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Chapter 8: Problem 26

Solve each equation. $$\frac{r-2}{2}=\frac{3 r-5}{4}$$

Short Answer

Expert verified
The short answer based on the step-by-step solution is: To solve the equation \(\frac{r-2}{2}=\frac{3r-5}{4}\), find the least common denominator (LCD), which is 4. Multiply both sides of the equation by 4 to eliminate the fractions: \(4(\frac{r-2}{2}) = 4(\frac{3r-5}{4})\). Simplify both sides and apply the distributive property: \(2(r-2) = 3r - 5\). Then, solve for r: \(r = 1\).

Step by step solution

01

Find the least common denominator (LCD)

In this case, the LCD is the least common multiple of 2 and 4, which is 4. #Step 2: Multiply both sides by the LCD#
02

Multiply the equation by 4

To eliminate the fractions, multiply both sides of the equation by 4: \(4(\frac{r-2}{2}) = 4(\frac{3r-5}{4})\) #Step 3: Simplify both sides#
03

Distribute the 4 to both sides

Distribute the 4 to both sides of the equation: \((4)\frac{r-2}{2} = (4)\frac{3r-5}{4}\) The 4s will cancel out on both sides, giving us: \(2(r-2) = 3r-5\) #Step 4: Distribute and combine like terms#
04

Apply the distributive property and combine like terms

Start by distributing the 2 on the left side of the equation: \(2r - 4 = 3r - 5\) Now, we'll move the r terms to one side and the constant terms to the other: #Step 5: Solve for r#
05

Subtract 2r from both sides and add 4 to both sides

Subtract 2r from both sides to eliminate the r terms: \(-4 = r - 5\) Finally, add 5 to both sides to solve for r: \(r = 1\) The solution of the equation is \(r = 1\).

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Key Concepts

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

Least Common Denominator
When working with equations that involve fractions, it's often helpful to eliminate these fractions to simplify the solving process. This is where the concept of the Least Common Denominator (LCD) comes into play.
To clear an equation of fractions, we find the LCD of all the fractional denominators involved.
  • The least common denominator is simply the smallest number that each of the denominators divides into without leaving a remainder.
  • Finding the LCD allows us to multiply each term by this common value, removing all fractions from the equation.
In this exercise, we have denominators of 2 and 4. The smallest number that both 2 and 4 can divide into is 4. Hence, 4 is our LCD. Multiplying each term in the equation by 4 helps us to simplify the problem significantly, eliminating fractions and making the equation easier to work with.
Distributive Property
The distributive property is a fundamental algebraic principle that helps in solving equations, especially when variables are multiplied by a common factor. This property ensures that we can manage terms within parentheses by distributing a multiplying factor over them individually.
The basic form of the distributive property is:
  • If you have an equation like \(a(b + c)\), applying the distributive property allows you to expand it to \(ab + ac\).
  • This is handy because it allows you to rewrite expressions without parentheses, simplifying calculations.
In our equation, when we multiply each side by the least common denominator, we use the distributive property to distribute the factor across terms inside the parentheses. For example, distributing 2 across \(r-2\) gives \(2r - 4\). This helps in simplifying the problem further, setting the stage for combining like terms.
Combining Like Terms
Combining like terms is an essential step in simplifying and solving linear equations. It involves grouping terms with the same variables and exponents together. This consolidates the equation and allows for easier manipulation and resolution.
Here's how to combine like terms within an equation:
  • Identify terms that have the same variable raised to the same power.
  • Sum or subtract these terms by adding or subtracting their coefficients.
In this exercise, after distributing the terms, we had:
  • \(2r - 4 = 3r - 5\).
  • Notice the variable terms \(2r\) and \(3r\), and the constants \(-4\) and \(-5\).
  • By subtracting \(2r\) from both sides, we isolate the variable term on one side.
This process firms up the equation for solving the variable.
Linear Equation Solution Steps
Solving linear equations involves a series of logical steps that simplify the equation to find the value of the unknown variable. Let's break down the steps used in the exercise to find the solution for the given equation:
  • First, identify and apply the least common denominator to clear any fractions, as shown with the denominator of 4 in our example.
  • Next, expand the terms using the distributive property, allowing each term within the parentheses to be accounted for separately.
  • Then, combine like terms on both sides of the equation to simplify it.
  • Finally, manipulate the equation by isolating the variable. This may involve adding or subtracting terms on both sides, such as moving constants to one side and variable terms to the other.
  • The last step is solving for the unknown variable, in this case resulting in \(r = 1\).
These structured steps are crucial for achieving the correct and simplified solution to any linear equation.

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