Chapter 1.5, Problem 6LC

To determine

To write an algebraic expression for the given verbal expression and then simplify.

Answer to Problem 6LC

  $5\cdot \left(a^2+b\right)-4\cdot \left(a^2+b\right)=a^2+b$

Explanation of Solution

Given information:

7 times the sum of a squared and b minus 4 times the sum of a squared and b .

Concept Used:

• Like terms are the terms that contain the same variable with similar exponents on it. Constants are like terms as well.
Distributive property:
• Left Distributive property: $a\left(b\pm c\right)=a\cdot b\pm a\cdot c$
• Right Distributive property: $\left(a\pm b\right)c=a\cdot c\pm b\cdot c$

Rules of Addition/ Subtraction:

• Two numbers with similar sign always get added and the resulting number will carry the similar sign.
• Two numbers with opposite signs always get subtracted and the resulting number will carry the sign of larger number.

Rules of Multiplication/ Division:

• The product/quotient of two similar sign numbers is always positive.
• The product/quotient of two numbers with opposite signs is always negative.

Calculation:

In order to write an algebraic expression for the given verbal expression, observe that the 7 times the sum of a squared and b , that is, $5\cdot \left(a^2+b\right)$ , minus 4 times the sum of a squared and b , that is, $-4\cdot \left(a^2+b\right)$ . So, in algebraic expression, it can be written as

  $\begin{array}{c}5\cdot \left(a^2+b\right)\text{ minus }4\cdot \left(a^2+b\right)\\ =5\cdot \left(a^2+b\right)-4\cdot \left(a^2+b\right)\end{array}$

Now, to simplify further combine the like terms as shown below,

  $\begin{array}{c}5\cdot \left(a^2+b\right)-4\cdot \left(a^2+b\right)=5\cdot a^2+5\cdot b-4\cdot a^2-4\cdot b\\ =5a^2+5b-4a^2-4b\\ =5a^2-4a^2+5b-4b\\ =\left(5-4\right)a^2+\left(5-4\right)b\\ =a^2+b\end{array}$

Thus, the algebraic expression for the given verbal expression is $5\cdot \left(a^2+b\right)-4\cdot \left(a^2+b\right)$ and simplify to $a^2+b$ .