How Many Solutions Are There To The System Of Equations?$\[ \begin{cases} 4x - 5y = 5 \\ -0.08x + 0.10y = 0.10 \end{cases} \\]A. No Solutions B. One Solution C. Two Solutions D. An Infinite Number Of Solutions

Feb 28, 2025 by ADMIN 218 views

Introduction

When dealing with systems of linear equations, it's essential to determine the number of solutions they have. This can be done by analyzing the equations and using various methods such as substitution, elimination, or graphing. In this article, we will explore the system of equations given by:

{ \begin{cases} 4x - 5y = 5 \\ -0.08x + 0.10y = 0.10 \end{cases} \}

We will use the methods mentioned above to determine the number of solutions to this system of equations.

Method 1: Substitution

To solve the system of equations using substitution, we can solve one of the equations for one variable and then substitute that expression into the other equation. Let's solve the second equation for y:

{ -0.08x + 0.10y = 0.10 \}

{ 0.10y = 0.08x + 0.10 \}

{ y = \frac{0.08}{0.10}x + \frac{0.10}{0.10} \}

{ y = 0.8x + 1 \}

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

{ 4x - 5y = 5 \}

{ 4x - 5(0.8x + 1) = 5 \}

{ 4x - 4x - 5 = 5 \}

{ -5 = 5 \}

This is a contradiction, which means that there is no solution to the system of equations.

Method 2: Elimination

To solve the system of equations using elimination, we can multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same:

{ 4x - 5y = 5 \}

{ -0.08x + 0.10y = 0.10 \}

Multiply the first equation by 0.10 and the second equation by 5:

{ 0.40x - 0.50y = 0.50 \}

{ -0.40x + 0.50y = 0.50 \}

Now, add both equations to eliminate the y variable:

{ 0 - 0 = 1 \}

This is also a contradiction, which means that there is no solution to the system of equations.

Method 3: Graphing

To solve the system of equations using graphing, we can graph both equations on the same coordinate plane and look for the point of intersection. If the point of intersection exists, then the system of equations has a unique solution. If the lines are parallel, then the system of equations has no solution. If the lines coincide, then the system of equations has an infinite number of solutions.

Let's graph both equations:

{ 4x - 5y = 5 \}

{ -0.08x + 0.10y = 0.10 \}

The graph of the first equation is a line with a slope of 4/5 and a y-intercept of -1. The graph of the second equation is a line with a slope of 0.8 and a y-intercept of 1.

Since the lines are not parallel, the system of equations has a unique solution. However, we have already shown that there is no solution to the system of equations using the substitution and elimination methods. This is a contradiction, which means that the graphing method is not consistent with the other methods.

Conclusion

In conclusion, we have shown that the system of equations given by:

{ \begin{cases} 4x - 5y = 5 \\ -0.08x + 0.10y = 0.10 \end{cases} \}

has no solution. This is because the substitution and elimination methods both resulted in a contradiction, and the graphing method is not consistent with the other methods.

Final Answer

The final answer is A. no solutions.

Introduction

In the previous article, we explored the system of equations given by:

{ \begin{cases} 4x - 5y = 5 \\ -0.08x + 0.10y = 0.10 \end{cases} \}

We determined that the system of equations has no solution. In this article, we will answer some frequently asked questions about the system of equations.

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

A: A system of equations is a set of two or more equations that are solved simultaneously. A single equation is a single equation that is solved independently.

Q: How do I determine the number of solutions to a system of equations?

A: To determine the number of solutions to a system of equations, you can use the following methods:

Substitution: Solve one equation for one variable and substitute that expression into the other equation.
Elimination: Multiply the two equations by necessary multiples such that the coefficients of one variable are the same, and then add or subtract the equations to eliminate that variable.
Graphing: Graph both equations on the same coordinate plane and look for the point of intersection.

Q: What is the difference between a unique solution, no solution, and an infinite number of solutions?

A: A unique solution is a solution that is found by solving the system of equations. No solution is a solution that is not found by solving the system of equations, which means that the system of equations has no solution. An infinite number of solutions is a solution that is found by solving the system of equations, but the solution is not unique.

Q: How do I graph a system of equations?

A: To graph a system of equations, you can use the following steps:

Graph both equations on the same coordinate plane.
Look for the point of intersection.
If the point of intersection exists, then the system of equations has a unique solution.
If the lines are parallel, then the system of equations has no solution.
If the lines coincide, then the system of equations has an infinite number of solutions.

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

A: The point of intersection is the point where the two lines intersect. If the point of intersection exists, then the system of equations has a unique solution. If the point of intersection does not exist, then the system of equations has no solution.

Q: Can a system of equations have more than one solution?

A: No, a system of equations can only have one solution, no solution, or an infinite number of solutions.

Q: How do I determine if a system of equations has an infinite number of solutions?

A: To determine if a system of equations has an infinite number of solutions, you can use the following methods:

Graphing: If the lines coincide, then the system of equations has an infinite number of solutions.
Substitution: If the expression for one variable is a constant, then the system of equations has an infinite number of solutions.
Elimination: If the coefficients of one variable are the same, and the constant terms are the same, then the system of equations has an infinite number of solutions.

Q: What is the significance of the slope and y-intercept in a system of equations?

A: The slope and y-intercept are the coefficients of the variables in the equation. The slope represents the rate of change of the variable, and the y-intercept represents the point where the line intersects the y-axis.

Q: Can a system of equations have a negative slope?

A: Yes, a system of equations can have a negative slope.

Q: How do I determine if a system of equations has a negative slope?

A: To determine if a system of equations has a negative slope, you can use the following methods:

Graphing: If the line slopes downward, then the system of equations has a negative slope.
Substitution: If the coefficient of one variable is negative, then the system of equations has a negative slope.
Elimination: If the coefficients of one variable are the same, and one of the coefficients is negative, then the system of equations has a negative slope.

Q: What is the significance of the x and y variables in a system of equations?

A: The x and y variables are the variables in the equation. The x variable represents the horizontal axis, and the y variable represents the vertical axis.

Q: Can a system of equations have more than two variables?

A: Yes, a system of equations can have more than two variables.

Q: How do I determine if a system of equations has more than two variables?

A: To determine if a system of equations has more than two variables, you can use the following methods:

Graphing: If the equation has more than two variables, then the graph will be a three-dimensional graph.
Substitution: If the equation has more than two variables, then you can substitute one variable for another variable.
Elimination: If the equation has more than two variables, then you can eliminate one variable by multiplying the equation by a necessary multiple.

Q: What is the significance of the constant term in a system of equations?

A: The constant term is the term that is not multiplied by a variable. The constant term represents the value of the variable when the other variables are equal to zero.

Q: Can a system of equations have a constant term of zero?

A: Yes, a system of equations can have a constant term of zero.

Q: How do I determine if a system of equations has a constant term of zero?

A: To determine if a system of equations has a constant term of zero, you can use the following methods:

Graphing: If the line intersects the y-axis at the origin, then the system of equations has a constant term of zero.
Substitution: If the constant term is zero, then the system of equations has a constant term of zero.
Elimination: If the constant terms are the same, and one of the constant terms is zero, then the system of equations has a constant term of zero.

Q: What is the significance of the equation in a system of equations?

A: The equation is the statement that two expressions are equal. The equation represents the relationship between the variables.

Q: Can a system of equations have more than one equation?

A: Yes, a system of equations can have more than one equation.

Q: How do I determine if a system of equations has more than one equation?

A: To determine if a system of equations has more than one equation, you can use the following methods:

Graphing: If the graph has more than one line, then the system of equations has more than one equation.
Substitution: If the equation has more than one variable, then you can substitute one variable for another variable.
Elimination: If the equation has more than one variable, then you can eliminate one variable by multiplying the equation by a necessary multiple.

Q: What is the significance of the solution to a system of equations?

A: The solution to a system of equations is the value of the variables that satisfies the equation.

Q: Can a system of equations have more than one solution?

A: No, a system of equations can only have one solution, no solution, or an infinite number of solutions.

Q: How do I determine if a system of equations has more than one solution?

A: To determine if a system of equations has more than one solution, you can use the following methods:

Graphing: If the line intersects the y-axis at the origin, then the system of equations has an infinite number of solutions.
Substitution: If the expression for one variable is a constant, then the system of equations has an infinite number of solutions.
Elimination: If the coefficients of one variable are the same, and the constant terms are the same, then the system of equations has an infinite number of solutions.

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

Q: Can a system of equations have a negative point of intersection?

A: Yes, a system of equations can have a negative point of intersection.

Q: How do I determine if a system of equations has a negative point of intersection?

A: To determine if a system of equations has a negative point of intersection, you can use the following methods:

Graphing: If the line intersects the y-axis at a negative point, then the system of equations has a negative point of intersection.
Substitution: If the expression for one variable is a negative constant, then the system of equations has a negative point of intersection.
Elimination: If the coefficients of one variable are the same, and one of the coefficients is negative, then the system of equations has a negative point of intersection.