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Chapter 1: Problem 80

$$ 4 x+5 y=20 $$

Short Answer

Expert verified
The solution to the linear equation is \(x = 5 - \frac{5y}{4}\).

Step by step solution

01

Rewrite the Equation to Isolate x

The equation can be rewritten as \(4x = 20 - 5y\), by subtracting \(5y\) from both sides. The result is a new version of the equation with \(x\) on one side and everything else on the other.
02

Solve for x

Divide both sides of the equation from Step 1 by 4. This will isolate \(x\) on one side of the equation. The equation then becomes \(x = \frac{20 - 5y}{4}\).
03

Simplify the Right Side of the Equation

We can simplify the right side of the equation by distributing the division by 4 to each term inside the brackets, this gives us \(x = \frac{20}{4} - \frac{5y}{4}\). So \(x = 5 - \frac{5y}{4}\).

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation is the magical tool that allows us to rearrange equations. It's like solving a puzzle by shifting the pieces around, but with numbers and variables instead. In the given equation, \( 4x + 5y = 20 \), the aim is to manipulate the terms to reach a simpler form. We start by moving terms around to either side of the equation. This involves basic operations such as:
  • Addition or Subtraction: Moving terms from one side of the equation to the other by adding or subtracting.
  • Multiplication or Division: Changing the size of terms to set up for solving for a specific variable.
In Step 1, we subtracted \( 5y \) from both sides to start isolating \( x \), resulting in the manipulated equation \( 4x = 20 - 5y \). The ability to rearrange equations by algebraic manipulation is a foundational skill in algebra.
Solving for a Variable
Solving for a variable means getting that variable by itself on one side of the equation. It focuses on finding the unknown quantity. Once we've manipulated the equation into a preferred form, we can isolate the variable. With the rearranged equation \( 4x = 20 - 5y \), the goal is to have \( x \) alone on one side, which we accomplish by:
  • Dividing: To eliminate coefficients attached to the variable. Here, we divide both sides by 4.
This operation results in \( x = \frac{20 - 5y}{4} \). Now, \( x \), the variable we were solving for, is isolated, making this step crucial in understanding the relationship between \( x \) and \( y \). It's important to carry out these operations accurately to avoid errors in solving.
Equation Simplification
Simplifying an equation is like decluttering a room. It makes solutions clearer and easier to handle by breaking them down into the simplest possible parts. For the expression \( x = \frac{20 - 5y}{4} \), further simplification can be done by distributing:
  • Dividing each term within the parenthesis by 4.
This leads us to \( x = \frac{20}{4} - \frac{5y}{4} \). Simplified, it is \( x = 5 - \frac{5y}{4} \). Simplification helps in visualizing the solution more clearly, and also aids in checking that the steps taken are correct. The key is to apply straightforward arithmetic to reach the most reduced form of the equation, thus making further calculations or interpretations much easier.

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Most popular questions from this chapter

Consumer Awareness A shipping company's charge for sending an overnight package from New York to Atlanta is \(\$ 12.80\) for the first pound and \(\$ 2.50\) for each additional pound or fraction thereof. Use the greatest integer function to create a model for the charge \(C\) for overnight delivery of a package weighing \(x\) pounds. Use a graphing utility to graph the function, and discuss its continuity.

Find the limit (if it exists). \(\lim _{\Delta x \rightarrow 0} \frac{2(x+\Delta x)-2 x}{\Delta x}\)

Find the limit. \(\lim _{x \rightarrow 2} x^{2}\)

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. \(\lim _{x \rightarrow 0^{-}} \frac{\frac{1}{x+4}-\frac{1}{4}}{x}\) \(\begin{array}{|c|c|c|c|c|c|}\hline x & {-0.5} & {-0.1} & {-0.01} & {-0.001} & {0} \\ \hline f(x) & {} & {} & {} & {?} \\ \hline\end{array}\)

Find the limit. \(\lim _{x \rightarrow 1} \frac{\frac{1}{x+4}-\frac{1}{4}}{x}\)
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