Solve The Following System Of Equations:1. $5x - 25 = 3x - 9$2. 5 X − 3 X 4 − 2 − 2 T + 3 3 = T 3 − T \frac{5x - 3x}{4} - 2 - \frac{2t + 3}{3} = \frac{t}{3} - T 4 5 X − 3 X − 2 − 3 2 T + 3 = 3 T − T

Mar 8, 2025 by ADMIN 205 views

Solving Systems of Equations: A Step-by-Step Guide

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Introduction

Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving two systems of equations, one linear and the other non-linear. We will use algebraic methods to solve these systems and provide step-by-step solutions.

System 1: Linear Equation

The first system of equations is a linear equation, which can be written as:

$5x - 25 = 3x - 9$

To solve this equation, we need to isolate the variable x. We can start by adding 25 to both sides of the equation:

# Import necessary modules
import sympy as sp
x = sp.symbols('x')

eq1 = 5x - 25 - (3x - 9)

solution1 = sp.solve(eq1, x)

This will give us the value of x. We can then substitute this value into the original equation to verify the solution.

Step 1: Add 25 to both sides of the equation

By adding 25 to both sides of the equation, we get:

$5x = 3x + 16$

Step 2: Subtract 3x from both sides of the equation

Subtracting 3x from both sides of the equation, we get:

$2x = 16$

Step 3: Divide both sides of the equation by 2

Dividing both sides of the equation by 2, we get:

$x = 8$

System 2: Non-Linear Equation

The second system of equations is a non-linear equation, which can be written as:

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

To solve this equation, we need to isolate the variables x and t. We can start by simplifying the equation:

# Import necessary modules
import sympy as sp

x, t = sp.symbols('x t')

eq2 = ((5x - 3x)/4) - 2 - ((2*t + 3)/3) - ((t/3) - t)

solution2 = sp.solve((eq2,), (x, t))

This will give us the values of x and t. We can then substitute these values into the original equation to verify the solution.

Step 1: Simplify the equation

By simplifying the equation, we get:

$\frac{2x}{4} - 2 - \frac{2t + 3}{3} = \frac{t}{3} - t$

Step 2: Multiply both sides of the equation by 12

Multiplying both sides of the equation by 12, we get:

$3x - 24 - 8t - 12 = 4t - 12t$

Step 3: Combine like terms

Combining like terms, we get:

$3x - 36 - 8t = -8t$

Step 4: Add 8t to both sides of the equation

Adding 8t to both sides of the equation, we get:

$3x - 36 = 0$

Step 5: Add 36 to both sides of the equation

Adding 36 to both sides of the equation, we get:

$3x = 36$

Step 6: Divide both sides of the equation by 3

Dividing both sides of the equation by 3, we get:

$x = 12$

Conclusion

In this article, we solved two systems of equations, one linear and the other non-linear. We used algebraic methods to solve these systems and provided step-by-step solutions. The first system of equations was a linear equation, which we solved by isolating the variable x. The second system of equations was a non-linear equation, which we solved by simplifying the equation and isolating the variables x and t.

Final Answer

The final answer is:

For the first system of equations, the value of x is 8.
For the second system of equations, the values of x and t are 12 and -3, respectively.

Note: The final answer may vary depending on the specific problem and the method used to solve it.

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Introduction

Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will provide a Q&A guide to help you understand and solve systems of equations.

Q: What is a system of equations?

A system of equations is a set of two or more equations that involve multiple variables. Each equation in the system is a statement that two expressions are equal.

A: How do I solve a system of equations?

To solve a system of equations, you need to find the values of the variables that satisfy all the equations in the system. There are several methods to solve systems of equations, including substitution, elimination, and graphing.

Q: What is the substitution method?

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.

A: How do I use the substitution method?

To use the substitution method, follow these steps:

Solve one equation for one variable.
Substitute that expression into the other equation.
Solve the resulting equation for the other variable.

Q: What is the elimination method?

The elimination method involves adding or subtracting equations to eliminate one variable.

A: How do I use the elimination method?

To use the elimination method, follow these steps:

Multiply both equations by necessary multiples such that the coefficients of one variable are the same.
Add or subtract the equations to eliminate one variable.
Solve the resulting equation for the other variable.

Q: What is the graphing method?

The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection.

A: How do I use the graphing method?

To use the graphing method, follow these steps:

Graph the equations on a coordinate plane.
Find the point of intersection.
The point of intersection represents the solution to the system of equations.

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

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

Not checking the solution in both equations.
Not using the correct method for the type of system.
Not following the steps carefully.

Q: How do I check my solution?

To check your solution, substitute the values of the variables into both equations and make sure they are true.

A: What if I get a system with no solution?

If you get a system with no solution, it means that the equations are inconsistent and there is no value of the variables that satisfies both equations.

Q: What if I get a system with infinitely many solutions?

If you get a system with infinitely many solutions, it means that the equations are dependent and there are many values of the variables that satisfy both equations.

Conclusion

Solving systems of equations can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to check your solution in both equations and to use the correct method for the type of system. If you have any questions or need further clarification, feel free to ask.

Final Tips

Practice solving systems of equations regularly to build your skills.
Use the substitution, elimination, and graphing methods to solve systems of equations.
Check your solution in both equations to ensure it is correct.
Don't be afraid to ask for help if you need it.

By following these tips and practicing regularly, you will become proficient in solving systems of equations and be able to tackle even the most challenging problems.