Is ( 3 , 5 (3,5 ( 3 , 5 ] A Solution To This System Of Equations? Y = 5 Y = − 5 3 X + 10 \begin{array}{l} y = 5 \\ y = -\frac{5}{3}x + 10 \end{array} Y = 5 Y = − 3 5 ​ X + 10 ​ A. Yes B. No

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Introduction

When dealing with systems of equations, it's essential to determine whether a given point satisfies both equations. In this case, we're presented with a system of two linear equations and asked to verify if the point (3,5)(3,5) is a solution. To do this, we'll substitute the coordinates of the point into each equation and check if the resulting statement is true.

Understanding the System of Equations

The given system consists of two linear equations:

  1. y=5y = 5
  2. y=−53x+10y = -\frac{5}{3}x + 10

These equations represent two lines on a coordinate plane. The first equation is a horizontal line with a constant y-value of 5, while the second equation is a linear equation with a slope of −53-\frac{5}{3} and a y-intercept of 10.

Substituting the Point into the First Equation

Let's start by substituting the coordinates of the point (3,5)(3,5) into the first equation:

y=5y = 5

Since the y-coordinate of the point is 5, which matches the constant y-value in the equation, we can conclude that the point (3,5)(3,5) satisfies the first equation.

Substituting the Point into the Second Equation

Now, let's substitute the coordinates of the point (3,5)(3,5) into the second equation:

y=−53x+10y = -\frac{5}{3}x + 10

To do this, we'll replace the variable x with the x-coordinate of the point (3) and the variable y with the y-coordinate of the point (5):

5=−53(3)+105 = -\frac{5}{3}(3) + 10

Simplifying the equation, we get:

5=−5+105 = -5 + 10

Combining like terms, we get:

5=55 = 5

This statement is true, which means that the point (3,5)(3,5) satisfies the second equation as well.

Conclusion

Since the point (3,5)(3,5) satisfies both equations in the system, we can conclude that it is indeed a solution to the system of equations.

Final Answer

The final answer is: A. Yes

Discussion

In this discussion, we've demonstrated how to verify whether a given point satisfies a system of linear equations. By substituting the coordinates of the point into each equation, we can determine if the point lies on both lines represented by the equations. This is an essential skill in mathematics, particularly in algebra and geometry.

Related Topics

  • Systems of linear equations
  • Linear equations
  • Coordinate geometry
  • Algebra
  • Geometry

Further Reading

For more information on systems of linear equations and coordinate geometry, you can refer to the following resources:

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Geometry: Seeing, Doing, Understanding" by Harold R. Jacobs
  • [3] "Mathematics for the Nonmathematician" by Morris Kline

Introduction

Systems of linear equations are a fundamental concept in mathematics, particularly in algebra and geometry. In this article, we'll address some common questions and provide detailed answers to help you better understand systems of linear equations.

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 determine if a point is a solution to a system of linear equations?

To determine if a point is a solution to a system of linear equations, substitute the coordinates of the point into each equation and check if the resulting statement is true.

Q3: What is the difference between a system of linear equations and a single linear equation?

A single linear equation represents a line on a coordinate plane, while a system of linear equations represents the intersection of two or more lines.

Q4: How do I solve a system of linear equations?

There are several methods to solve a system of linear equations, including:

  • Substitution method: Substitute the expression for one variable from one equation into the other equation.
  • Elimination method: Add or subtract the equations to eliminate one variable.
  • Graphical method: Graph the lines represented by the equations and find the point of intersection.

Q5: What is the significance of the point of intersection in a system of linear equations?

The point of intersection represents the solution to the system of linear equations, where the two lines meet.

Q6: Can a system of linear equations have no solution?

Yes, a system of linear equations can have no solution if the lines represented by the equations are parallel and do not intersect.

Q7: Can a system of linear equations have infinitely many solutions?

Yes, a system of linear equations can have infinitely many solutions if the lines represented by the equations are identical.

Q8: How do I determine if a system of linear equations has a unique solution?

To determine if a system of linear equations has a unique solution, check if the lines represented by the equations intersect at a single point.

Q9: What is the role of the y-intercept in a system of linear equations?

The y-intercept represents the point where the line intersects the y-axis.

Q10: Can a system of linear equations have a negative y-intercept?

Yes, a system of linear equations can have a negative y-intercept, which represents the point where the line intersects the y-axis below the origin.

Q11: How do I graph a system of linear equations?

To graph a system of linear equations, graph the lines represented by the equations and find the point of intersection.

Q12: What is the significance of the slope in a system of linear equations?

The slope represents the rate of change of the line, which can be positive, negative, or zero.

Q13: Can a system of linear equations have a slope of zero?

Yes, a system of linear equations can have a slope of zero, which represents a horizontal line.

Q14: Can a system of linear equations have a negative slope?

Yes, a system of linear equations can have a negative slope, which represents a line that slopes downward from left to right.

Q15: How do I determine if a system of linear equations is consistent?

To determine if a system of linear equations is consistent, check if the lines represented by the equations intersect at a single point.

Q16: Can a system of linear equations be inconsistent?

Yes, a system of linear equations can be inconsistent if the lines represented by the equations are parallel and do not intersect.

Q17: What is the significance of the point of discontinuity in a system of linear equations?

The point of discontinuity represents the point where the line is not defined.

Q18: Can a system of linear equations have a point of discontinuity?

Yes, a system of linear equations can have a point of discontinuity, which represents the point where the line is not defined.

Q19: How do I determine if a system of linear equations has a point of discontinuity?

To determine if a system of linear equations has a point of discontinuity, check if the line is not defined at a particular point.

Q20: Can a system of linear equations have multiple points of discontinuity?

Yes, a system of linear equations can have multiple points of discontinuity, which represents the points where the line is not defined.

Final Answer

The final answer is: Systems of linear equations are a fundamental concept in mathematics, and understanding them is essential for solving problems in algebra and geometry.

Discussion

In this Q&A article, we've addressed some common questions and provided detailed answers to help you better understand systems of linear equations. By mastering these concepts, you'll be able to solve problems in algebra and geometry with confidence.

Related Topics

  • Systems of linear equations
  • Linear equations
  • Coordinate geometry
  • Algebra
  • Geometry

Further Reading

For more information on systems of linear equations and coordinate geometry, you can refer to the following resources:

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Geometry: Seeing, Doing, Understanding" by Harold R. Jacobs
  • [3] "Mathematics for the Nonmathematician" by Morris Kline