Write your own system of two linear inequalities.

System of Linear Inequalities

Inequalities:

1. \(2x + 3y \leq 6\)
2. \(x - y > 1\)

Determining If a Point is a Solution

To determine whether a given point is a solution to the system of linear inequalities, follow these steps:

1. Substitute the \(x\) and \(y\) coordinates of the point into each inequality.
2. Check if the point satisfies both inequalities at the same time.

Example: Let's consider the point \( (2, 1) \).

Checking the first inequality: \[ 2x + 3y \leq 6 \] Substituting \(x = 2\) and \(y = 1\): \[ 2(2) + 3(1) = 4 + 3 = 7 \] Since \(7 \leq 6\) is false, this point does not satisfy the first inequality.

Checking the second inequality: \[ x - y > 1 \] Substituting \(x = 2\) and \(y = 1\): \[ 2 - 1 = 1 \] Since \(1 > 1\) is also false, the point \( (2, 1) \) does not satisfy the second inequality either.

Thus, \( (2, 1) \) is not a solution to the system of inequalities.

Could a System of Linear Inequalities Have No Solutions?

Yes, a system of linear inequalities can have no solutions. This occurs when the regions described by the inequalities do not overlap.

Example of No Solutions: Consider the inequalities:

1. \(y < -x + 2\)
2. \(y > -x + 4\)

In this case, the first inequality represents the region below the line \(y = -x + 2\), and the second inequality represents the region above the line \(y = -x + 4\). Since these two lines are parallel and distinct, there is no area where both inequalities are satisfied simultaneously.

Graphical Representation:

• The line \(y = -x + 2\) would be drawn as a dashed line (since it's a strict inequality).
• The line \(y = -x + 4\) would also be drawn as a dashed line.

The resulting graph would show two parallel dashed lines with no shaded region in between, indicating that there are no points that satisfy both inequalities.

This configuration clearly illustrates that the system of inequalities has no solution, as the areas defined by the inequalities are completely separated from each other.

Certainly! Here's the response with the question numbers indicated:

1. System of Linear Inequalities

Inequalities:

1. \(2x + 3y \leq 6\)
2. \(x - y > 1\)

2. Determining If a Point is a Solution

To determine whether a given point is a solution to the system of linear inequalities, follow these steps:

1. Substitute the \(x\) and \(y\) coordinates of the point into each inequality.
2. Check if the point satisfies both inequalities at the same time.

Example: Let's consider the point \( (2, 1) \).

Checking the first inequality: \[ 2x + 3y \leq 6 \] Substituting \(x = 2\) and \(y = 1\): \[ 2(2) + 3(1) = 4 + 3 = 7 \] Since \(7 \leq 6\) is false, this point does not satisfy the first inequality.

Checking the second inequality: \[ x - y > 1 \] Substituting \(x = 2\) and \(y = 1\): \[ 2 - 1 = 1 \] Since \(1 > 1\) is also false, the point \( (2, 1) \) does not satisfy the second inequality either.

Thus, \( (2, 1) \) is not a solution to the system of inequalities.

3. Could a System of Linear Inequalities Have No Solutions?

Yes, a system of linear inequalities can have no solutions. This occurs when the regions described by the inequalities do not overlap.

Example of No Solutions: Consider the inequalities:

1. \(y < -x + 2\)
2. \(y > -x + 4\)

In this case, the first inequality represents the region below the line \(y = -x + 2\), and the second inequality represents the region above the line \(y = -x + 4\). Since these two lines are parallel and distinct, there is no area where both inequalities are satisfied simultaneously.

Graphical Representation:

• The line \(y = -x + 2\) would be drawn as a dashed line (since it's a strict inequality).
• The line \(y = -x + 4\) would also be drawn as a dashed line.

The resulting graph would show two parallel dashed lines with no shaded region in between, indicating that there are no points that satisfy both inequalities.

This configuration clearly illustrates that the system of inequalities has no solution, as the areas defined by the inequalities are completely separated from each other.

Sure! Let's use a different example to determine whether a given point is a solution to the system of linear inequalities.

2. Determining If a Point is a Solution (New Example)

System of Linear Inequalities:

1. \(x + y > 3\)
2. \(2x - y \leq 4\)

Example Point: \( (1, 3) \)

Now, we will check if the point \( (1, 3) \) satisfies both inequalities in the system.

Checking the first inequality: \[ x + y > 3 \] Substituting \(x = 1\) and \(y = 3\): \[ 1 + 3 = 4 \] Since \(4 > 3\) is true, this point satisfies the first inequality.

Checking the second inequality: \[ 2x - y \leq 4 \] Substituting \(x = 1\) and \(y = 3\): \[ 2(1) - 3 = 2 - 3 = -1 \] Since \(-1 \leq 4\) is true, this point also satisfies the second inequality.

Conclusion

Since the point \( (1, 3) \) satisfies both inequalities, it is a solution to the system of linear inequalities.