Question

Explain the mistake and work the problem correctly. $$\begin{array}{l} \left(10 y^{2}-5 y\right)-\left(y^{2}+3 y\right) \\ \quad=\left(10 y^{2}-5 y\right)+\left(-y^{2}-3 y\right) \\ \quad=9 y^{2}+8 y \end{array}$$

Short answer

Answer: The correct solution is $9y^2 - 8y$.

Step-by-step solution

Write down the given expressions.

Write down the original expressions that we want to subtract: $$\begin{array}{l} \left(10 y^{2} - 5y\right) - \left(y^{2} + 3y\right) \end{array}$$

Apply the distributive property to the subtraction.

Apply the distributive property to the negative sign across the second expression, without changing the signs of its terms: $$\begin{array}{l} \left(10 y^{2} - 5y\right) + \left(-y^{2} - 3y\right) \end{array}$$

Combine like terms.

We can now combine the like terms in the expression: $$\begin{array}{l} \left(10 y^{2} - y^{2}\right) + \left(-5y - 3y\right) \end{array}$$

Perform the subtraction of like terms.

Subtract the respective coefficients of the like terms, and rewrite the simplified expression: $$\begin{array}{l} (9y^{2}) + (-8y) = 9y^{2} - 8y \end{array}$$ Hence, the corrected solution for the given problem is: $$\begin{array}{l} 9y^2 - 8y \end{array}$$

Key concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us handle expressions, especially when dealing with subtraction. When you have an expression like \( (a - b) \), it's crucial to distribute the negative sign to both terms inside the parenthesis. This changing of signs can often be overlooked but is necessary for accurate calculations.

In our example, we started with:

• \( (10y^2 - 5y) - (y^2 + 3y) \)

We then applied the distributive property to transform the subtraction into addition by distributing the negative sign across the second expression:

• \( (10y^2 - 5y) + (-y^2 - 3y) \)

The negative sign affected each term inside the parenthesis, ensuring our future steps in the process would remain accurate.

Combining Like Terms

Combining like terms is about simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. This process allows us to condense polynomial expressions into a more manageable form.

In our exercise, we looked at the expression:

• \( (10y^2 - y^2) + (-5y - 3y) \)

Here, we are identifying terms that can be combined:

• \( 10y^2 \) and \( -y^2 \)
• \( -5y \) and \( -3y \)

By adding these pairs of like terms together, we end up with:

• \( 9y^2 \)
• \( -8y \)

This simplification helps to see the expression clearly.

Algebraic Expressions

Algebraic expressions are made up of variables, numbers, and operations (addition, subtraction, multiplication, division). Understanding how to manipulate these expressions is fundamental in algebra. They can represent real-world situations and various mathematical relationships.

In our problem, the algebraic expression given is already a bit complex:

• \( 10y^2 - 5y \) and \( y^2 + 3y \)

Each part of this expression can be broken down into more manageable components. The expression \( 10y^2 - 5y \) includes terms that can be used to represent a variety of scenarios or patterns.

Knowing how to work with algebraic expressions means being able to understand and manipulate these components. It includes using operations like the distributive property or combining like terms, as we've done in the exercise, to simplify and solve problems accurately.