Examine The System Of Equations.$\[ \begin{array}{l} y = 2x - 3 \\ y = -3 \end{array} \\]Which Statement About The System Of Linear Equations Is True?A. The Lines Have Different Slopes.B. There Is No Solution To The System.C. The Lines Have
Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will examine a system of linear equations and determine which statement about the system is true.
The System of Linear Equations
The system of linear equations we will be examining is given by:
{ \begin{array}{l} y = 2x - 3 \\ y = -3 \end{array} \}
Analyzing the Equations
The first equation in the system is:
This equation represents a line in the coordinate plane. The slope of this line is 2, which means that for every 1 unit increase in x, the value of y increases by 2 units.
The second equation in the system is:
This equation represents a horizontal line in the coordinate plane. The value of y is constant at -3, which means that the line is a horizontal line that intersects the y-axis at -3.
Comparing the Lines
To determine which statement about the system of linear equations is true, we need to compare the two lines represented by the equations. The first line has a slope of 2, while the second line is a horizontal line with a constant value of y.
Statement A: The Lines Have Different Slopes
The first line has a slope of 2, while the second line is a horizontal line with a constant value of y. Since the slope of the first line is not equal to the slope of the second line, we can conclude that the lines have different slopes.
Statement B: There is No Solution to the System
To determine if there is a solution to the system, we need to find the point of intersection between the two lines. If the lines intersect at a single point, then there is a solution to the system. However, if the lines do not intersect, then there is no solution to the system.
In this case, the second line is a horizontal line with a constant value of y, while the first line has a slope of 2. Since the first line is not horizontal, it will intersect the second line at a single point. Therefore, there is a solution to the system.
Statement C: The Lines Have the Same Slope
Since the first line has a slope of 2 and the second line is a horizontal line with a constant value of y, we can conclude that the lines do not have the same slope.
Conclusion
In conclusion, the statement about the system of linear equations that is true is:
- The lines have different slopes.
This is because the first line has a slope of 2, while the second line is a horizontal line with a constant value of y. Therefore, the lines have different slopes.
Solving the System
To solve the system of linear equations, we need to find the point of intersection between the two lines. We can do this by setting the two equations equal to each other and solving for x.
Adding 3 to both sides of the equation, we get:
Dividing both sides of the equation by 2, we get:
Substituting x = 0 into one of the original equations, we get:
Simplifying the equation, we get:
Therefore, the solution to the system of linear equations is x = 0 and y = -3.
Graphing the Lines
To visualize the system of linear equations, we can graph the two lines on a coordinate plane. The first line has a slope of 2 and a y-intercept of -3, while the second line is a horizontal line with a constant value of y = -3.
The graph of the first line is a line with a slope of 2 that intersects the y-axis at -3. The graph of the second line is a horizontal line that intersects the y-axis at -3.
The two lines intersect at a single point, which is the solution to the system of linear equations.
Conclusion
In conclusion, the system of linear equations consists of two lines with different slopes. The first line has a slope of 2 and a y-intercept of -3, while the second line is a horizontal line with a constant value of y = -3. The solution to the system is x = 0 and y = -3.
Final Answer
The final answer is:
- The lines have different slopes.
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 the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system.
Q: How do I determine if a system of linear equations has a solution?
A: To determine if a system of linear equations has a solution, you need to find the point of intersection between the two lines represented by the equations. If the lines intersect at a single point, then there is a solution to the system. However, if the lines do not intersect, then there is no solution to the system.
Q: What is the difference between a system of linear equations and a system of nonlinear equations?
A: A system of linear equations consists of two or more linear equations, while a system of nonlinear equations consists of two or more nonlinear equations. Nonlinear equations are equations that are not linear, meaning they do not have a constant slope.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you need to find the point of intersection between the two lines represented by the equations. You can do this by setting the two equations equal to each other and solving for x. Once you have found the value of x, you can substitute it into one of the original equations to find the value of y.
Q: What is the significance of the point of intersection in a system of linear equations?
A: The point of intersection in a system of linear equations represents the solution to the system. It is the point where the two lines represented by the equations intersect, and it is the value of the variables that satisfy all the equations in the system.
Q: Can a system of linear equations have more than one solution?
A: No, a system of linear equations can only have one solution. If a system of linear equations has more than one solution, then it is not a system of linear equations, but rather a system of nonlinear equations.
Q: How do I graph a system of linear equations?
A: To graph a system of linear equations, you need to graph the two lines represented by the equations on a coordinate plane. The first line is graphed by finding two points on the line and drawing a line through them. The second line is graphed by finding two points on the line and drawing a line through them.
Q: What is the difference between a system of linear equations and a system of inequalities?
A: A system of linear equations consists of two or more linear equations, while a system of inequalities consists of two or more inequalities. Inequalities are statements that compare two expressions, but do not state that they are equal.
Q: How do I solve a system of inequalities?
A: To solve a system of inequalities, you need to find the region of the coordinate plane that satisfies all the inequalities in the system. You can do this by graphing the inequalities on a coordinate plane and finding the region where all the inequalities are true.
Q: What is the significance of the region of the coordinate plane in a system of inequalities?
A: The region of the coordinate plane in a system of inequalities represents the solution to the system. It is the region where all the inequalities in the system are true, and it is the value of the variables that satisfy all the inequalities in the system.
Conclusion
In conclusion, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. A system of linear equations can have one solution, and it can be graphed on a coordinate plane. A system of inequalities is a set of two or more inequalities that involve the same set of variables, and it can be solved by finding the region of the coordinate plane that satisfies all the inequalities in the system.