Question 7The Solution Point Of The System { X + Y = 3 X = Y + 1 \left\{\begin{array}{l}x+y=3 \\ X=y+1\end{array}\right. { X + Y = 3 X = Y + 1 ​ Is:A. ( 4 , 1 (4,1 ( 4 , 1 ]B. ( 3 , − 2 (3,-2 ( 3 , − 2 ]C. ( − 2 , − 1 (-2,-1 ( − 2 , − 1 ]D. ( 2 , 1 (2,1 ( 2 , 1 ]

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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. 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 to demonstrate the steps involved in solving such a system.

The Given System of Equations

The given system of equations is:

{x+y=3x=y+1\left\{\begin{array}{l}x+y=3 \\ x=y+1\end{array}\right.

Understanding the System of Equations

To solve this system of equations, we need to understand the concept of a solution point. The solution point is the point of intersection of the two lines represented by the two equations. In other words, it is the point where the two lines meet.

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Let's start by solving the second equation for x:

x=y+1x = y + 1

Now, substitute this expression for x into the first equation:

x+y=3x + y = 3

y+1+y=3y + 1 + y = 3

Combine like terms:

2y+1=32y + 1 = 3

Subtract 1 from both sides:

2y=22y = 2

Divide both sides by 2:

y=1y = 1

Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use the second equation:

x=y+1x = y + 1

x=1+1x = 1 + 1

x=2x = 2

Therefore, the solution point is (2, 1).

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. This method involves adding or subtracting the two equations to eliminate one variable.

Let's start by adding the two equations:

x+y=3x + y = 3

x=y+1x = y + 1

Add the two equations:

2x+y=42x + y = 4

Now, subtract the second equation from the first equation:

x+y=3x + y = 3

x−(y+1)=0x - (y + 1) = 0

Simplify:

x−y−1=0x - y - 1 = 0

Add 1 to both sides:

x−y=1x - y = 1

Now, add this equation to the first equation:

x+y=3x + y = 3

(x−y)+(x+y)=3+1(x - y) + (x + y) = 3 + 1

Combine like terms:

2x=42x = 4

Divide both sides by 2:

x=2x = 2

Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y. Let's use the first equation:

x+y=3x + y = 3

2+y=32 + y = 3

Subtract 2 from both sides:

y=1y = 1

Therefore, the solution point is (2, 1).

Conclusion

In this article, we have demonstrated two methods for solving a system of linear equations with two variables. 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 two equations to eliminate one variable. We have used the given system of equations as an example to demonstrate the steps involved in solving such a system. The solution point of the system is (2, 1).

Answer

The correct answer is:

  • D. (2,1)

This is the solution point of the system of equations.

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Q1: What is a system of linear equations?

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.

Q2: How do I know if a system of linear equations has a solution?

A system of linear equations has a solution if the two lines represented by the two equations intersect at a single point. If the lines are parallel, the system has no solution.

Q3: What is the difference between the substitution method and the elimination method?

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 two equations to eliminate one variable.

Q4: How do I choose which method to use?

You can choose either method, but the substitution method is often easier to use when one of the equations is already solved for one variable. The elimination method is often easier to use when the coefficients of the variables are the same.

Q5: What if I have a system of linear equations with three variables?

To solve a system of linear equations with three variables, you can use the same methods as before, but you will need to solve for two variables first and then substitute those values into the third equation.

Q6: Can I use a graphing calculator to solve a system of linear equations?

Yes, you can use a graphing calculator to solve a system of linear equations by graphing the two lines and finding the point of intersection.

Q7: What if I have a system of linear equations with no solution?

If a system of linear equations has no solution, it means that the two lines represented by the two equations are parallel and will never intersect.

Q8: Can I use a system of linear equations to model real-world problems?

Yes, systems of linear equations can be used to model real-world problems such as finding the intersection of two lines, representing a system of constraints, and solving optimization problems.

Q9: How do I know if a system of linear equations is consistent or inconsistent?

A system of linear equations is consistent if it has a solution. A system of linear equations is inconsistent if it has no solution.

Q10: Can I use a system of linear equations to solve a system of inequalities?

Yes, you can use a system of linear equations to solve a system of inequalities by finding the intersection of the two regions represented by the inequalities.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of linear equations. We have covered topics such as the definition of a system of linear equations, the difference between the substitution method and the elimination method, and how to choose which method to use. We have also discussed how to solve systems of linear equations with three variables, how to use a graphing calculator to solve a system of linear equations, and how to model real-world problems using systems of linear equations.