Problem 16 Solve each proportion. See Examp... [FREE SOLUTION]

Chapter 3: Problem 16

Solve each proportion. See Example 2 . $$\frac{5 d-1}{2}=\frac{6 d+3}{3}$$

Short Answer

Expert verified

First, write the given proportion as $\frac{5d-1}{2}=\frac{6d+3}{3}$. Cross-multiply to get $(5d-1) \times 3 = (6d+3) \times 2$. Simplify to get $15d - 3 = 12d + 6$. Move all variables to one side and constants to the other side, resulting in $15d - 12d = 6 + 3$. Simplify and solve for d: $3d = 9$, $d = \frac{9}{3}$, and $d = 3$. The solution is $d = 3$.

Step by step solution

Write the Proportion

First, let's write the given proportion as it is: $\frac{5d-1}{2}=\frac{6d+3}{3}$

Cross-Multiply

Now, we will cross-multiply to eliminate the denominators and make it easier to work with the equation. $(5d-1) \times 3 = (6d+3) \times 2$

Simplify the Equation

Let's simplify the equation by performing the multiplications: $15d - 3 = 12d + 6$

Move All Variables to One Side and Constants to the Other

To solve for the variable d, we need to bring all the variables to one side and the constants to the other side: $15d - 12d = 6 + 3$

Simplify and Solve for d

Now we can simplify and solve for d: $3d = 9$ $d = \frac{9}{3}$ $d = 3$ So the solution to the proportion is $d = 3$.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Cross-Multiplication

Cross-multiplication is a powerful technique used to solve equations involving proportions. It helps to remove fractions by equating the products of the means and extremes of the proportion.
This method is particularly useful when dealing with equations like $ \frac{a}{b} = \frac{c}{d} $.
Here, you can cross-multiply to get $ a \times d = b \times c $. This eliminates the fractions and allows you to work with a simpler equation.
In our example, we have the proportion $ \frac{5d-1}{2} = \frac{6d+3}{3} $. By cross-multiplying, we obtain:

$(5d-1) \times 3 = (6d+3) \times 2$

This action transforms the proportion into a straightforward algebraic equation that is much easier to handle.

Algebraic Equations

Algebraic equations consist of mathematical expressions set equal to each other. They are foundational to many areas of mathematics and enable us to represent and solve various problems.
In the context of proportions, we turn each side into an equation, often involving variables like $d$ in our example.
After cross-multiplying the initial proportion, we convert it into the algebraic equation:

$15d - 3 = 12d + 6$

This equation contains variables and constants, which we need to manipulate as we solve for $d$.
Understanding how algebraic equations work is essential for solving problems using mathematical reasoning.

Variable Isolation

Variable isolation is the process of moving all instances of the variable to one side of the equation. This is an important step in solving for unknowns.
Once you have your algebraic equation, you need to start rearranging terms. Our goal is to have the variable, $d$, all alone on one side.
From the equation $15d - 3 = 12d + 6$, follow these steps to isolate $d$:

First, subtract $12d$ from both sides: $15d - 12d = 6 + 3$

This leaves us with $3d = 9$, which simplifies our task of solving for $d$. Variable isolation is crucial because it helps to focus on solving the primary unknown in a consistent manner.

Simplifying Equations

Simplifying equations involves reducing an expression to its simplest form to make it easier to solve. This requires combining like terms and performing basic arithmetic operations.
In our exercise, once the equation $3d = 9$ is reached, simplifying involves straightforward division:

Divide both sides by 3 to solve for $d$.
This gives us $d = \frac{9}{3} = 3$.

Simplifying equations helps remove complexity, making it possible to reach a solution quickly and correctly.
It's a valuable skill to develop, as it is applicable across a broad range of mathematical problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Solve each proportion. See Example 2 . $$\frac{5 d-1}{2}=\frac{6 d+3}{3}$$

Short Answer

Step by step solution

Write the Proportion

Cross-Multiply

Simplify the Equation

Move All Variables to One Side and Constants to the Other

Simplify and Solve for d

Key Concepts

Cross-Multiplication

Algebraic Equations

Variable Isolation

Simplifying Equations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Applied Mathematics

Logic and Functions

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.