Graph The Solution To The Following System Of Inequalities:$\[ \begin{array}{l} 4x + 7y \ \textless \ 14 \\ -3x + 5y \geq 15 \end{array} \\]Then Give The Coordinates Of One Point In The Solution Set.

Feb 26, 2025 by ADMIN 202 views

Graph the Solution to the System of Inequalities and Find the Coordinates of One Point in the Solution Set

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Introduction

In this article, we will explore the concept of graphing the solution to a system of inequalities and finding the coordinates of one point in the solution set. We will use the given system of inequalities:

{ \begin{array}{l} 4x + 7y \ \textless \ 14 \\ -3x + 5y \geq 15 \end{array} \}

Understanding the Inequalities

To graph the solution to the system of inequalities, we need to understand the individual inequalities and their corresponding graphs.

Inequality 1: $4x + 7y < 14$

The first inequality is a linear inequality in two variables, $x$ and $y$ . To graph this inequality, we need to find the boundary line, which is given by the equation $4x + 7y = 14$ . We can rewrite this equation in slope-intercept form as $y = -\frac{4}{7}x + 2$ . This is a non-vertical line with a slope of $-\frac{4}{7}$ and a $y$ -intercept of $2$ .

To graph the inequality, we need to shade the region on one side of the boundary line. Since the inequality is of the form $y < mx + b$ , we will shade the region below the boundary line.

Inequality 2: $-3x + 5y \geq 15$

The second inequality is also a linear inequality in two variables, $x$ and $y$ . To graph this inequality, we need to find the boundary line, which is given by the equation $-3x + 5y = 15$ . We can rewrite this equation in slope-intercept form as $y = \frac{3}{5}x + 3$ . This is a non-vertical line with a slope of $\frac{3}{5}$ and a $y$ -intercept of $3$ .

To graph the inequality, we need to shade the region on one side of the boundary line. Since the inequality is of the form $y \geq mx + b$ , we will shade the region above the boundary line.

Graphing the Solution Set

To graph the solution set, we need to find the intersection of the two shaded regions. The solution set is the region that satisfies both inequalities.

Since the first inequality is shaded below the boundary line, and the second inequality is shaded above the boundary line, the solution set will be the region that is below the boundary line of the first inequality and above the boundary line of the second inequality.

Finding the Coordinates of One Point in the Solution Set

To find the coordinates of one point in the solution set, we need to find the intersection of the two boundary lines. We can do this by solving the system of equations:

{ \begin{array}{l} 4x + 7y = 14 \\ -3x + 5y = 15 \end{array} \}

We can solve this system of equations using substitution or elimination. Let's use substitution.

Rearranging the first equation, we get:

$y = -\frac{4}{7}x + 2$

Substituting this expression for $y$ into the second equation, we get:

$-3x + 5(-\frac{4}{7}x + 2) = 15$

Simplifying and solving for $x$ , we get:

$x = \frac{7}{13}$

Substituting this value of $x$ into the expression for $y$ , we get:

$y = -\frac{4}{7}(\frac{7}{13}) + 2$

Simplifying, we get:

$y = \frac{25}{13}$

Therefore, the coordinates of one point in the solution set are $(\frac{7}{13}, \frac{25}{13})$ .

Conclusion

In this article, we graphed the solution to the system of inequalities and found the coordinates of one point in the solution set. We used the given system of inequalities:

{ \begin{array}{l} 4x + 7y \ \textless \ 14 \\ -3x + 5y \geq 15 \end{array} \}

We graphed the individual inequalities and found the intersection of the two shaded regions. We then found the coordinates of one point in the solution set by solving the system of equations.

The coordinates of one point in the solution set are $(\frac{7}{13}, \frac{25}{13})$ .

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Q: What is a system of inequalities?

A: A system of inequalities is a set of two or more inequalities that are related to each other. In this article, we graphed the solution to a system of two inequalities.

Q: How do I graph the solution to a system of inequalities?

A: To graph the solution to a system of inequalities, you need to graph each inequality separately and then find the intersection of the two shaded regions. The solution set is the region that satisfies both inequalities.

Q: What is the difference between a linear inequality and a non-linear inequality?

A: A linear inequality is an inequality that can be written in the form $ax + by < c$ , where $a$ , $b$ , and $c$ are constants. A non-linear inequality is an inequality that cannot be written in this form.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to find the boundary line, which is given by the equation $ax + by = c$ . You can then shade the region on one side of the boundary line, depending on the direction of the inequality.

Q: What is the boundary line?

A: The boundary line is the line that separates the region that satisfies the inequality from the region that does not satisfy the inequality.

Q: How do I find the boundary line?

A: To find the boundary line, you need to rewrite the inequality in the form $ax + by = c$ . You can then graph the line and shade the region on one side of the line.

Q: What is the solution set?

A: The solution set is the region that satisfies both inequalities in the system.

Q: How do I find the solution set?

A: To find the solution set, you need to graph each inequality separately and then find the intersection of the two shaded regions.

Q: What is the intersection of two regions?

A: The intersection of two regions is the region that is common to both regions.

Q: How do I find the intersection of two regions?

A: To find the intersection of two regions, you need to find the points that are common to both regions.

Q: What is the significance of the intersection of two regions?

A: The intersection of two regions represents the solution set, which is the region that satisfies both inequalities in the system.

Q: Can I have multiple solutions to a system of inequalities?

A: Yes, you can have multiple solutions to a system of inequalities. The solution set can be a single point, a line, a region, or even the entire plane.

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

A: To determine the number of solutions to a system of inequalities, you need to graph each inequality separately and then find the intersection of the two shaded regions.

Q: What if the solution set is empty?

A: If the solution set is empty, it means that there are no points that satisfy both inequalities in the system.

Q: What if the solution set is the entire plane?

A: If the solution set is the entire plane, it means that every point in the plane satisfies both inequalities in the system.

Q: Can I have a system of inequalities with no solution?

A: Yes, you can have a system of inequalities with no solution. This occurs when the solution set is empty.

Q: How do I determine if a system of inequalities has no solution?

A: To determine if a system of inequalities has no solution, you need to graph each inequality separately and then find the intersection of the two shaded regions. If the intersection is empty, then the system has no solution.