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Chapter 1: Problem 53

For Problems 41-60, simplify each numerical expression. $$ \frac{-7(10)+6(-9)}{-4} $$

Short Answer

Expert verified
The simplified result is 31.

Step by step solution

01

Apply the Distributive Property

First, apply the distributive property to the terms within the expression \(-7(10)\) and \(6(-9)\), calculating them separately.For \(-7(10)\):\[-7 \times 10 = -70\]For \(6(-9)\):\[6 \times -9 = -54\].
02

Combine the Results

Add the results from the previous step to simplify the numerator of the fraction. So, \(-70 + (-54) = -70 - 54 = -124\).
03

Divide the Numerator by the Denominator

Take the simplified result from Step 2 and divide by the denominator to find the final simplified expression.Divide \(-124\) by \(-4\):\[-124 \div -4 = 31\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in elementary algebra. It allows us to multiply a single term by terms inside a bracket. This is useful for simplifying complex expressions where terms are grouped together.
When you see something like \(-7(10)\), it simply means \(-7\) should be multiplied by \(10\). This calculation gives us \(-70\). Similarly, for \(6(-9)\), you multiply \(6\) by \(-9\) to get \(-54\).
  • Instead of doing everything at once, break it down. Multiply each term in the brackets separately.
  • Afterward, combine the results. This simplifies your expression step by step.
The distributive property is handy because it ensures that you do not miss out on any term while multiplying, especially when dealing with negative numbers.
Simplifying Expressions
Simplifying expressions involves combining like terms to make a mathematical expression as simple as possible. It is all about making an expression easier to handle.
In our example with \(-70 + (-54)\), you combine these into a single number. Treat addition of negatives like subtraction, which here becomes \(-70 - 54\).
  • Identify all the like terms. These are terms that have the same variable parts if any.
  • Combine these terms together to get a simplified result, which is often a single number or a simpler expression.
Once calculations are simplified to the smallest form, the last step is much easier. Simplification reduces the risk of mistakes in later steps.
Numerical Expressions
A numerical expression contains numbers and operations like addition, subtraction, multiplication, or division. These are math problems that, when solved, give a number.
Our task was to simplify \(\frac{-7(10)+6(-9)}{-4}\). Start by solving inside the brackets: Multiply, add or subtract as needed. The goal is to have one number on the top and one on the bottom.
  • Perform operations like multiplication and addition in the correct order. Follow the order of operations: Parentheses, Exponents, Multiplication, and Division, Addition and Subtraction.
  • Then, divide the simplified numerator by the denominator. Remember, dividing two negatives results in a positive.
The final result here is \(31\). Breaking down each step ensures you do not miss any part of the expression. Remembering the order of operations is key with numerical expressions.

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Most popular questions from this chapter

Simplify each numerical expression. Don't forget to take advantage of the properties if they can be used to simplify the computation. $$17+(-18)-19-14+13-17$$

Explain the difference between a numerical expression and an algebraic expression.

Simplify each algebraic expression by combining similar terms. $$6 x y-x-13 x y+4 x$$

State whether the expressions in each problem are equivalent and explain why or why not. \(a+5(x+y)\) and \((a+5) x+y\)

Find the value of \(\frac{h\left(b_{1}+b_{2}\right)}{2}\) for each set of values for the variables \(h, b_{1}\), and \(b_{2}\). (Subscripts are used to indicate that \(b_{1}\) and \(b_{2}\) are different variables.) \(h=18, b_{1}=6\), and \(b_{2}=11\)
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