/*! 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 4 Solve each equation for \(y\). ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 4

Solve each equation for \(y\). \(x+7 y=14\)

Short Answer

Expert verified
The solution for \( y \) is \( y = 2 - \frac{x}{7} \).

Step by step solution

01

Isolate Term with y

Start with the given equation: \( x + 7y = 14 \). We need to solve for \( y \), so our first step is to isolate the term with \( y \) on one side of the equation. Subtract \( x \) from both sides to get the equation in terms of \( y \): \( 7y = 14 - x \).
02

Solve for y

Now that we have \( 7y = 14 - x \), we need to solve for \( y \) by dividing each term by 7. This results in: \( y = \frac{14 - x}{7} \).
03

Simplify the Expression

Simplify the expression \( y = \frac{14 - x}{7} \). This can be broken down into two separate fractions: \( y = \frac{14}{7} - \frac{x}{7} \), which simplifies to \( y = 2 - \frac{x}{7} \).

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Key Concepts

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

Solving Equations
To solve equations for a specific variable means to find the value(s) for that variable which make the equation true. In this exercise, we are tasked with solving the equation for \( y \) in the given equation \( x + 7y = 14 \).

Here are a few core principles of equation solving:
  • Always identify the variable you need to solve for: In our example, it is \( y \).
  • Make sure to perform the same operation on both sides of the equation. This keeps the equation balanced.
  • Step-by-step simplification will lead you to the solution. Start with known quantities and rearrange them to isolate the variable you are interested in.
  • Check your solution by plugging it back into the original equation to ensure it holds true.
Through practice, solving equations becomes a systematic process.
Isolation of Variables
The concept of isolation of variables is crucial when tackling algebraic equations. To isolate a variable means to rearrange the equation so that the variable appears by itself on one side of the equation. For instance, in the equation \( x + 7y = 14 \), we want to isolate \( y \) on one side.

Here's how we go about it:
  • First, identify the terms involving the variable you want to isolate. In the example, that term is \( 7y \).
  • To isolate \( y \), eliminate other terms from its side by performing inverse operations. Subtract \( x \) from both sides to achieve this: \( 7y = 14 - x \).
  • Finally, divide all terms on both sides by the coefficient of \( y \) (which is 7) to solve for it: \( y = \frac{14 - x}{7} \).
Isolation allows us to see the direct relationship between variables, making it easier to interpret their connection.
Simplifying Fractions
Once the variable is isolated, the final step often involves simplifying fractions. Simplifying fractions makes them easier to read and interpret. From the expression \( y = \frac{14 - x}{7} \), we can break this down to simplify it further.

Here鈥檚 the process of simplification:
  • Recognize that each term in the numerator can be divided separately: \( y = \frac{14}{7} - \frac{x}{7} \).
  • Carry out the division for the constant term: \( \frac{14}{7} \) simplifies to 2.
  • Keep the variable fraction as is, or simplify it further if possible, resulting in \( y = 2 - \frac{x}{7} \).
When fractions are simplified, it becomes easier to substitute values, analyze results, and graph the equation. It's always good practice to check if simplification leads to any noteworthy insights in your solution.

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Most popular questions from this chapter

Critical Thinking Explain how you could find an equation of a line if you are only given the coordinates of two points on the line.

Complete each sentence with sometimes, always, or never. If two lines are parallel, then they ____ lie in the same plane.

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Write an equation of the line satisfying the given conditions. parallel to the graph of \(4 x+y=9\), passes through the point at \((0,-5)\)

Given each set of points, determine if \(\overleftrightarrow{J K}\) and \(\overleftrightarrow{L M}\) are parallel, perpendicular, or neither. \(J(-4,11), K(-6,3), L(7,7), M(6,3)\)
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