Which Method Correctly Solves The Equation Using The Distributive Property?1. { -0.2(x-4)=-1.7$}$ ${ \begin{align*} -0.2(x-4) &= -1.7 \ -0.2x - 4 &= -1.7 \ -0.2x &= 2.3 \ x &= -11.5 \end{align*} }$2.
Introduction
The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions by multiplying each term inside the parentheses with the term outside. In this article, we will explore how to use the distributive property to solve equations, focusing on the correct method to apply this property.
Understanding the Distributive Property
The distributive property states that for any real numbers a, b, and c:
a(b + c) = ab + ac
This property can be applied to both positive and negative numbers, and it is a crucial tool for simplifying expressions and solving equations.
Method 1: Incorrect Application of the Distributive Property
Let's examine the first method, which is incorrect:
{\begin{align*} -0.2(x-4) &= -1.7 \\ -0.2x - 4 &= -1.7 \\ -0.2x &= 2.3 \\ x &= -11.5 \end{align*}}
In this method, the distributive property is applied incorrectly. The correct application of the distributive property would be to multiply -0.2 by each term inside the parentheses, resulting in:
However, the correct method is not shown in this example.
Method 2: Correct Application of the Distributive Property
Now, let's examine the correct method:
{\begin{align*} -0.2(x-4) &= -1.7 \\ -0.2x + 0.8 &= -1.7 \\ -0.2x &= -2.5 \\ x &= 12.5 \end{align*}}
In this method, the distributive property is applied correctly. The -0.2 is multiplied by each term inside the parentheses, resulting in -0.2x + 0.8. Then, the equation is simplified by subtracting 0.8 from both sides, resulting in -0.2x = -2.5. Finally, the value of x is found by dividing both sides by -0.2, resulting in x = 12.5.
Step-by-Step Solution
To solve the equation using the distributive property, follow these steps:
- Apply the distributive property: Multiply the term outside the parentheses by each term inside the parentheses.
- Simplify the equation: Combine like terms and simplify the equation.
- Isolate the variable: Add or subtract the same value from both sides to isolate the variable.
- Solve for the variable: Divide both sides by the coefficient of the variable to find its value.
Conclusion
In conclusion, the correct method to solve the equation using the distributive property is to apply the property correctly, multiply the term outside the parentheses by each term inside the parentheses, and then simplify the equation. By following these steps, you can solve equations using the distributive property with confidence.
Common Mistakes to Avoid
When applying the distributive property, it's essential to avoid common mistakes such as:
- Not applying the distributive property correctly: Failing to multiply the term outside the parentheses by each term inside the parentheses.
- Not simplifying the equation: Failing to combine like terms and simplify the equation.
- Not isolating the variable: Failing to add or subtract the same value from both sides to isolate the variable.
By avoiding these common mistakes, you can ensure that you are applying the distributive property correctly and solving equations with confidence.
Real-World Applications
The distributive property has numerous real-world applications in various fields, including:
- Mathematics: The distributive property is used to simplify expressions and solve equations in algebra, geometry, and calculus.
- Science: The distributive property is used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
- Engineering: The distributive property is used to design and optimize systems, such as electrical circuits and mechanical systems.
Conclusion
Q: What is the distributive property?
A: The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions by multiplying each term inside the parentheses with the term outside. It states that for any real numbers a, b, and c:
a(b + c) = ab + ac
Q: How do I apply the distributive property to solve an equation?
A: To apply the distributive property to solve an equation, follow these steps:
- Apply the distributive property: Multiply the term outside the parentheses by each term inside the parentheses.
- Simplify the equation: Combine like terms and simplify the equation.
- Isolate the variable: Add or subtract the same value from both sides to isolate the variable.
- Solve for the variable: Divide both sides by the coefficient of the variable to find its value.
Q: What is the difference between the distributive property and the commutative property?
A: The distributive property and the commutative property are two separate concepts in algebra. The distributive property states that for any real numbers a, b, and c:
a(b + c) = ab + ac
The commutative property states that for any real numbers a and b:
a + b = b + a
Q: Can I use the distributive property to solve equations with fractions?
A: Yes, you can use the distributive property to solve equations with fractions. When working with fractions, be sure to multiply the numerator and denominator of each fraction by the term outside the parentheses.
Q: How do I handle negative numbers when applying the distributive property?
A: When applying the distributive property to negative numbers, remember that the negative sign is distributed to each term inside the parentheses. For example:
-2(x + 3) = -2x - 6
Q: Can I use the distributive property to solve equations with variables on both sides?
A: Yes, you can use the distributive property to solve equations with variables on both sides. When working with variables on both sides, be sure to isolate the variable on one side of the equation before applying the distributive property.
Q: What are some common mistakes to avoid when applying the distributive property?
A: Some common mistakes to avoid when applying the distributive property include:
- Not applying the distributive property correctly: Failing to multiply the term outside the parentheses by each term inside the parentheses.
- Not simplifying the equation: Failing to combine like terms and simplify the equation.
- Not isolating the variable: Failing to add or subtract the same value from both sides to isolate the variable.
Q: How do I check my work when solving equations using the distributive property?
A: To check your work when solving equations using the distributive property, plug the solution back into the original equation and verify that it is true. If the solution is not true, re-evaluate your work and make any necessary corrections.
Q: Can I use the distributive property to solve equations with exponents?
A: Yes, you can use the distributive property to solve equations with exponents. When working with exponents, be sure to multiply the exponent by the term outside the parentheses.
Conclusion
In conclusion, the distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions by multiplying each term inside the parentheses with the term outside. By understanding how to apply the distributive property, you can solve equations with confidence and apply this concept to real-world problems in various fields.