/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 94 Solve for \(y: 7 x+3 y=18\)... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 94

Solve for \(y: 7 x+3 y=18\)

Short Answer

Expert verified
\(y = 6 - \frac{7}{3}x\)

Step by step solution

01

Isolate the y-term

First, move the term with \(x\) (which is \(7x\)) to the other side of the equation to isolate \(y\). Achieve this by subtracting \(7x\) from both sides of the equation to maintain equality. This gives: \(3y = 18 - 7x\).
02

Get y on its own

Next, to completely isolate \(y\), divide every term in the equation by 3. This gives: \(y = (18 - 7x) / 3\).
03

Simplify

Lastly, simplify the equation by dividing each term inside the parenthesis by 3. The simplified equation is: \(y = 6 - \frac{7x}{3}\).

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Key Concepts

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

Solving Equations
Solving linear equations is a fundamental skill in math that involves finding the value of unknown variables.
To solve the equation provided in the exercise, the main goal is to determine the value of "y" such that the equation holds true. Often, this requires multiple steps that involve making the equation simpler and more workable.
In this exercise, each step focuses on maintaining balance鈥攚hat you do to one side must be done to the other. This ensures that the equation remains valid through the entire process.
Understanding these steps helps you develop skills for solving more complex equations and strength your logical reasoning.
Isolating Variables
Isolating a variable means rewriting the equation so that the variable of interest stands alone on one side of the equation.
To isolate "y" in the equation given, the first step involves removing any other terms or coefficients that are on the same side as "y".
Here's how to effectively isolate a variable like "y":
  • Identify the term you want to isolate (in this case, 3y).
  • Move other terms to the opposite side of the equation by adding, subtracting, multiplying, or dividing.
Once the desired variable is alone on one side, you can easily find its value.
Algebraic Manipulation
Algebraic manipulation involves using arithmetic operations to rearrange equations or expressions. This skill is crucial for isolating variables and simplifying expressions.
For instance, the equation initially was: \[7x + 3y = 18\]. By subtracting \(7x\) from both sides, we performed algebraic manipulation.
This operation changed our equation to \[3y = 18 - 7x\], moving us closer to finding "y".
The next manipulation involved dividing each term by 3.
The result led us to: \[y = 6 - \frac{7x}{3}\].
  • This reflects division of each separated term by 3, making it easier to read.
  • Simplification through these steps results in an equation where the relationship between variables is clear.
These methods showcase how algebraic manipulation assists in reshaping equations to reveal solutions clearly.

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