What Is The Solution To The System Of Equations?${ \begin{array}{l} y = 1.5x - 4 \ y = -x \end{array} }$A. { (-1.6, 1.6)$}$B. { (-1.5, 1.5)$}$C. { (1.5, -1.5)$}$D. { (1.6, -1.6)$}$

Mar 11, 2025 by ADMIN 182 views

What is the Solution to the System of Equations?

Introduction to Systems of Equations

Systems of equations are a fundamental concept in mathematics, particularly in algebra and geometry. They involve multiple equations that are related to each other, and the goal is to find the values of the variables that satisfy all the equations simultaneously. In this article, we will explore a specific system of equations and find its solution.

The System of Equations

The given system of equations is:

y = 1.5x - 4

y = -x

We are asked to find the solution to this system, which means finding the values of $x$ and $y$ that satisfy both equations.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the substitution method.

First, we can rewrite the second equation as:

y = -x

x = -y

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

y = 1.5(-y) - 4

y = -1.5y - 4

Next, we can add $1.5y$ to both sides of the equation to get:

1.5y + y = -4

2.5y = -4

Now, we can divide both sides of the equation by $2.5$ to get:

y = -\frac{4}{2.5}

y = -1.6

Finding the Value of x

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 $x$ . We will use the second equation:

y = -x

-1.6 = -x

x = 1.6

Conclusion

Therefore, the solution to the system of equations is:

(x, y) = (1.6, -1.6)

This solution satisfies both equations, and it is the only solution to the system.

Discussion

The solution to the system of equations is a point in the coordinate plane, and it represents the intersection of the two lines. The first line has a slope of $1.5$ and a y-intercept of $-4$ , while the second line has a slope of $-1$ and a y-intercept of $0$ . The solution to the system is the point where these two lines intersect.

Final Answer

The final answer is:

(x, y) = (1.6, -1.6)

This is the solution to the system of equations, and it is the only solution to the system.

Comparison of Options

Let's compare the solution we found with the options given:

A. $(-1.6, 1.6)$

B. $(-1.5, 1.5)$

C. $(1.5, -1.5)$

D. $(1.6, -1.6)$

Only option D matches the solution we found, which is:

(x, y) = (1.6, -1.6)

Therefore, the correct answer is:

D. $(1.6, -1.6)$

Introduction

Systems of equations are a fundamental concept in mathematics, particularly in algebra and geometry. In this article, we will answer some frequently asked questions about systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other. The goal is to find the values of the variables that satisfy all the equations simultaneously.

Q: How do I solve a system of equations?

A: There are several methods to solve a system of equations, including the substitution method, elimination method, and graphing method. The choice of method depends on the type of equations and the number of variables.

Q: What is the substitution method?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one equation is linear and the other equation is quadratic.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one variable. This method is useful when the equations have the same coefficient for one variable.

Q: What is the graphing method?

A: The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful when the equations are linear.

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

A: A system of equations has a solution if the equations are consistent, meaning that they have at least one common solution. If the equations are inconsistent, meaning that they have no common solution, then the system has no solution.

Q: How do I know if a system of equations has multiple solutions?

A: A system of equations has multiple solutions if the equations are dependent, meaning that they are equivalent to each other. If the equations are independent, meaning that they are not equivalent to each other, then the system has a unique solution.

Q: Can a system of equations have no solution?

A: Yes, a system of equations can have no solution if the equations are inconsistent. This means that the equations have no common solution.

Q: Can a system of equations have multiple solutions?

A: Yes, a system of equations can have multiple solutions if the equations are dependent. This means that the equations are equivalent to each other and have multiple common solutions.

Q: How do I find the solution to a system of equations?

A: To find the solution to a system of equations, you can use the substitution method, elimination method, or graphing method. The choice of method depends on the type of equations and the number of variables.

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

A: The solution to a system of equations is the set of values that satisfy all the equations simultaneously. This can be a single point, a line, or a plane, depending on the type of equations and the number of variables.

Q: Can a system of equations have a solution that is not a point?

A: Yes, a system of equations can have a solution that is not a point. For example, if the equations are linear, then the solution can be a line or a plane.

Q: Can a system of equations have a solution that is not a line or a plane?

A: Yes, a system of equations can have a solution that is not a line or a plane. For example, if the equations are quadratic, then the solution can be a parabola or an ellipse.

Q: How do I graph a system of equations?

A: To graph a system of equations, you can use a coordinate plane and plot the equations on the plane. The point of intersection is the solution to the system.

Q: What is the point of intersection?

A: The point of intersection is the solution to the system of equations. It is the point where the two lines or curves intersect.

Q: Can the point of intersection be a single point?

A: Yes, the point of intersection can be a single point. This means that the two lines or curves intersect at a single point.

Q: Can the point of intersection be a line or a plane?

A: Yes, the point of intersection can be a line or a plane. This means that the two lines or curves intersect at a line or a plane.

Q: Can the point of intersection be a parabola or an ellipse?

A: Yes, the point of intersection can be a parabola or an ellipse. This means that the two curves intersect at a parabola or an ellipse.

Q: How do I find the point of intersection?

A: To find the point of intersection, you can use the substitution method, elimination method, or graphing method. The choice of method depends on the type of equations and the number of variables.

Q: What is the significance of the point of intersection?

A: The point of intersection is the solution to the system of equations. It represents the point where the two lines or curves intersect.

Q: Can the point of intersection be used to solve a system of equations?

A: Yes, the point of intersection can be used to solve a system of equations. It represents the solution to the system.