Solve The Following System Of Equations:$\[ \begin{aligned} 4x + Y &= -8 \\ 6x - 4y &= 10 \end{aligned} \\]
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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 typically represented in the form of ax + by = c, where a, b, and c are constants, and x and y are the variables. In this article, we will focus on solving a system of two linear equations using the method of substitution and elimination.
The System of Equations
The system of equations we will be solving is given by:
{ \begin{aligned} 4x + y &= -8 \\ 6x - 4y &= 10 \end{aligned} \}
Method of Substitution
One way to solve this system of equations is by using the method of substitution. This method involves solving one of the equations for one of the variables and then substituting that expression into the other equation.
Step 1: Solve the First Equation for y
To solve the first equation for y, we can isolate y by subtracting 4x from both sides of the equation.
{ \begin{aligned} 4x + y &= -8 \\ y &= -8 - 4x \end{aligned} \}
Step 2: Substitute the Expression for y into the Second Equation
Now that we have an expression for y, we can substitute it into the second equation.
{ \begin{aligned} 6x - 4y &= 10 \\ 6x - 4(-8 - 4x) &= 10 \end{aligned} \}
Step 3: Simplify the Equation
To simplify the equation, we can distribute the -4 to the terms inside the parentheses.
{ \begin{aligned} 6x - 4(-8 - 4x) &= 10 \\ 6x + 32 + 16x &= 10 \end{aligned} \}
Step 4: Combine Like Terms
Now we can combine like terms by adding 6x and 16x.
{ \begin{aligned} 6x + 32 + 16x &= 10 \\ 22x + 32 &= 10 \end{aligned} \}
Step 5: Isolate x
To isolate x, we can subtract 32 from both sides of the equation.
{ \begin{aligned} 22x + 32 &= 10 \\ 22x &= -22 \end{aligned} \}
Step 6: Solve for x
Finally, we can solve for x by dividing both sides of the equation by 22.
{ \begin{aligned} 22x &= -22 \\ x &= -1 \end{aligned} \}
Method of Elimination
Another way to solve this system of equations is by using the method of elimination. This method involves multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same.
Step 1: Multiply the First Equation by 4
To eliminate y, we can multiply the first equation by 4.
{ \begin{aligned} 4x + y &= -8 \\ 16x + 4y &= -32 \end{aligned} \}
Step 2: Multiply the Second Equation by 1
We can leave the second equation as it is.
{ \begin{aligned} 6x - 4y &= 10 \end{aligned} \}
Step 3: Add Both Equations
Now we can add both equations to eliminate y.
{ \begin{aligned} 16x + 4y + 6x - 4y &= -32 + 10 \\ 22x &= -22 \end{aligned} \}
Step 4: Solve for x
Finally, we can solve for x by dividing both sides of the equation by 22.
{ \begin{aligned} 22x &= -22 \\ x &= -1 \end{aligned} \}
Conclusion
In this article, we have solved a system of two linear equations using the method of substitution and elimination. We have shown that both methods can be used to solve the system of equations and have obtained the same solution for x. The solution to the system of equations is x = -1.
Final Answer
The final answer is x = -1.
Solving for y
Now that we have the value of x, we can substitute it into one of the original equations to solve for y.
Step 1: Substitute x into the First Equation
We can substitute x = -1 into the first equation.
{ \begin{aligned} 4x + y &= -8 \\ 4(-1) + y &= -8 \end{aligned} \}
Step 2: Solve for y
To solve for y, we can add 4 to both sides of the equation.
{ \begin{aligned} 4(-1) + y &= -8 \\ y &= -8 + 4 \\ y &= -4 \end{aligned} \}
Final Answer for y
The final answer for y is -4.
Summary
In this article, we have solved a system of two linear equations using the method of substitution and elimination. We have shown that both methods can be used to solve the system of equations and have obtained the same solution for x and y. The solutions to the system of equations are x = -1 and y = -4.
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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.
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 method of substitution and the method of elimination.
Q: What is the method of substitution?
A: The method of substitution involves solving one of the equations for one of the variables and then substituting that expression into the other equation.
Q: What is the method of elimination?
A: The method of elimination involves multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same, and then adding both equations to eliminate y.
Q: How do I know which method to use?
A: You can use either method, but the method of elimination is often easier to use when the coefficients of y's in both equations are the same.
Q: What if I have a system of three or more linear equations?
A: In that case, you can use the method of substitution or elimination, or you can use a combination of both methods.
Q: Can I use a calculator to solve systems of linear equations?
A: Yes, you can use a calculator to solve systems of linear equations. Many calculators have a built-in function for solving systems of linear equations.
Q: What if I get a system of linear equations with no solution?
A: If you get a system of linear equations with no solution, it means that the equations are inconsistent, and there is no value of x that satisfies both equations.
Q: What if I get a system of linear equations with infinitely many solutions?
A: If you get a system of linear equations with infinitely many solutions, it means that the equations are dependent, and there are many values of x that satisfy both equations.
Q: Can I use systems of linear equations to model real-world problems?
A: Yes, you can use systems of linear equations to model real-world problems. For example, you can use systems of linear equations to model the cost of producing a product, or the amount of money you have in your bank account.
Q: What are some common applications of systems of linear equations?
A: Some common applications of systems of linear equations include:
- Modeling the cost of producing a product
- Modeling the amount of money you have in your bank account
- Modeling the population growth of a city
- Modeling the amount of water in a reservoir
- Modeling the amount of electricity used in a building
Q: Can I use systems of linear equations to solve problems in other areas of mathematics?
A: Yes, you can use systems of linear equations to solve problems in other areas of mathematics, such as algebra, geometry, and calculus.
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 following the order of operations
- Not simplifying the equations
- Not checking for consistency
- Not checking for dependency
Q: Can I use systems of linear equations to solve problems in other areas of science and engineering?
A: Yes, you can use systems of linear equations to solve problems in other areas of science and engineering, such as physics, chemistry, and computer science.
Q: What are some common applications of systems of linear equations in science and engineering?
A: Some common applications of systems of linear equations in science and engineering include:
- Modeling the motion of an object
- Modeling the flow of a fluid
- Modeling the behavior of a circuit
- Modeling the behavior of a system of springs
- Modeling the behavior of a system of pendulums
Q: Can I use systems of linear equations to solve problems in other areas of business and economics?
A: Yes, you can use systems of linear equations to solve problems in other areas of business and economics, such as finance, marketing, and management.
Q: What are some common applications of systems of linear equations in business and economics?
A: Some common applications of systems of linear equations in business and economics include:
- Modeling the cost of producing a product
- Modeling the amount of money you have in your bank account
- Modeling the population growth of a city
- Modeling the amount of water in a reservoir
- Modeling the amount of electricity used in a building