Question

Verify that the values of the variables listed are solutions of the system of equations. $$ \begin{array}{l} \left\\{\begin{array}{l} x-y=3 \\ \frac{1}{2} x+y=3 \end{array}\right. \\ x=4, y=1 ;(4,1) \end{array} $$

Short answer

(4, 1) is a solution to the system of equations.

Step-by-step solution

- Substitute values into the first equation

Take the values of the variables given, which are \(x = 4\) and \(y = 1\). Substitute these values into the first equation: \(x - y = 3\). This gives us: \(4 - 1 = 3\).

- Verify the first equation

Simplify the equation after substituting: \(4 - 1 = 3\). This simplifies to \(3 = 3\), which is a true statement. Therefore, the values satisfy the first equation.

- Substitute values into the second equation

Next, substitute \(x = 4\) and \(y = 1\) into the second equation: \(\frac{1}{2} x + y = 3\). This gives us: \(\frac{1}{2} \times 4 + 1 = 3\).

- Verify the second equation

Simplify the equation after substituting: \(2 + 1 = 3\). This simplifies to \(3 = 3\), which is a true statement. Therefore, the values also satisfy the second equation.

- Conclusion

Since the values \(x = 4\) and \(y = 1\) satisfy both equations in the system, we can conclude that \((4, 1)\) is indeed a solution to the system of equations.

Key concepts

Substitution Method

The substitution method is a common technique used to solve a system of linear equations. This method involves solving one of the equations for one variable in terms of the others, and then substituting this expression into the other equations.
In the original exercise and solution, the system of equations given is: 1. \( x - y = 3 \) 2. \( \frac{1}{2} x + y = 3 \)
We are given the values: \( x = 4 \) and \( y = 1 \).
By substituting these values into each equation, we verify if they satisfy both equations.
This step-by-step substitution ensures that both equations hold true, proving that \( (4, 1) \) is a solution.

Linear Equations

Linear equations are equations where the highest power of the variable is always one. They graph as straight lines in a coordinate system. The general form of a linear equation in one variable is: \( ax + b = c \). When dealing with systems of linear equations, we are looking at two or more linear equations that need to be solved simultaneously.
In our given system: - For the equation \( x - y = 3 \), simplifying it shows the relationship between \(x\) and \(y\).
- For the equation \( \frac{1}{2}x + y = 3 \), it also describes a linear relationship.
Solving a system of linear equations means finding a set of values for the variables that make both equations true at the same time. This is often visualized as finding the intersection point of the lines represented by these equations.

Solution Verification

Solution verification is crucial to ensure that the obtained values for variables are indeed correct. For the given system of equations, we verify the solution \((4, 1)\) by substituting back into the original equations.

• First, substitute \(x = 4\) and \(y = 1\) into the first equation \( x - y = 3 \). This simplifies to \( 4 - 1 = 3 \), which is true.
• Next, substitute \(x = 4\) and \(y = 1\) into the second equation \( \frac{1}{2} x + y = 3 \). This simplifies to \( \frac{1}{2} \times 4 + 1 = 3 \), which is also true.

Since both substitutions result in true statements, we can confirm that \((4, 1)\) is indeed a valid solution to the system of equations. This verification process provides the confidence that the solution is correct and consistent with both original equations.