A System Of Equations Has One Solution. If $4x - Y = 5$ Is One Of The Equations, Which Could Be The Other Equation?A. $y = -4x + 5$ B. $y = 4x - 5$ C. $2y = 8x - 10$ D. $-2y = -8x - 10$

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. When a system of equations has one solution, it means that there is only one set of values that satisfies all the equations in the system. In this article, we will explore the concept of a system of equations with one solution and examine the possibilities for the second equation in a system where one of the equations is given as $4x - y = 5$.

What is a System of Equations?

A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. Each equation in the system is a statement that two expressions are equal. For example, the system of equations:

$$\begin{cases} 4x - y = 5 \\ 2x + y = 3 \end{cases}$$

has two equations, each with two variables, x and y. To solve this system, we need to find the values of x and y that satisfy both equations simultaneously.

Systems of Equations with One Solution

A system of equations has one solution when the two equations are linearly independent, meaning that they are not parallel lines. In other words, the two equations intersect at exactly one point, which is the solution to the system. When a system of equations has one solution, it means that there is only one set of values that satisfies all the equations in the system.

The Given Equation: $4x - y = 5$

The given equation is $4x - y = 5$. This equation is a linear equation in two variables, x and y. To find the other equation in the system, we need to consider the possibilities for the second equation.

Possibilities for the Second Equation

The second equation in the system must be linearly independent of the given equation. In other words, the two equations must not be parallel lines. To find the possibilities for the second equation, we can start by rearranging the given equation to isolate y:

$$y = 4x - 5$$

This equation is in the form of a linear equation, where y is expressed in terms of x. To find the possibilities for the second equation, we can consider the following options:

• Option A: $y = -4x + 5$
  • This equation is a linear equation in two variables, x and y. However, it is not linearly independent of the given equation, as it is a parallel line to the given equation.
• Option B: $y = 4x - 5$
  • This equation is the same as the given equation, which means that it is not a possibility for the second equation in the system.
• Option C: $2y = 8x - 10$
  • This equation is a linear equation in two variables, x and y. However, it is not linearly independent of the given equation, as it is a parallel line to the given equation.
• Option D: $-2y = -8x - 10$
  • This equation is a linear equation in two variables, x and y. However, it is not linearly independent of the given equation, as it is a parallel line to the given equation.

Conclusion

In conclusion, a system of equations has one solution when the two equations are linearly independent, meaning that they are not parallel lines. The given equation is $4x - y = 5$. To find the other equation in the system, we need to consider the possibilities for the second equation. The possibilities for the second equation are:

• Option A: $y = -4x + 5$
  • This equation is not linearly independent of the given equation, as it is a parallel line to the given equation.
• Option B: $y = 4x - 5$
  • This equation is the same as the given equation, which means that it is not a possibility for the second equation in the system.
• Option C: $2y = 8x - 10$
  • This equation is not linearly independent of the given equation, as it is a parallel line to the given equation.
• Option D: $-2y = -8x - 10$
  • This equation is not linearly independent of the given equation, as it is a parallel line to the given equation.

Therefore, none of the options A, B, C, or D are correct possibilities for the second equation in the system.

Final Answer

In the previous article, we explored the concept of a system of equations with one solution and examined the possibilities for the second equation in a system where one of the equations is given as $4x - y = 5$. In this article, we will answer some frequently asked questions about systems of equations with one solution.

Q: What is a system of equations?

A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. Each equation in the system is a statement that two expressions are equal.

Q: What is the difference between a system of equations with one solution and a system of equations with no solution?

A system of equations has one solution when the two equations are linearly independent, meaning that they are not parallel lines. In other words, the two equations intersect at exactly one point, which is the solution to the system. On the other hand, a system of equations has no solution when the two equations are parallel lines, meaning that they never intersect.

Q: How can I determine if a system of equations has one solution or no solution?

To determine if a system of equations has one solution or no solution, you can use the following steps:

1. Graph the equations: Graph the two equations on a coordinate plane to see if they intersect at exactly one point or never intersect.
2. Check if the equations are parallel: Check if the two equations are parallel by comparing their slopes. If the slopes are the same, then the equations are parallel and the system has no solution.
3. Check if the equations are linearly independent: Check if the two equations are linearly independent by checking if one equation is a multiple of the other. If one equation is a multiple of the other, then the equations are linearly dependent and the system has no solution.

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

A system of equations with one solution is significant because it means that there is only one set of values that satisfies all the equations in the system. This is useful in many real-world applications, such as solving optimization problems, finding the maximum or minimum value of a function, and determining the best solution to a problem.

Q: How can I find the solution to a system of equations with one solution?

To find the solution to a system of equations with one solution, you can use the following steps:

1. Solve one equation for one variable: Solve one equation for one variable, such as x or y.
2. Substitute the expression into the other equation: Substitute the expression into the other equation to get a new equation with one variable.
3. Solve the new equation: Solve the new equation to find the value of the variable.
4. Substitute the value back into one of the original equations: Substitute the value back into one of the original equations to find the value of the other variable.

Q: What are some common mistakes to avoid when solving systems of equations with one solution?

Some common mistakes to avoid when solving systems of equations with one solution include:

• Not checking if the equations are linearly independent: Make sure to check if the equations are linearly independent before solving the system.
• Not solving one equation for one variable: Make sure to solve one equation for one variable before substituting the expression into the other equation.
• Not substituting the value back into one of the original equations: Make sure to substitute the value back into one of the original equations to find the value of the other variable.

Conclusion

In conclusion, a system of equations with one solution is a set of two or more equations that are solved simultaneously to find the values of the variables. The system has one solution when the two equations are linearly independent, meaning that they are not parallel lines. To determine if a system of equations has one solution or no solution, you can use the following steps: graph the equations, check if the equations are parallel, and check if the equations are linearly independent. To find the solution to a system of equations with one solution, you can use the following steps: solve one equation for one variable, substitute the expression into the other equation, solve the new equation, and substitute the value back into one of the original equations.