Estimate The Solution To The System Of Equations:$\left\{\begin{array}{l} Y=-x+2 \\ Y=3x-4 \end{array}\right.$Choose 1 Answer:A. $x=\frac{1}{2}, Y=\frac{3}{2}$B. $x=\frac{5}{2}, Y=\frac{1}{2}$C. $x=\frac{3}{2},

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 method of substitution to find the solution to the system of equations.

The System of Equations

The system of equations we will be solving is:

$\left\{\begin{array}{l} y=-x+2 \\ y=3x-4 \end{array}\right.$

Step 1: Write Down the Equations

The first step in solving a system of linear equations is to write down the equations. In this case, we have two equations:

1. $y=-x+2$
2. $y=3x-4$

Step 2: Solve One Equation for One Variable

To solve the system of equations, we need to solve one equation for one variable. Let's solve the first equation for $y$:

$y=-x+2$

We can rewrite this equation as:

$y=2-x$

Step 3: Substitute the Expression into the Second Equation

Now that we have an expression for $y$ in terms of $x$, we can substitute this expression into the second equation:

$y=3x-4$

Substituting $y=2-x$ into this equation, we get:

$2-x=3x-4$

Step 4: Solve for $x$

Now that we have a single equation with one variable, we can solve for $x$. Let's start by adding $x$ to both sides of the equation:

$2=4x-4$

Next, we can add 4 to both sides of the equation:

$6=4x$

Finally, we can divide both sides of the equation by 4 to solve for $x$:

$x=\frac{6}{4}=\frac{3}{2}$

Step 5: Find the Value of $y$

Now that we have the value of $x$, we can find the value of $y$ by substituting $x$ into one of the original equations. Let's use the first equation:

$y=-x+2$

Substituting $x=\frac{3}{2}$ into this equation, we get:

$y=-\frac{3}{2}+2$

Simplifying this expression, we get:

$y=\frac{1}{2}$

Conclusion

In this article, we solved a system of two linear equations with two variables using the method of substitution. We found that the solution to the system of equations is:

$x=\frac{3}{2}, y=\frac{1}{2}$

This solution satisfies both equations in the system.

Answer

The correct answer is:

C. $x=\frac{3}{2}, y=\frac{1}{2}$

Discussion

This problem is a classic example of a system of linear equations with two variables. The method of substitution is a powerful tool for solving systems of linear equations, and it is often used in a variety of applications, including physics, engineering, and economics.

In this problem, we used the method of substitution to solve the system of equations. We started by solving one equation for one variable, and then we substituted this expression into the second equation. We then solved for the value of $x$, and finally, we found the value of $y$ by substituting $x$ into one of the original equations.

The solution to the system of equations is $x=\frac{3}{2}, y=\frac{1}{2}$. This solution satisfies both equations in the system, and it is the only solution to the system of equations.

Related Problems

If you are interested in solving more systems of linear equations, here are a few related problems:

• Solve the system of equations: $\left\{\begin{array}{l} y=2x-1 \\ y=3x+2 \end{array}\right.$
• Solve the system of equations: $\left\{\begin{array}{l} y=x+1 \\ y=2x-3 \end{array}\right.$
• Solve the system of equations: $\left\{\begin{array}{l} y=3x-2 \\ y=2x+1 \end{array}\right.$

Introduction

In our previous article, we solved a system of two linear equations with two variables using the method of substitution. In this article, we will answer some frequently asked questions about solving systems of linear equations.

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 different methods for solving systems of linear equations?

A: There are several methods for solving systems of linear equations, including:

• Substitution method: This method involves solving one equation for one variable and then substituting this expression into the other equation.
• Elimination method: This method involves adding or subtracting the equations to eliminate one of the variables.
• Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method is a method for solving systems of linear equations that involves solving one equation for one variable and then substituting this expression into the other equation.

Q: How do I use the substitution method to solve a system of linear equations?

A: To use the substitution method, follow these steps:

1. Solve one equation for one variable.
2. Substitute this expression into the other equation.
3. Solve for the value of the other variable.
4. Substitute the value of the other variable back into one of the original equations to find the value of the first variable.

Q: What is the elimination method?

A: The elimination method is a method for solving systems of linear equations that involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I use the elimination method to solve a system of linear equations?

A: To use the elimination method, follow these steps:

1. Multiply both equations by necessary multiples such that the coefficients of one of the variables are the same in both equations.
2. Add or subtract the equations to eliminate one of the variables.
3. Solve for the value of the other variable.
4. Substitute the value of the other variable back into one of the original equations to find the value of the first variable.

Q: What is the graphical method?

A: The graphical method is a method for solving systems of linear equations that involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: How do I use the graphical method to solve a system of linear equations?

A: To use the graphical method, follow these steps:

1. Graph the equations on a coordinate plane.
2. Find the point of intersection of the two lines.
3. The point of intersection is the solution to the system of equations.

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 the solution: Make sure to check the solution to the system of equations to ensure that it satisfies both equations.
• Not using the correct method: Make sure to use the correct method for solving the system of equations.
• Not following the steps: Make sure to follow the steps for the method you are using.

Conclusion

Solving systems of linear equations is an important skill in mathematics and is used in a variety of applications. By understanding the different methods for solving systems of linear equations and avoiding common mistakes, you can become proficient in solving these types of problems.

Related Problems

If you are interested in practicing solving systems of linear equations, here are a few related problems:

• Solve the system of equations: $\left\{\begin{array}{l} y=2x-1 \\ y=3x+2 \end{array}\right.$
• Solve the system of equations: $\left\{\begin{array}{l} y=x+1 \\ y=2x-3 \end{array}\right.$
• Solve the system of equations: $\left\{\begin{array}{l} y=3x-2 \\ y=2x+1 \end{array}\right.$

These problems can be solved using the same methods that we discussed in this article.