Solve By Substitution:$\[ \begin{array}{l} y = 4x + 10 \\ y = 6 \end{array} \\]

Introduction

In algebra, solving equations by substitution is a powerful technique used to find the solution to a system of equations. This method involves substituting the expression for one variable from one equation into the other equation, allowing us to solve for the remaining variable. In this article, we will explore how to solve equations by substitution using a step-by-step approach, with a focus on solving linear equations.

What is Substitution?

Substitution is a method of solving equations by replacing one variable with an expression that is equal to it. This expression is obtained from one of the equations in the system. By substituting this expression into the other equation, we can solve for the remaining variable.

Step-by-Step Guide to Solving Equations by Substitution

Step 1: Identify the Equations

The first step in solving equations by substitution is to identify the two equations in the system. In this case, we have two linear equations:

{ \begin{array}{l} y = 4x + 10 \\ y = 6 \end{array} \}

Step 2: Choose One Equation to Substitute

Next, we need to choose one equation to substitute into the other equation. Let's choose the first equation, $y = 4x + 10$. We will substitute this expression into the second equation, $y = 6$.

Step 3: Substitute the Expression

Now, we substitute the expression $y = 4x + 10$ into the second equation, $y = 6$. This gives us:

$${ 4x + 10 = 6 }$$

Step 4: Solve for the Variable

The next step is to solve for the variable $x$. To do this, we need to isolate $x$ on one side of the equation. We can do this by subtracting 10 from both sides of the equation:

$${ 4x = -4 }$$

Step 5: Solve for the Variable (continued)

Now, we need to solve for $x$ by dividing both sides of the equation by 4:

$${ x = -1 }$$

Step 6: Verify the Solution

Finally, we need to verify that our solution is correct. We can do this by substituting $x = -1$ back into one of the original equations. Let's use the first equation, $y = 4x + 10$. Substituting $x = -1$ gives us:

$${ y = 4(-1) + 10 }$$

$${ y = -4 + 10 }$$

$${ y = 6 }$$

This confirms that our solution is correct.

Conclusion

Solving equations by substitution is a powerful technique used to find the solution to a system of equations. By following the step-by-step guide outlined in this article, you can learn how to solve equations by substitution using a simple and effective approach. Remember to identify the equations, choose one equation to substitute, substitute the expression, solve for the variable, and verify the solution.

Real-World Applications

Solving equations by substitution has many real-world applications. For example, in physics, you may need to solve a system of equations to determine the position and velocity of an object. In engineering, you may need to solve a system of equations to determine the stress and strain on a material. In economics, you may need to solve a system of equations to determine the demand and supply of a product.

Common Mistakes to Avoid

When solving equations by substitution, there are several common mistakes to avoid. These include:

• Not identifying the equations correctly
• Not choosing the correct equation to substitute
• Not substituting the expression correctly
• Not solving for the variable correctly
• Not verifying the solution

By avoiding these common mistakes, you can ensure that your solution is correct and accurate.

Tips and Tricks

Here are some tips and tricks to help you solve equations by substitution:

• Make sure to identify the equations correctly before starting to solve.
• Choose the correct equation to substitute based on the variables involved.
• Substitute the expression carefully to avoid errors.
• Solve for the variable step-by-step to ensure accuracy.
• Verify the solution by substituting the value back into one of the original equations.

Q: What is substitution in algebra?

A: Substitution is a method of solving equations by replacing one variable with an expression that is equal to it. This expression is obtained from one of the equations in the system.

Q: How do I choose which equation to substitute?

A: You should choose the equation that has the variable you want to solve for. If you want to solve for x, choose the equation that has x in it.

Q: What if I have two equations with the same variable?

A: In this case, you can choose either equation to substitute. The result will be the same.

Q: Can I substitute an expression that has more than one variable?

A: Yes, you can substitute an expression that has more than one variable. However, you will need to solve for the variable you want to substitute first.

Q: What if I get a quadratic equation after substituting?

A: If you get a quadratic equation after substituting, you can solve it using the quadratic formula or factoring.

Q: Can I use substitution to solve systems of nonlinear equations?

A: Yes, you can use substitution to solve systems of nonlinear equations. However, the process may be more complex and require additional steps.

Q: What are some common mistakes to avoid when using substitution?

A: Some common mistakes to avoid when using substitution include:

• Not identifying the equations correctly
• Not choosing the correct equation to substitute
• Not substituting the expression correctly
• Not solving for the variable correctly
• Not verifying the solution

Q: How do I verify my solution?

A: To verify your solution, substitute the value back into one of the original equations. If the equation is true, then your solution is correct.

Q: Can I use substitution to solve systems of equations with fractions?

A: Yes, you can use substitution to solve systems of equations with fractions. However, you will need to simplify the fractions before substituting.

Q: What if I have a system of equations with variables on both sides?

A: If you have a system of equations with variables on both sides, you can use substitution to solve for one variable and then substitute that value into the other equation.

Q: Can I use substitution to solve systems of equations with absolute values?

A: Yes, you can use substitution to solve systems of equations with absolute values. However, you will need to consider both the positive and negative cases.

Conclusion

Solving equations by substitution is a powerful technique used to find the solution to a system of equations. By following the step-by-step guide and avoiding common mistakes, you can become proficient in solving equations by substitution and apply this technique to a wide range of problems.

Additional Resources

For more information on solving equations by substitution, check out the following resources:

• Khan Academy: Solving Systems of Equations by Substitution
• Mathway: Solving Systems of Equations by Substitution
• Wolfram Alpha: Solving Systems of Equations by Substitution

By practicing and applying the concepts learned in this article, you can become proficient in solving equations by substitution and tackle a wide range of problems with confidence.