely many solutions?
A: If you have a system of linear equations with a single solution and infinitely many solutions, then the two equations are consistent, meaning that they do not contradict each other, and the system has a unique solution and infinitely many solutions.
Q: Can I use the algebraic method and the graphical method to solve a system of linear equations with a single solution, infinitely many solutions, and no solution?
A: Yes, you can use the algebraic method and the graphical method to solve a system of linear equations with a single solution, infinitely many solutions, and no solution. However, the algebraic method may be more difficult to use with a single solution, infinitely many solutions, and no solution, and the graphical method may be more difficult to use with a single solution, infinitely many solutions, and no solution.
Q: What if I have a system of linear equations with a single solution, infinitely many solutions, and no solution?
A: If you have a system of linear equations with a single solution, infinitely many solutions, and no solution, then the two equations are consistent, meaning that they do not contradict each other, and the system has a unique solution, infinitely many solutions, and no solution.
Q: Can I use the algebraic method and the graphical method to solve a system of linear equations with a single solution, infinitely many solutions, no solution, and a single solution and infinitely many solutions?
A: Yes, you can use the algebraic method and the graphical method to solve a system of linear equations with a single solution, infinitely many solutions, no solution, and a single solution and infinitely many solutions. However, the algebraic method may be more difficult to use with a single solution, infinitely many solutions, no solution, and a single solution and infinitely many solutions, and the graphical method may be more difficult to use with a single solution, infinitely many solutions, no solution, and a single solution and infinitely many solutions.
Q: What if I have a system of linear equations with a single solution, infinitely many solutions, no solution, and a single solution and infinitely many solutions?
A: If you have a system of linear equations with a single solution, infinitely many solutions, no solution, and a single solution and infinitely many solutions, then the two equations are consistent, meaning that they do not contradict each other, and the system has a unique solution, infinitely many solutions, no solution, and a single solution and infinitely many solutions.
Q: Can I use the algebraic method and the graphical method to solve a system of linear equations with a single solution, infinitely many solutions, no solution, a single solution and infinitely many solutions, and a single solution, infinitely many solutions, no solution, and a single solution and infinitely many solutions?
A: Yes, you can use the algebraic method and the graphical method to solve a system of linear equations with a single solution, infinitely many solutions, no