Solve The System Of Equations:${ \begin{array}{l} 9x + 10y \geq 14 \ 7x + 5y = -3 \end{array} }$

Introduction

In mathematics, solving systems of linear inequalities and equations is a crucial concept that involves finding the solution set of a system of linear inequalities and equations. This concept is widely used in various fields such as economics, engineering, and computer science. In this article, we will focus on solving the system of equations:

$$\begin{array}{l} 9x + 10y \geq 14 \\ 7x + 5y = -3 \end{array}$$

Understanding the Problem

The given system of equations consists of two linear inequalities and one linear equation. The first inequality is $9x + 10y \geq 14$, and the second inequality is $7x + 5y \geq -3$. The linear equation is $7x + 5y = -3$. Our goal is to find the solution set of this system of equations.

Methodology

To solve this system of equations, we will use the method of substitution and elimination. We will first solve the linear equation $7x + 5y = -3$ for one variable in terms of the other variable. Then, we will substitute this expression into the first inequality $9x + 10y \geq 14$ to obtain a new inequality in one variable. We will then solve this inequality to find the solution set of the system of equations.

Step 1: Solve the Linear Equation

We will solve the linear equation $7x + 5y = -3$ for $y$ in terms of $x$. We can do this by subtracting $7x$ from both sides of the equation and then dividing both sides by $5$. This gives us:

$$y = \frac{-3 - 7x}{5}$$

Step 2: Substitute into the First Inequality

We will substitute the expression for $y$ into the first inequality $9x + 10y \geq 14$. We can do this by replacing $y$ with $\frac{-3 - 7x}{5}$ in the inequality. This gives us:

$$9x + 10\left(\frac{-3 - 7x}{5}\right) \geq 14$$

Simplifying the Inequality

We will simplify the inequality by multiplying both sides by $5$ to eliminate the fraction. This gives us:

$$45x - 30 - 70x \geq 70$$

We will then combine like terms by adding $45x$ and $70x$ to get:

$$-25x - 30 \geq 70$$

Solving the Inequality

We will solve the inequality by adding $30$ to both sides to get:

$$-25x \geq 100$$

We will then divide both sides by $-25$ to get:

$$x \leq -4$$

Conclusion

We have solved the system of equations by using the method of substitution and elimination. We first solved the linear equation for one variable in terms of the other variable. Then, we substituted this expression into the first inequality to obtain a new inequality in one variable. We then solved this inequality to find the solution set of the system of equations.

Solution Set

The solution set of the system of equations is $x \leq -4$. This means that the solution set is all values of $x$ that are less than or equal to $-4$.

Graphical Representation

We can represent the solution set graphically by plotting the line $7x + 5y = -3$ and the region $x \leq -4$. The solution set is the region that satisfies both the line and the inequality.

Real-World Applications

Solving systems of linear inequalities and equations has many real-world applications. For example, in economics, it can be used to determine the optimal production levels of a company. In engineering, it can be used to design and optimize systems such as bridges and buildings. In computer science, it can be used to solve problems such as scheduling and resource allocation.

Conclusion

In conclusion, solving systems of linear inequalities and equations is a crucial concept in mathematics that has many real-world applications. We have used the method of substitution and elimination to solve the system of equations:

$$\begin{array}{l} 9x + 10y \geq 14 \\ 7x + 5y = -3 \end{array}$$

We have found the solution set to be $x \leq -4$. We have also represented the solution set graphically and discussed its real-world applications.

References

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Introduction to Linear Algebra" by Jim Hefferon
• [3] "Solving Systems of Linear Equations" by Khan Academy

Additional Resources

• [1] Khan Academy: Solving Systems of Linear Equations
• [2] MIT OpenCourseWare: Linear Algebra
• [3] Wolfram Alpha: Solving Systems of Linear Equations
  Solving Systems of Linear Inequalities and Equations: Q&A =====================================================

Introduction

In our previous article, we discussed solving systems of linear inequalities and equations. We used the method of substitution and elimination to solve the system of equations:

$$\begin{array}{l} 9x + 10y \geq 14 \\ 7x + 5y = -3 \end{array}$$

We found the solution set to be $x \leq -4$. In this article, we will answer some frequently asked questions about solving systems of linear inequalities and equations.

Q&A

Q: What is the difference between a system of linear equations and a system of linear inequalities?

A: A system of linear equations consists of two or more linear equations that are equal to each other. A system of linear inequalities consists of two or more linear inequalities that are greater than or less than each other.

Q: How do I know which method to use to solve a system of linear inequalities and equations?

A: You can use the method of substitution and elimination to solve a system of linear inequalities and equations. However, if the system has multiple variables, you may need to use other methods such as graphing or matrices.

Q: Can I use a calculator to solve a system of linear inequalities and equations?

A: Yes, you can use a calculator to solve a system of linear inequalities and equations. However, it's always a good idea to check your work by hand to make sure you understand the solution.

Q: How do I represent the solution set graphically?

A: You can represent the solution set graphically by plotting the line or inequality and shading the region that satisfies the inequality.

Q: What are some real-world applications of solving systems of linear inequalities and equations?

A: Solving systems of linear inequalities and equations has many real-world applications, including economics, engineering, and computer science. For example, it can be used to determine the optimal production levels of a company, design and optimize systems such as bridges and buildings, and solve problems such as scheduling and resource allocation.

Q: Can I use a computer program to solve a system of linear inequalities and equations?

A: Yes, you can use a computer program such as MATLAB or Python to solve a system of linear inequalities and equations. However, it's always a good idea to check your work by hand to make sure you understand the solution.

Q: How do I know if a system of linear inequalities and equations has a solution?

A: A system of linear inequalities and equations has a solution if the solution set is not empty. If the solution set is empty, then the system has no solution.

Q: Can I use a system of linear inequalities and equations to model real-world problems?

A: Yes, you can use a system of linear inequalities and equations to model real-world problems. For example, you can use it to model the production levels of a company, the cost of a product, or the demand for a product.

Conclusion

In conclusion, solving systems of linear inequalities and equations is a crucial concept in mathematics that has many real-world applications. We have answered some frequently asked questions about solving systems of linear inequalities and equations, including how to represent the solution set graphically and how to use a computer program to solve the system.

Additional Resources

• [1] Khan Academy: Solving Systems of Linear Equations
• [2] MIT OpenCourseWare: Linear Algebra
• [3] Wolfram Alpha: Solving Systems of Linear Equations

References

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Introduction to Linear Algebra" by Jim Hefferon
• [3] "Solving Systems of Linear Equations" by Khan Academy

Practice Problems

1. Solve the system of linear inequalities and equations:

$$\begin{array}{l} 2x + 3y \geq 5 \\ x + 2y = 3 \end{array}$$

2. Solve the system of linear inequalities and equations:

$$\begin{array}{l} x + 2y \geq 4 \\ 2x + 3y = 5 \end{array}$$

3. Solve the system of linear inequalities and equations:

$$\begin{array}{l} x + 3y \geq 6 \\ 2x + 2y = 4 \end{array}$$

Answer Key

1. The solution set is $x \geq 1$.
2. The solution set is $x \leq 1$.
3. The solution set is $x \geq 2$.