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Chapter 1: Problem 14

Solve each equation. $$\frac{7}{4}+\frac{1}{5} x-\frac{3}{2}=\frac{4}{5} x$$

Short Answer

Expert verified
x = \frac{5}{12}

Step by step solution

01

Simplify the equation

Rewrite the equation to isolate the terms involving variable on one side and constant terms on the other side. $$ \frac{7}{4} + \frac{1}{5}x - \frac{3}{2} = \frac{4}{5}x $$ Subtract \frac{1}{5}x from both sides: $$ \frac{7}{4} - \frac{3}{2} = \frac{4}{5}x - \frac{1}{5}x $$
02

Combine like terms

First, simplify the left side by combining the fractions: Find a common denominator for \frac{7}{4} and -\frac{3}{2}: $$ \frac{7}{4} = \frac{7}{4} $$ $$ \frac{3}{2} = \frac{6}{4} $$ Now subtract: $$ \frac{7}{4} - \frac{6}{4} = \frac{1}{4} $$ Combine the terms on the right side: $$ \frac{4}{5}x - \frac{1}{5}x = \frac{3}{5}x $$ The equation becomes: $$ \frac{1}{4} = \frac{3}{5}x $$
03

Solve for x

To isolate x, multiply both sides by the reciprocal of \frac{3}{5}: $$ \frac{1}{4} \times \frac{5}{3} = x $$ Simplify: $$ x = \frac{5}{12} $$

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Key Concepts

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

Combine Like Terms
Combining like terms means simplifying an equation by adding or subtracting terms that have the same variables. For example, in the equation \(\frac{7}{4} + \frac{1}{5}x - \frac{3}{2} = \frac{4}{5}x\), we combine all constants (numbers without variables) together and all terms with the variable \(x\) together.
Adding or subtracting like terms helps to simplify the equation to make it easier to solve. Let's take \(\frac{4}{5}x - \frac{1}{5}x = \frac{3}{5}x\). Here, both terms are 'like' since they contain the variable \(x\) with a common denominator, \(5\). Hence, they can be directly combined.
Always:
  • Identify the terms with variables.
  • Identify the constant terms.
  • Add/subtract terms with like terms only.
Common Denominators
Using common denominators makes adding or subtracting fractions easier. By having the same denominator, you can combine fractions directly. In the exercise, we have fractions such as \(\frac{7}{4}\) and \(-\frac{3}{2}\). These fractions need to have the same denominator before they can be added or subtracted.
Here's how to find a common denominator:
  • Identify the denominators of your fractions.
  • Find the Least Common Denominator (LCD).
  • Rewrite each fraction with the LCD as the new denominator.
For \(\frac{7}{4}\) and \(-\frac{3}{2}\), the denominators are 4 and 2. The LCD for 4 and 2 is 4. So, rewrite \(-\frac{3}{2}\) as \(-\frac{6}{4}\). Now you can easily subtract \(\frac{7}{4} - \frac{6}{4} = \frac{1}{4}\).
Isolating Variables
To isolate the variable in an equation means to get the variable on one side of the equation by itself. This is usually our goal because it allows us to find the value of the variable. In the equation \(\frac{7}{4} + \frac{1}{5}x - \frac{3}{2} = \frac{4}{5}x\), we want to isolate \(x\).
How do you isolate \(x\)?
  • Move terms that do not contain \(x\) to the other side of the equation.
  • Combine like terms if necessary.
  • Perform the same operation on both sides of the equation.
For example, subtract \(\frac{1}{5}x\) from both sides: \(\frac{7}{4} - \frac{3}{2} = \frac{4}{5}x - \frac{1}{5}x\), simplifying left side and combining right side to get \(\frac{1}{4} = \frac{3}{5}x\).
Reciprocals in Equations
A reciprocal is simply the flipped version of a fraction. For example, the reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\). Using reciprocals helps to isolate variables when they are in a fraction. In the equation \(\frac{1}{4} = \frac{3}{5}x\), we multiply both sides by the reciprocal of \(\frac{3}{5}\) to solve for \(x\).
Steps to solve with reciprocals:
  • Identify the fraction containing the variable.
  • Find its reciprocal.
  • Multiply both sides of the equation by this reciprocal.
In our example, multiplying both sides by \(\frac{5}{3}\), we get \(\frac{1}{4} \times \frac{5}{3} = x\), simplifying gives \(\frac{5}{12} = x\). Therefore, \(x = \frac{5}{12}\).

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Most popular questions from this chapter

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