Requires knowledge of matrices

Oct 13, 2025 by windailys.com 6 views

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Introduction

Solving a system of equations without graphing is a crucial skill in mathematics, particularly in algebra. It involves using various methods to find the solution to a system of linear equations. In this article, we will explore one such method, which is the substitution method. We will use the given system of equations to demonstrate this method.

The System of Equations

The given system of equations is:

{ \begin{cases} 2y = 4 - x \\ 4x + 3y = 5 \end{cases} \}

Step 1: Isolate y in the First Equation

To begin solving the system of equations, we need to isolate one of the variables. Let's start by isolating y in the first equation.

{ 2y = 4 - x \}

We can isolate y by dividing both sides of the equation by 2.

{ y = \frac{4 - x}{2} \}

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

Now that we have isolated y in the first equation, we can substitute this expression into the second equation.

{ 4x + 3y = 5 \}

Substituting the expression for y, we get:

{ 4x + 3\left(\frac{4 - x}{2}\right) = 5 \}

Step 3: Simplify the Equation

To simplify the equation, we can start by multiplying the 3 by the fraction.

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

This simplifies to:

{ 4x + \frac{12 - 3x}{2} = 5 \}

Step 4: Multiply Both Sides by 2

To eliminate the fraction, we can multiply both sides of the equation by 2.

{ 2(4x) + (12 - 3x) = 2(5) \}

This simplifies to:

{ 8x + 12 - 3x = 10 \}

Step 5: Combine Like Terms

Now that we have eliminated the fraction, we can combine like terms.

{ 8x - 3x + 12 = 10 \}

This simplifies to:

{ 5x + 12 = 10 \}

Step 6: Subtract 12 from Both Sides

To isolate the term with the variable, we can subtract 12 from both sides of the equation.

{ 5x + 12 - 12 = 10 - 12 \}

This simplifies to:

{ 5x = -2 \}

Step 7: Divide Both Sides by 5

Finally, we can divide both sides of the equation by 5 to solve for x.

{ \frac{5x}{5} = \frac{-2}{5} \}

This simplifies to:

{ x = -\frac{2}{5} \}

Step 8: Substitute the Value of x into the First Equation

Now that we have found the value of x, we can substitute it into the first equation to find the value of y.

{ 2y = 4 - x \}

Substituting the value of x, we get:

{ 2y = 4 - \left(-\frac{2}{5}\right) \}

Step 9: Simplify the Equation

To simplify the equation, we can start by adding the fraction to 4.

{ 2y = 4 + \frac{2}{5} \}

This simplifies to:

{ 2y = \frac{20}{5} + \frac{2}{5} \}

Step 10: Combine Like Terms

Now that we have added the fractions, we can combine like terms.

{ 2y = \frac{22}{5} \}

Step 11: Divide Both Sides by 2

Finally, we can divide both sides of the equation by 2 to solve for y.

{ \frac{2y}{2} = \frac{\frac{22}{5}}{2} \}

This simplifies to:

{ y = \frac{11}{5} \}

Conclusion

In this article, we have demonstrated how to solve a system of equations without graphing using the substitution method. We started by isolating y in the first equation and then substituted this expression into the second equation. We then simplified the equation, combined like terms, and solved for x and y. The final solution is x = -\frac{2}{5} and y = \frac{11}{5}.

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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 a common variable or variables. In this article, we solved a system of two linear equations with two variables.

Q: Why do we need to solve systems of equations?

A: Solving systems of equations is essential in various fields, including mathematics, physics, engineering, and economics. It helps us to find the solution to a problem that involves multiple variables and equations.

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 substituting the expression for one variable from one equation into the other equation.
Elimination method: This method involves eliminating one variable by adding or subtracting the equations.
Graphing method: This method involves graphing 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 are the advantages and disadvantages of the substitution method?

A: The advantages of the substitution method include:

Easy to understand: This method is easy to understand and implement, especially for simple systems of equations.
Flexible: This method can be used to solve systems of equations with any number of variables.

The disadvantages of the substitution method include:

Time-consuming: This method can be time-consuming for complex systems of equations.
Error-prone: This method can be error-prone if not done correctly.

Q: What are the advantages and disadvantages of the elimination method?

A: The advantages of the elimination method include:

Fast: This method is fast and efficient, especially for systems of equations with two variables.
Accurate: This method is accurate and reliable, as long as the equations are linear.

The disadvantages of the elimination method include:

Difficult to understand: This method can be difficult to understand and implement, especially for complex systems of equations.
Limited: This method is limited to systems of equations with two variables.

Q: What are the advantages and disadvantages of the graphing method?

A: The advantages of the graphing method include:

Visual: This method is visual and can help to understand the relationship between the variables.
Easy to understand: This method is easy to understand and implement, especially for simple systems of equations.

The disadvantages of the graphing method include:

Time-consuming: This method can be time-consuming for complex systems of equations.
Error-prone: This method can be error-prone if not done correctly.

Q: What are the advantages and disadvantages of the matrix method?

A: The advantages of the matrix method include:

Fast: This method is fast and efficient, especially for complex systems of equations.
Accurate: This method is accurate and reliable, as long as the matrices are correct.

The disadvantages of the matrix method include:

Difficult to understand: This method can be difficult to understand and implement, especially for complex systems of equations.
Requires knowledge of matrices: This method requires knowledge of matrices and linear algebra.

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 complexity of the system and the variables involved. If the system is simple and has two variables, the substitution or elimination method may be the best choice. If the system is complex and has multiple variables, the matrix method may be the best choice.

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

A: Some common mistakes to avoid when solving systems of equations include:

Not checking the solution: Make sure to check the solution to ensure that it satisfies both equations.
Not using the correct method: Choose the correct method for the system of equations, based on the complexity and variables involved.
Not following the steps: Follow the steps carefully and accurately to avoid errors.

Q: How do I practice solving systems of equations?

A: To practice solving systems of equations, try the following:

Start with simple systems: Begin with simple systems of equations and gradually move to more complex ones.
Use online resources: Use online resources, such as worksheets and practice problems, to practice solving systems of equations.
Work with a partner: Work with a partner or tutor to practice solving systems of equations and get feedback on your work.

By following these tips and practicing regularly, you can become proficient in solving systems of equations and apply this skill to real-world problems.