Which Statement About This System Of Equations Is True

Which Statement About This System of Equations Is True?

Solving systems of equations can be a daunting task, especially when you’re not sure which statement about them is true. Fear not, for we’re here to shed some light on this puzzling topic and guide you towards solving these equations with ease.

The Struggle with Conflicting Statements

When faced with a system of equations, it’s common to encounter conflicting statements that can leave you feeling lost and confused. Some may claim the solution is unique, while others insist there are no solutions at all. This uncertainty can lead to frustration and hinder your progress.

The Truth Revealed

The key to unlocking the truth about systems of equations lies in understanding their underlying structure. A system of equations can have one of three possible outcomes: a unique solution, no solution, or infinitely many solutions. Which outcome occurs depends on the relationships between the equations.

Main Points

  • A unique solution exists when the lines represented by the equations intersect at a single point.
  • No solution exists when the lines are parallel and do not intersect.
  • Infinitely many solutions exist when the lines coincide, meaning they are the same line.
Which Statement About This System Of Equations Is True

System of Equations: Identifying True Statements

Introduction

A system of equations consists of two or more equations that involve the same variables. Solving a system of equations is finding values for the variables that satisfy all the equations simultaneously. In this article, we will examine a particular system of equations and identify which statement about it is true.

System of Equations:

2x + 3y = 11
x - y = 5

Identifying the True Statement

Statement 1: The system has a unique solution.

This statement is true. The system has a unique solution because the two equations are independent and consistent. The first equation does not depend on the second equation, and there is only one pair of values for x and y that satisfy both equations.

Statement 2: The system has infinitely many solutions.

This statement is false. The system does not have infinitely many solutions because the two equations intersect at a single point. The point (x = 5, y = 0) is the only solution to the system.

Statement 3: The system has no solution.

This statement is false. The system has a unique solution, as stated in Statement 1.

Statement 4: The system is equivalent to the equation 3x = 16.

This statement is false. The system is not equivalent to the equation 3x = 16. The equation 3x = 16 only represents one of the equations in the system.

Statement 5: The value of y is 2 times the value of x.

This statement is true. From the second equation, x – y = 5, we can deduce that y = x – 5. Substituting this into the first equation, 2x + 3y = 11, gives us 2x + 3(x – 5) = 11. Solving for x, we get x = 5. Therefore, y = x – 5 = 5 – 5 = 0.

Conclusion

The correct statement about the given system of equations is: The system has a unique solution. The system has one pair of values for x and y that satisfy both equations simultaneously.

FAQs

1. What is a system of equations?

A system of equations is a set of two or more equations that involve the same variables.

2. How do you solve a system of equations?

There are several methods for solving a system of equations, such as substitution, elimination, and graphing.

3. What does it mean for a system to have a unique solution?

A system has a unique solution when there is only one pair of values for the variables that satisfies all the equations.

4. What does it mean for a system to have infinitely many solutions?

A system has infinitely many solutions when there are an infinite number of pairs of values for the variables that satisfy all the equations.

5. What does it mean for a system to have no solution?

A system has no solution when there are no pairs of values for the variables that satisfy all the equations.

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