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Chapter 1: Problem 28

Evaluate each following expression for \(x=2, y=5,\) and \(z=3 .\) \(\frac{2 z+6}{3}\)

Short Answer

Expert verified
The expression evaluates to 4.

Step by step solution

01

Substitute Values

Start by substituting the given values for \(x\), \(y\), and \(z\). In this expression, only \(z\) is used, so substitute \(z = 3\) into the expression. The expression becomes \(\frac{2(3) + 6}{3}\).
02

Simplify the Numerator

Calculate the expression in the numerator. Multiply \(2\) by \(3\) to get \(6\), then add \(6\) to get \(12\). The expression is now \(\frac{12}{3}\).
03

Simplify the Fraction

Finally, divide the numerator by the denominator. Divide \(12\) by \(3\) to get \(4\).

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Key Concepts

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

Substitution in Expressions
Substitution is a powerful tool in math that allows us to replace variables with specific values. This process makes evaluating expressions much more manageable. Think of substitution as filling in the blanks. When you're given values for variables like in our exercise, you simply plug them in. For example, if you have an expression with a variable \(z\), and you're told \(z = 3\), you substitute \(3\) for \(z\). This changes your variable expression into something you can directly compute.
  • Identifying variables: Look for letters that stand for unknown numbers.
  • Replacing with values: Put in the given number wherever the variable appears.
  • Compute: Once the "blank" is filled in, you calculate the expression.
Simplifying Expressions
Simplifying expressions is where math starts to look elegant. Once you've substituted your values into an expression, it's time to simplify. Simplifying means taking a complicated expression and expressing it in the simplest terms possible. When you simplify, you're basically performing all the possible operations, like addition, subtraction, multiplication, or division, to end up with a single number or a simpler expression.
Some tips to effectively simplify include:
  • Follow the order of operations (PEMDAS/BODMAS).
  • Combine like terms: Add or subtract coefficients of terms with the same variables.
  • Convert complex fractions or decimals into simpler forms.

In our example, you multiply and add inside the numerator before dividing, ensuring each step is clear and straightforward.
Understanding Fractions
Working with fractions seems tricky at first, but with a little practice, they become a vital part of problem-solving. A fraction is simply a division problem between a numerator and a denominator. The numerator is the top part of the fraction, representing the number of parts you have. The denominator is the bottom, showing into how many parts the whole is divided.
  • Simplification: If both the numerator and denominator can be divided by the same number, this should be done to simplify the fraction.
  • Multiplication and division: Perform operations on the fractions as you would with any other numbers, but remember to apply them to both the numerator and the denominator.
  • Final Result: Check if the fraction gives a whole number when divided. If it does, you can express the fraction as this whole number.

In our exercise, the fraction \(\frac{12}{3}\) simplifies directly to \(4\), illustrating how fractions can resolve into clean and easy-to-understand numbers.

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Most popular questions from this chapter

Write each phrase as a variable expression. Use \(x\) to represent "a number." The sum of a number and 7

Perform each indicated operation. Watch the operation symbol. $$228 \div 24$$

Perform each indicated operation. $$ \begin{array}{r} 134 \\ \times \quad 16 \\ \hline \end{array} $$

Check to see whether each number written in standard form matches the number written in words. If not, correct the number in words.How large is a quadrillion? To get an idea, if one quadrillion dollars were divided among the population of the United States, we would each receive over 9 million dollars. Look up "quadrillion" (in the American system) and write this number in standard form.

Write each whole number in expanded form. $$20,215$$
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