Graph The Solution To The Following System Of Inequalities In The Coordinate Plane.$\[ \begin{align*} 2x - 3y &\ \textless \ 15 \\ y &\leq X + 2 \end{align*} \\]

Introduction

Graphing the solution to a system of inequalities in the coordinate plane is a fundamental concept in mathematics, particularly in algebra and geometry. It involves representing the solution set of a system of linear inequalities as a region in the coordinate plane. In this article, we will explore how to graph the solution to the system of inequalities given by:

${ \begin{align*} 2x - 3y &\ \textless \ 15 \\ y &\leq x + 2 \end{align*} \}$

Understanding the Inequalities

Before we proceed to graph the solution, let's understand the two inequalities involved in the system.

Inequality 1: $2x - 3y < 15$

This inequality represents a linear inequality in two variables, $x$ and $y$. The coefficient of $x$ is $2$, and the coefficient of $y$ is $-3$. The constant term is $15$. To graph this inequality, we need to find the boundary line and determine the direction of the inequality.

Inequality 2: $y \leq x + 2$

This inequality represents another linear inequality in two variables, $x$ and $y$. The coefficient of $x$ is $1$, and the coefficient of $y$ is $-1$. The constant term is $2$. To graph this inequality, we need to find the boundary line and determine the direction of the inequality.

Graphing the Inequalities

To graph the solution to the system of inequalities, we need to graph each inequality separately and then find the intersection of the two solution sets.

Graphing Inequality 1: $2x - 3y < 15$

To graph this inequality, we need to find the boundary line and determine the direction of the inequality. The boundary line is given by the equation $2x - 3y = 15$. We can rewrite this equation in slope-intercept form as $y = \frac{2}{3}x - 5$. This is a line with a slope of $\frac{2}{3}$ and a $y$-intercept of $-5$.

Since the inequality is of the form $y < mx + b$, where $m$ is the slope and $b$ is the $y$-intercept, we know that the solution set lies below the boundary line. Therefore, we graph the boundary line and shade the region below it.

Graphing Inequality 2: $y \leq x + 2$

To graph this inequality, we need to find the boundary line and determine the direction of the inequality. The boundary line is given by the equation $y = x + 2$. This is a line with a slope of $1$ and a $y$-intercept of $2$.

Since the inequality is of the form $y \leq mx + b$, where $m$ is the slope and $b$ is the $y$-intercept, we know that the solution set lies on or below the boundary line. Therefore, we graph the boundary line and shade the region on or below it.

Finding the Intersection of the Solution Sets

To find the intersection of the two solution sets, we need to find the points of intersection between the two boundary lines.

Finding the Points of Intersection

To find the points of intersection, we need to solve the system of equations formed by the two boundary lines.

${ \begin{align*} y &= \frac{2}{3}x - 5 \\ y &= x + 2 \end{align*} \}$

We can solve this system of equations by equating the two expressions for $y$ and solving for $x$.

${ \begin{align*} \frac{2}{3}x - 5 &= x + 2 \\ \frac{2}{3}x - x &= 2 + 5 \\ -\frac{1}{3}x &= 7 \\ x &= -21 \end{align*} \}$

Now that we have found the value of $x$, we can substitute it into one of the boundary equations to find the corresponding value of $y$.

${ \begin{align*} y &= \frac{2}{3}(-21) - 5 \\ y &= -14 - 5 \\ y &= -19 \end{align*} \}$

Therefore, the point of intersection is $(-21, -19)$.

Graphing the Solution Set

To graph the solution set, we need to graph the two boundary lines and shade the region that satisfies both inequalities.

The solution set is the region that lies below the boundary line $y = \frac{2}{3}x - 5$ and on or below the boundary line $y = x + 2$. We can graph this region by shading the area below the line $y = \frac{2}{3}x - 5$ and on or below the line $y = x + 2$.

Conclusion

In this article, we have graphed the solution to the system of inequalities given by:

${ \begin{align*} 2x - 3y &\ \textless \ 15 \\ y &\leq x + 2 \end{align*} \}$

We have graphed each inequality separately and then found the intersection of the two solution sets. We have also graphed the solution set by shading the region that satisfies both inequalities. This article has provided a step-by-step guide on how to graph the solution to a system of inequalities in the coordinate plane.

References

• [1] "Graphing Linear Inequalities" by Math Open Reference
• [2] "Systems of Linear Inequalities" by Khan Academy
• [3] "Graphing Systems of Linear Inequalities" by Purplemath

Additional Resources

• [1] "Graphing Linear Inequalities" by IXL
• [2] "Systems of Linear Inequalities" by Mathway
• [3] "Graphing Systems of Linear Inequalities" by Algebra.com
  Graphing the Solution to a System of Inequalities in the Coordinate Plane: Q&A ================================================================================

Introduction

Graphing the solution to a system of inequalities in the coordinate plane is a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we explored how to graph the solution to the system of inequalities given by:

${ \begin{align*} 2x - 3y &\ \textless \ 15 \\ y &\leq x + 2 \end{align*} \}$

In this article, we will answer some frequently asked questions about graphing the solution to a system of inequalities in the coordinate plane.

Q&A

Q: What is the first step in graphing the solution to a system of inequalities?

A: The first step in graphing the solution to a system of inequalities is to graph each inequality separately.

Q: How do I graph a linear inequality in the coordinate plane?

A: To graph a linear inequality in the coordinate plane, you need to find the boundary line and determine the direction of the inequality. If the inequality is of the form $y < mx + b$, where $m$ is the slope and $b$ is the $y$-intercept, you know that the solution set lies below the boundary line. If the inequality is of the form $y \leq mx + b$, you know that the solution set lies on or below the boundary line.

Q: How do I find the boundary line of a linear inequality?

A: To find the boundary line of a linear inequality, you need to rewrite the inequality in the form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Q: How do I determine the direction of the inequality?

A: To determine the direction of the inequality, you need to look at the inequality sign. If the inequality sign is $<$, the solution set lies below the boundary line. If the inequality sign is $\leq$, the solution set lies on or below the boundary line.

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

A: To graph the solution set of a system of inequalities, you need to graph the boundary lines of each inequality and shade the region that satisfies both inequalities.

Q: What is the point of intersection of two boundary lines?

A: The point of intersection of two boundary lines is the point where the two lines meet. To find the point of intersection, you need to solve the system of equations formed by the two boundary lines.

Q: How do I find the point of intersection of two boundary lines?

A: To find the point of intersection of two boundary lines, you need to solve the system of equations formed by the two boundary lines. You can do this by equating the two expressions for $y$ and solving for $x$. Then, you can substitute the value of $x$ into one of the boundary equations to find the corresponding value of $y$.

Q: What is the solution set of a system of inequalities?

A: The solution set of a system of inequalities is the region that satisfies both inequalities. It is the region that lies below the boundary line of the first inequality and on or below the boundary line of the second inequality.

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

A: To graph the solution set of a system of inequalities, you need to graph the boundary lines of each inequality and shade the region that satisfies both inequalities.

Conclusion

In this article, we have answered some frequently asked questions about graphing the solution to a system of inequalities in the coordinate plane. We have covered topics such as graphing linear inequalities, finding the boundary line, determining the direction of the inequality, graphing the solution set, finding the point of intersection, and defining the solution set. This article has provided a comprehensive guide on how to graph the solution to a system of inequalities in the coordinate plane.

References

• [1] "Graphing Linear Inequalities" by Math Open Reference
• [2] "Systems of Linear Inequalities" by Khan Academy
• [3] "Graphing Systems of Linear Inequalities" by Purplemath

Additional Resources

• [1] "Graphing Linear Inequalities" by IXL
• [2] "Systems of Linear Inequalities" by Mathway
• [3] "Graphing Systems of Linear Inequalities" by Algebra.com