Solve The Following System Of Equations:${ \begin{array}{l} 4x - 2y = -48 \ 10x + 10y = -60 \end{array} }$
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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. We will use the given system of equations as an example and provide a step-by-step guide on how to solve it.
The System of Equations
The given system of equations is:
{ \begin{array}{l} 4x - 2y = -48 \\ 10x + 10y = -60 \end{array} \}
This system consists of two linear equations with two variables, x and y. The first equation is 4x - 2y = -48, and the second equation is 10x + 10y = -60.
Method 1: Substitution Method
One way to solve this system is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.
Step 1: Solve the First Equation for x
We can start by solving the first equation for x. We can do this by isolating x on one side of the equation.
{ 4x - 2y = -48 \}
Add 2y to both sides:
{ 4x = -48 + 2y \}
Divide both sides by 4:
{ x = \frac{-48 + 2y}{4} \}
Simplify the expression:
{ x = -12 + \frac{1}{2}y \}
Step 2: Substitute the Expression for x into the Second Equation
Now that we have an expression for x, we can substitute it into the second equation.
{ 10x + 10y = -60 \}
Substitute x = -12 + 1/2y:
{ 10(-12 + \frac{1}{2}y) + 10y = -60 \}
Expand and simplify:
{ -120 + 5y + 10y = -60 \}
Combine like terms:
{ -120 + 15y = -60 \}
Add 120 to both sides:
{ 15y = 60 \}
Divide both sides by 15:
{ y = 4 \}
Step 3: Find the Value of x
Now that we have the value of y, we can substitute it back into the expression for x.
{ x = -12 + \frac{1}{2}y \}
Substitute y = 4:
{ x = -12 + \frac{1}{2}(4) \}
Simplify:
{ x = -12 + 2 \}
{ x = -10 \}
Method 2: Elimination Method
Another way to solve this system is by using the elimination method. This method involves adding or subtracting the equations to eliminate one variable.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one variable, we need to make the coefficients of either x or y the same in both equations. We can do this by multiplying the equations by necessary multiples.
{ \begin{array}{l} 4x - 2y = -48 \\ 10x + 10y = -60 \end{array} \}
Multiply the first equation by 5 and the second equation by 2:
{ \begin{array}{l} 20x - 10y = -240 \\ 20x + 20y = -120 \end{array} \}
Step 2: Add the Equations
Now that we have the same coefficients for x in both equations, we can add them to eliminate x.
{ 20x - 10y = -240 \\ 20x + 20y = -120 \end{array} \}
Add the equations:
{ -10y + 20y = -240 - 120 \}
Combine like terms:
{ 10y = -360 \}
Divide both sides by 10:
{ y = -36 \}
Step 3: Find the Value of x
Now that we have the value of y, we can substitute it back into one of the original equations to find the value of x.
{ 4x - 2y = -48 \}
Substitute y = -36:
{ 4x - 2(-36) = -48 \}
Simplify:
{ 4x + 72 = -48 \}
Subtract 72 from both sides:
{ 4x = -120 \}
Divide both sides by 4:
{ x = -30 \}
Conclusion
In this article, we have solved a system of two linear equations with two variables using the substitution method and the elimination method. We have shown that both methods can be used to solve the system, and we have provided step-by-step guides on how to do it. We have also emphasized the importance of checking the solutions to ensure that they satisfy both equations.
Final Answer
The final answer is:
x = -30 y = -36
Note: The final answer is the same for both methods.
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Introduction
Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we provided a step-by-step guide on how to solve a system of two linear equations with two variables using the substitution method and the elimination method. In this article, we will answer some frequently asked questions related to solving systems of linear equations.
Q&A
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, which means 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 is the difference between the substitution method and the elimination method?
A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable.
Q: How do I choose between the substitution method and the elimination method?
A: You can choose between the substitution method and the elimination method based on the coefficients of the variables in the two equations. If the coefficients of one variable are the same in both equations, then the elimination method is a good choice. If the coefficients of one variable are different in both equations, then the substitution method is a good choice.
Q: What if I get a fraction or a decimal as a solution?
A: If you get a fraction or a decimal as a solution, then you can simplify it by dividing both the numerator and the denominator by their greatest common divisor (GCD) or by rounding it to a certain number of decimal places.
Q: Can I use a calculator to solve a system of linear equations?
A: Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations, such as the "Solve" function on a graphing calculator.
Q: How do I check if my solution is correct?
A: To check if your solution is correct, you can substitute the values of the variables back into both equations and see if they are true. If they are true, then your solution is correct.
Q: What if I get a solution that involves complex numbers?
A: If you get a solution that involves complex numbers, then you can simplify it by using the fact that complex numbers can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit.
Conclusion
In this article, we have answered some frequently asked questions related to solving systems of linear equations. We have provided explanations and examples to help you understand the concepts and methods involved in solving systems of linear equations.
Final Tips
- Always check your solutions to ensure that they satisfy both equations.
- Use a calculator or a computer program to check your solutions if you are unsure.
- Practice solving systems of linear equations to become more comfortable with the concepts and methods involved.
- Use the substitution method and the elimination method to solve systems of linear equations, depending on the coefficients of the variables in the two equations.
By following these tips and practicing solving systems of linear equations, you will become more confident and proficient in solving these types of problems.