Solve The System Of Equations:${ \begin{array}{l} y = -2x - 1 \ 3x - 4y = -40 \end{array} }$

Mar 10, 2025 by ADMIN 94 views

Solve the System of Equations

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Introduction

Solving a system of equations is a fundamental concept in mathematics, particularly in algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables. We will use the given equations:

{ \begin{array}{l} y = -2x - 1 \\ 3x - 4y = -40 \end{array} \}

Understanding the Equations

The first equation is a linear equation in the slope-intercept form, where $y$ is equal to $-2x$ minus $1$ . This equation represents a straight line with a slope of $-2$ and a y-intercept of $-1$ .

The second equation is also a linear equation, but it is in the general form, where $3x$ minus $4y$ is equal to $-40$ . This equation can be rewritten as $y = \frac{3}{4}x + 10$ , which represents a straight line with a slope of $\frac{3}{4}$ and a y-intercept of $10$ .

Substitution Method

One of the methods to solve a system of equations is the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation. In this case, we can solve the first equation for $y$ and substitute it into the second equation.

Step 1: Solve the First Equation for y

We can solve the first equation for $y$ by isolating $y$ on one side of the equation.

{ y = -2x - 1 \}

Step 2: Substitute the Expression for y into the Second Equation

Now, we can substitute the expression for $y$ into the second equation.

{ 3x - 4(-2x - 1) = -40 \}

Step 3: Simplify the Equation

Next, we can simplify the equation by distributing the $-4$ and combining like terms.

{ 3x + 8x + 4 = -40 \}

{ 11x + 4 = -40 \}

Step 4: Solve for x

Now, we can solve for $x$ by isolating $x$ on one side of the equation.

{ 11x = -44 \}

{ x = -4 \}

Step 5: Find the Value of y

Now that we have the value of $x$ , we can find the value of $y$ by substituting $x$ into one of the original equations.

{ y = -2(-4) - 1 \}

{ y = 8 - 1 \}

{ y = 7 \}

Conclusion

In this article, we solved a system of two linear equations with two variables using the substitution method. We first solved the first equation for $y$ and then substituted that expression into the second equation. We simplified the equation and solved for $x$ , and then found the value of $y$ by substituting $x$ into one of the original equations. The solution to the system of equations is $x = -4$ and $y = 7$ .

Applications

Solving systems of equations has numerous applications in various fields, including:

Physics and Engineering: Systems of equations are used to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
Computer Science: Systems of equations are used in computer graphics, game development, and machine learning.
Economics: Systems of equations are used to model economic systems, including supply and demand, inflation, and unemployment.
Biology: Systems of equations are used to model population dynamics, epidemiology, and ecology.

Tips and Tricks

Here are some tips and tricks to help you solve systems of equations:

Use the substitution method: The substitution method is a powerful tool for solving systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation.
Use the elimination method: The elimination method involves adding or subtracting equations to eliminate one variable and solve for the other variable.
Use graphing: Graphing is a visual method for solving systems of equations. It involves plotting the equations on a coordinate plane and finding the point of intersection.
Check your work: Always check your work by plugging the solution back into the original equations to ensure that it is true.

Practice Problems

Here are some practice problems to help you practice solving systems of equations:

Problem 1: Solve the system of equations:

{ \begin{array}{l} y = 2x + 3 \\ x + 2y = 7 \end{array} \}

Problem 2: Solve the system of equations:

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

Problem 3: Solve the system of equations:

{ \begin{array}{l} y = x - 1 \\ x + 2y = 9 \end{array} \}

Conclusion

Solving systems of equations is a fundamental concept in mathematics, particularly in algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we solved a system of two linear equations with two variables using the substitution method. We first solved the first equation for $y$ and then substituted that expression into the second equation. We simplified the equation and solved for $x$ , and then found the value of $y$ by substituting $x$ into one of the original equations. The solution to the system of equations is $x = -4$ and $y = 7$ .

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Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other through the variables in the equations. In other words, it is a collection of equations that are solved simultaneously to find the values of the variables.

Q: What are the different methods for solving systems of equations?

A: There are several methods for solving systems of equations, including:

Substitution method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
Elimination method: This method involves adding or subtracting equations to eliminate one variable and solve for the other variable.
Graphing method: This method involves plotting the equations on a coordinate plane and finding the point of intersection.
Matrix method: This method involves using matrices to solve the system of equations.

Q: What is the substitution method?

A: The substitution method is a method for solving systems of equations that involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one of the equations is already solved for one variable.

Q: What is the elimination method?

A: The elimination method is a method for solving systems of equations that involves adding or subtracting equations to eliminate one variable and solve for the other variable. This method is useful when the coefficients of the variables in the two equations are additive inverses.

Q: What is the graphing method?

A: The graphing method is a method for solving systems of equations that involves plotting the equations on a coordinate plane and finding the point of intersection. This method is useful when the equations are linear and the system has a unique solution.

Q: What is the matrix method?

A: The matrix method is a method for solving systems of equations that involves using matrices to solve the system of equations. This method is useful when the system of equations is large and complex.

Q: How do I choose the best method for solving a system of equations?

A: The best method for solving a system of equations depends on the specific system and the variables involved. Here are some general guidelines for choosing the best method:

Use the substitution method: If one of the equations is already solved for one variable, use the substitution method.
Use the elimination method: If the coefficients of the variables in the two equations are additive inverses, use the elimination method.
Use the graphing method: If the equations are linear and the system has a unique solution, use the graphing method.
Use the matrix method: If the system of equations is large and complex, use the matrix method.

Q: What are some common mistakes to avoid when solving systems of equations?

A: Here are some common mistakes to avoid when solving systems of equations:

Not checking the solution: Always check the solution by plugging it back into the original equations to ensure that it is true.
Not using the correct method: Choose the best method for solving the system of equations based on the specific system and the variables involved.
Not simplifying the equations: Simplify the equations before solving them to make it easier to find the solution.
Not using the correct order of operations: Use the correct order of operations when solving the equations to avoid errors.

Q: How do I check my work when solving systems of equations?

A: Here are some steps to check your work when solving systems of equations:

Plug the solution back into the original equations: Plug the solution back into the original equations to ensure that it is true.
Check the solution for each equation: Check the solution for each equation to ensure that it is true.
Use a calculator or computer: Use a calculator or computer to check the solution and ensure that it is true.
Check the solution graphically: Check the solution graphically by plotting the equations on a coordinate plane and finding the point of intersection.

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

A: Solving systems of equations has numerous real-world applications, including:

Physics and engineering: Solving systems of equations is used to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
Computer science: Solving systems of equations is used in computer graphics, game development, and machine learning.
Economics: Solving systems of equations is used to model economic systems, including supply and demand, inflation, and unemployment.
Biology: Solving systems of equations is used to model population dynamics, epidemiology, and ecology.

Q: How do I practice solving systems of equations?

A: Here are some ways to practice solving systems of equations:

Practice problems: Practice solving systems of equations using practice problems.
Online resources: Use online resources, such as Khan Academy and Mathway, to practice solving systems of equations.
Textbooks: Use textbooks to practice solving systems of equations.
Workbooks: Use workbooks to practice solving systems of equations.

Q: What are some common systems of equations that I should know?

A: Here are some common systems of equations that you should know:

Linear systems: Linear systems are systems of equations where the variables are linear.
Quadratic systems: Quadratic systems are systems of equations where the variables are quadratic.
Polynomial systems: Polynomial systems are systems of equations where the variables are polynomial.
Rational systems: Rational systems are systems of equations where the variables are rational.

Q: How do I use technology to solve systems of equations?

A: Here are some ways to use technology to solve systems of equations:

Graphing calculators: Use graphing calculators to plot the equations and find the point of intersection.
Computer algebra systems: Use computer algebra systems, such as Mathematica and Maple, to solve systems of equations.
Online calculators: Use online calculators, such as Wolfram Alpha and Symbolab, to solve systems of equations.
Software: Use software, such as MATLAB and Python, to solve systems of equations.