Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ \left(5 y^{2}-2 y+1\right)-\left(y^{2}-3 y-7\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(4y^2 + y + 8\).

Step 1: Remove the Parentheses

We will remove the parentheses from the expression while also remembering to change the signs of the terms inside the second parentheses since we are subtracting: \[ \begin{aligned} & \left(5 y^{2} -2 y + 1\right) - \left(y^{2} -3 y -7\right) \\ &= 5y^2 - 2y + 1 - y^2 + 3y + 7 \end{aligned} \]

Step 2: Combine Like Terms

Combine the terms with the same variable and exponent: \[ \begin{aligned} &= (5y^2 - y^2) + (-2y + 3y) + (1 + 7) \end{aligned} \]

Step 3: Simplify the Expression

Finally, simplify the expression by combining the terms from the previous step: \[ \begin{aligned} &= 4y^2 + 1y + 8 \end{aligned} \] So, the simplified expression is: \[ \boxed{4y^2 + y + 8}. \]

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols like addition and subtraction. In our exercise, the expression involves two sets of terms within parentheses. Each term in an algebraic expression can have variables (like \( y \)) and coefficients (numerical values attached to a variable, such as 5 in \( 5y^2 \)).

Algebraic expressions are foundational in algebra because they represent general relationships between quantities.

• They often require manipulation, such as adding or subtracting similar expressions.
• They serve as a basis for equations when set equal to something else, usually for solving problems.

Understanding these components helps with organizing and simplifying different expressions.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. "Like terms" are terms that have the same variables raised to the same power. In our exercise, such terms need to be grouped together for simplification.

By combining like terms, we add or subtract their coefficients:

• In \(5y^2 - y^2\), both terms are like terms because they have \(y^2\) as a variable and can be combined into \(4y^2\).
• Similarly, \(-2y + 3y\) are like terms with \(y\) as a variable, resulting in \(1y\).
• Constants like \(1\) and \(7\) are also considered like terms, which combine into \(8\).

This process of combining helps reduce complexity and makes expressions easier to work with. It involves recognising common factors and coefficients, and always ensuring operations are correctly applied.

Simplification Process

Simplification makes an expression easier to read and work with. It means rewriting the expression in the simplest form possible after removing parentheses and combining like terms.

In the given exercise, the simplification process involved a few straightforward steps:

• Removing parentheses required changing signs of the terms within the second pair of parentheses due to subtraction.
• Next was combining like terms by adding or subtracting coefficients of terms with the same variables.
• The final step was stating the expression in the simplest form by noting \(4y^2 + y + 8\), which doesn't contain any further simplifiable terms.

Ultimately, the simplification process streamlines expressions into a neat and concise form, making further calculations more manageable or preparing them for solution in equations.