Solve Each Linear System By The Substitution Method.10. $\[ \begin{array}{l} y = -3x + 5 \\ 5x - 4y = -3 \end{array} \\]11. (Note: The Original Problem For Number 11 Is Unclear. Here Is A Potential Version For A Linear

Introduction

Linear systems are a set of two or more linear equations that involve the same variables. Solving linear systems is an essential concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. In this article, we will focus on solving linear systems using the substitution method.

What is the Substitution Method?

The substitution method is a technique used to solve linear systems by substituting one equation into another. This method involves solving one equation for one variable and then substituting that expression into the other equation. The substitution method is particularly useful when one of the equations is already solved for one variable.

Step-by-Step Guide to Solving Linear Systems by Substitution Method

To solve a linear system using the substitution method, follow these steps:

1. Identify the equations: Write down the two linear equations that make up the system.
2. Solve one equation for one variable: Choose one of the equations and solve it for one variable. This will give you an expression that you can substitute into the other equation.
3. Substitute the expression into the other equation: Take the expression you obtained in step 2 and substitute it into the other equation.
4. Solve for the remaining variable: Simplify the resulting equation and solve for the remaining variable.
5. Find the value of the other variable: Once you have found the value of one variable, substitute it back into one of the original equations to find the value of the other variable.

Example 1: Solving a Linear System with Two Equations

Let's consider the following linear system:

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

To solve this system using the substitution method, we will follow the steps outlined above.

Step 1: Identify the equations

The two linear equations that make up the system are:

$$y = -3x + 5$$

$$5x - 4y = -3$$

Step 2: Solve one equation for one variable

We will solve the first equation for y:

$$y = -3x + 5$$

Step 3: Substitute the expression into the other equation

Now, we will substitute the expression for y into the second equation:

$$5x - 4(-3x + 5) = -3$$

Step 4: Solve for the remaining variable

Simplifying the resulting equation, we get:

$$5x + 12x - 20 = -3$$

Combine like terms:

$$17x - 20 = -3$$

Add 20 to both sides:

$$17x = 17$$

Divide both sides by 17:

$$x = 1$$

Step 5: Find the value of the other variable

Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y. We will use the first equation:

$$y = -3x + 5$$

Substitute x = 1:

$$y = -3(1) + 5$$

Simplify:

$$y = 2$$

Therefore, the solution to the linear system is x = 1 and y = 2.

Example 2: Solving a Linear System with Two Equations (Note: The original problem for number 11 is unclear. Here is a potential version for a linear system)

Let's consider the following linear system:

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

To solve this system using the substitution method, we will follow the steps outlined above.

Step 1: Identify the equations

The two linear equations that make up the system are:

$$2x + 3y = 7$$

$$x - 2y = -3$$

Step 2: Solve one equation for one variable

We will solve the second equation for x:

$$x = -3 + 2y$$

Step 3: Substitute the expression into the other equation

Now, we will substitute the expression for x into the first equation:

$$2(-3 + 2y) + 3y = 7$$

Step 4: Solve for the remaining variable

Simplifying the resulting equation, we get:

$$-6 + 4y + 3y = 7$$

Combine like terms:

$$7y = 13$$

Divide both sides by 7:

$$y = \frac{13}{7}$$

Step 5: Find the value of the other variable

Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x. We will use the second equation:

$$x - 2y = -3$$

Substitute y = 13/7:

$$x - 2(\frac{13}{7}) = -3$$

Simplify:

$$x - \frac{26}{7} = -3$$

Add 26/7 to both sides:

$$x = -3 + \frac{26}{7}$$

Simplify:

$$x = \frac{-21 + 26}{7}$$

Simplify further:

$$x = \frac{5}{7}$$

Therefore, the solution to the linear system is x = 5/7 and y = 13/7.

Conclusion

Solving linear systems using the substitution method is a powerful technique that can be used to solve systems of two or more linear equations. By following the steps outlined above, you can solve linear systems and find the values of the variables. Remember to identify the equations, solve one equation for one variable, substitute the expression into the other equation, solve for the remaining variable, and find the value of the other variable. With practice, you will become proficient in solving linear systems using the substitution method.

Tips and Tricks

• Make sure to identify the equations correctly and solve one equation for one variable.
• Substitute the expression into the other equation carefully to avoid errors.
• Simplify the resulting equation and solve for the remaining variable.
• Find the value of the other variable by substituting the value of one variable back into one of the original equations.

Common Mistakes to Avoid

• Failing to identify the equations correctly.
• Solving the wrong equation for one variable.
• Substituting the expression into the other equation incorrectly.
• Failing to simplify the resulting equation and solve for the remaining variable.

Introduction

Solving linear systems using the substitution method is a powerful technique that can be used to solve systems of two or more linear equations. In this article, we will provide a Q&A section to help you better understand the concept and address any questions you may have.

Q: What is the substitution method?

A: The substitution method is a technique used to solve linear systems by substituting one equation into another. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Q: How do I choose which equation to solve for one variable?

A: You can choose either equation to solve for one variable. However, it is often easier to solve the equation that has the variable you want to eliminate.

Q: What if I have a system with three or more equations?

A: If you have a system with three or more equations, you can use the substitution method to solve for two variables and then use the third equation to find the value of the third variable.

Q: Can I use the substitution method to solve a system with fractions?

A: Yes, you can use the substitution method to solve a system with fractions. However, you will need to simplify the resulting equation and solve for the remaining variable.

Q: What if I get a quadratic equation when I substitute the expression into the other equation?

A: If you get a quadratic equation when you substitute the expression into the other equation, you can use the quadratic formula to solve for the remaining variable.

Q: Can I use the substitution method to solve a system with decimals?

A: Yes, you can use the substitution method to solve a system with decimals. However, you will need to simplify the resulting equation and solve for the remaining variable.

Q: What if I make a mistake when I substitute the expression into the other equation?

A: If you make a mistake when you substitute the expression into the other equation, you will need to go back and correct the mistake. This may involve re-solving the equation for one variable and re-substituting the expression into the other equation.

Q: Can I use the substitution method to solve a system with absolute values?

A: Yes, you can use the substitution method to solve a system with absolute values. However, you will need to simplify the resulting equation and solve for the remaining variable.

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

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

Q: Can I use the substitution method to solve a system with infinitely many solutions?

A: Yes, you can use the substitution method to solve a system with infinitely many solutions. This means that the two equations are dependent and there are infinitely many values of the variables that satisfy both equations.

Conclusion

Solving linear systems using the substitution method is a powerful technique that can be used to solve systems of two or more linear equations. By following the steps outlined above and addressing common questions, you can become proficient in solving linear systems using the substitution method.

Tips and Tricks

• Make sure to identify the equations correctly and solve one equation for one variable.
• Substitute the expression into the other equation carefully to avoid errors.
• Simplify the resulting equation and solve for the remaining variable.
• Find the value of the other variable by substituting the value of one variable back into one of the original equations.

Common Mistakes to Avoid

• Failing to identify the equations correctly.
• Solving the wrong equation for one variable.
• Substituting the expression into the other equation incorrectly.
• Failing to simplify the resulting equation and solve for the remaining variable.

By following the steps outlined above and avoiding common mistakes, you can solve linear systems using the substitution method with confidence.