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Chapter 4: Problem 82

$$\text { Evaluate each expression if } x=\frac{3}{4} \text { and } y=-\frac{4}{7}$$ $$4 x+y$$

Short Answer

Expert verified
The evaluated expression is \(\frac{17}{7}\).

Step by step solution

01

Substitute Values into the Expression

Start by substituting the given values of \(x\) and \(y\) into the expression \(4x + y\). So the expression becomes \(4 \left( \frac{3}{4} \right) + \left( -\frac{4}{7} \right)\).
02

Simplify the Expression

Calculate \(4 \times \frac{3}{4}\) to get \(3\), because multiplying by \(4\) cancels out the \(\frac{4}{4}\), then add \(-\frac{4}{7}\) to \(3\).
03

Combine the Terms

Add the integer \(3\) to the fraction \(-\frac{4}{7}\). To do this, convert \(3\) to a fraction with a denominator of 7: \(3 = \frac{21}{7}\). Thus, \(\frac{21}{7} + \left(-\frac{4}{7}\right) = \frac{21 - 4}{7}\).
04

Calculate the Result

Simplify \(\frac{21 - 4}{7}\) to get \(\frac{17}{7}\). This is the final result of the expression.

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Key Concepts

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

Substitution in Algebra
In algebra, substituting values for variables is an essential operation. Consider an expression with variables, like \(4x + y\). If values are provided for \(x\) and \(y\), such as \(x = \frac{3}{4}\) and \(y = -\frac{4}{7}\), you simply replace the variables.

Substitution involves plugging these numbers in place of the variables in the expression. For the expression given, you would replace \(x\) with \(\frac{3}{4}\) and \(y\) with \(-\frac{4}{7}\). This transforms the expression into a numerical problem:
  • Replace the \(x\): \(4 \times \frac{3}{4}\)
  • Replace the \(y\): add \(-\frac{4}{7}\)
By following substitution steps carefully, you can convert an algebraic expression into a more straightforward arithmetic calculation.
Fraction Operations
Working with fractions requires understanding their operations, especially multiplication and addition. When you multiply a whole number by a fraction, as in \(4 \times \frac{3}{4}\), you treat the whole number as a fraction with a denominator of 1. Here, the multiplication cancels the 4 in the denominator:
  • \(4 \times \frac{3}{4} = \frac{4 \times 3}{4} = 3\)
This simplifies to 3, as the 4s cancel each other out.

The next step involves adding fractions. To add \(3\) (which is \(\frac{21}{7}\)) to \(-\frac{4}{7}\), you need a common denominator. Since \(21\) is the product of \(3 \times 7\), you can rewrite \(3\) as \(\frac{21}{7}\), giving:
  • \(\frac{21}{7} + \left(-\frac{4}{7}\right) = \frac{21 - 4}{7} = \frac{17}{7}\)
Always ensure common denominators when working with addition or subtraction of fractions.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. After substituting and calculating initial operations, you're often left with a fraction or number that can be further simplified.

In the context of the given exercise, you transform \(4x + y\) into \(\frac{17}{7}\). Here are a few simplification steps to follow:
  • Start by calculating any multiplications or divisions, which in our case is \(4 \times \frac{3}{4} = 3\).
  • Next, combine terms by finding a common denominator, crucial for adding or subtracting fractions. For \(3\) and \(-\frac{4}{7}\), convert \(3\) to \(\frac{21}{7}\).
  • Simplify the result: \(\frac{21}{7} - \frac{4}{7} = \frac{17}{7}\).
By methodically simplifying expressions, you ensure your final result is in its clearest and most easily interpretable form.

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Most popular questions from this chapter

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