Select The Correct Answer From Each Drop-down Menu.The Solution Process Is Shown For An Equation. Justify Each Step In The Process With The Appropriate Property.$[ \begin{array}{|c|l|} \hline 2x + 5 = -7(x - 2) & \text{Original Equation}

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will guide you through the process of solving a linear equation using the correct properties and justify each step.

The Original Equation

The original equation is given as:

$$2x + 5 = -7(x - 2)$$

Our goal is to isolate the variable $x$ and find its value.

Step 1: Distribute the Negative 7

To begin solving the equation, we need to distribute the negative 7 to the terms inside the parentheses.

$$2x + 5 = -7x + 14$$

Justification: This step is justified by the Distributive Property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Step 2: Add 7x to Both Sides

Next, we add 7x to both sides of the equation to get all the x terms on one side.

$$2x + 7x + 5 = 14$$

Justification: This step is justified by the Addition Property of Equality, which states that for any real numbers $a$, $b$, and $c$, if $a = b$, then $a + c = b + c$.

Step 3: Combine Like Terms

Now, we combine the like terms on the left-hand side of the equation.

$$9x + 5 = 14$$

Justification: This step is justified by the Commutative Property of Addition, which states that for any real numbers $a$ and $b$, $a + b = b + a$.

Step 4: Subtract 5 from Both Sides

Next, we subtract 5 from both sides of the equation to isolate the term with the variable.

$$9x = 14 - 5$$

Justification: This step is justified by the Subtraction Property of Equality, which states that for any real numbers $a$, $b$, and $c$, if $a = b$, then $a - c = b - c$.

Step 5: Simplify the Right-Hand Side

Now, we simplify the right-hand side of the equation.

$$9x = 9$$

Justification: This step is justified by the Subtraction Property of Equality, which states that for any real numbers $a$, $b$, and $c$, if $a = b$, then $a - c = b - c$.

Step 6: Divide Both Sides by 9

Finally, we divide both sides of the equation by 9 to solve for $x$.

$$x = \frac{9}{9}$$

Justification: This step is justified by the Multiplication Property of Equality, which states that for any real numbers $a$, $b$, and $c$, if $a = b$, then $ac = bc$.

Conclusion

In this article, we have walked through the step-by-step process of solving a linear equation using the correct properties. We have justified each step with the appropriate property, ensuring that the solution is valid and accurate.

Key Takeaways

• Distributive Property: $a(b + c) = ab + ac$
• Addition Property of Equality: if $a = b$, then $a + c = b + c$
• Commutative Property of Addition: $a + b = b + a$
• Subtraction Property of Equality: if $a = b$, then $a - c = b - c$
• Multiplication Property of Equality: if $a = b$, then $ac = bc$

By mastering these properties and following the step-by-step process outlined in this article, you will be able to solve linear equations with confidence and accuracy.

Practice Problems

Try solving the following linear equations using the correct properties:

1. $3x + 2 = 5(x - 1)$
2. $2x - 3 = -4(x + 2)$
3. $x + 5 = 3(x - 2)$

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

1. Distribute any negative signs or coefficients to the terms inside the parentheses.
2. Add or subtract the same value to both sides of the equation to isolate the variable term.
3. Combine like terms on the same side of the equation.
4. Add or subtract the same value to both sides of the equation to isolate the constant term.
5. Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Q: What are the properties used to solve linear equations?

A: The properties used to solve linear equations are:

• Distributive Property: $a(b + c) = ab + ac$
• Addition Property of Equality: if $a = b$, then $a + c = b + c$
• Commutative Property of Addition: $a + b = b + a$
• Subtraction Property of Equality: if $a = b$, then $a - c = b - c$
• Multiplication Property of Equality: if $a = b$, then $ac = bc$

Q: How do I know which property to use?

A: To determine which property to use, identify the operation being performed on the equation. If you are adding or subtracting the same value to both sides, use the Addition Property of Equality or Subtraction Property of Equality. If you are multiplying or dividing both sides by the same value, use the Multiplication Property of Equality. If you are distributing a negative sign or coefficient to the terms inside the parentheses, use the Distributive Property.

Q: What if I have a fraction or decimal coefficient?

A: If you have a fraction or decimal coefficient, you can multiply both sides of the equation by the reciprocal of the coefficient to eliminate the fraction or decimal. For example, if you have the equation $x = \frac{1}{2}$, you can multiply both sides by 2 to get $2x = 1$.

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to show your work and use the properties to justify each step. This will help you understand the solution and ensure that it's accurate.

Q: What if I get stuck or make a mistake?

A: If you get stuck or make a mistake, don't worry! Take a deep breath and go back to the previous step. Check your work and make sure you used the correct properties. If you're still stuck, try re-reading the problem or asking a friend or teacher for help.

Conclusion

Solving linear equations can seem daunting at first, but with practice and patience, you'll become a pro in no time. Remember to use the properties to justify each step and show your work. If you have any more questions or need further clarification, don't hesitate to ask. Happy solving!