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Chapter 25: Problem 20

If \(x-4 y=1\), what is the value of \(x\) ? (1) \(x=4 y+1\) (2) \(y=x / 2+1\)

Short Answer

Expert verified
The value of \(x\) is \(4y + 1\).

Step by step solution

01

Analyze the given equation

The problem starts with the linear equation: \(x - 4y = 1\). The goal is to find the value of \(x\).
02

Solve for x using the equation

We rearrange the given equation \(x - 4y = 1\) to solve for \(x\). By adding \(4y\) to both sides, we obtain \(x = 4y + 1\). This is also option (1) provided in the problem, confirming our rearrangement is correct.
03

Explore if additional information is needed

Option (2) \(y = \frac{x}{2} + 1\) is provided but not immediately necessary since we've already expressed \(x\) in terms of \(y\) from the initial equation.
04

Decision based on given information

Given the rearrangement in Step 2 matches the first provided option, and there are no additional variables or parameters, we can determine \(x\) based on that alone without any further calculations or checks.

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Key Concepts

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

Understanding Linear Equations
Linear equations are fundamental in algebra. They typically have a form like \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In the given exercise, the equation \(x - 4y = 1\) is a simple linear equation.
Linear equations represent straight lines when graphed on a coordinate plane, and the solutions are any point \((x, y)\) that satisfies the equation. In this case, our task is slightly different because we are solving for one variable, \(x\), in terms of another, \(y\).
It is essential to recognize linear equations as they form the backbone of many mathematical models. By understanding their structure, you can easily manipulate and solve them.
Steps to Solving Equations
Solving equations involves finding the value of unknown variables that make the equation true. In our problem, the goal is to determine \(x\). Here's a structured approach to solving such equations:
  • Isolate the variable of interest (in this case, \(x\)) on one side of the equation.
  • Perform the same mathematical operations on both sides to maintain equality.
  • Simplify each side until the variable is isolated.
From the equation \(x - 4y = 1\), we isolate \(x\) by adding \(4y\) to both sides, resulting in \(x = 4y + 1\). This method ensures we perform reciprocal operations, preserving the equation's balance.
Remember, each step in solving an equation should logically follow from the previous step, leading to a clear solution.
Variable Manipulation Techniques
Variable manipulation is a crucial skill to solve equations effectively. It involves rearranging equations to isolate and solve for desired variables. In our exercise, we manipulate the original equation to express \(x\) in terms of \(y\).
To do this, we adjust both sides of the equation consistently. Start with \(x - 4y = 1\). By adding \(4y\) to both sides, we get \(x = 4y + 1\). Here, we've effectively moved \(4y\) from the left to the right side, in turn simplifying the expression for \(x\).
When manipulating variables, ensure each transformation stays valid under mathematical rules. This step keeps the equation balanced and the solution accurate.

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Most popular questions from this chapter

In \(\triangle P Q R\), if \(P Q=x, Q R=x+3\), and \(P R=y\), which one of the three angles of \(\triangle P Q R\) is the greatest? (1) \(x>y\) (2) \(x=y+3\)

Is \((x-2)(x-3)>0\) ? (1) \(x<2\) (2) \(x<3\)

In a factory, there are workers, executives, and clerks. How many employees are there in the factory? (1) \(41 \%\) of the employees are workers, 460 are executives, and the remaining 720 employees are clerks. (2) 460 of the employees are executives and account for \(23 \%\) of the employees in the factory.

The price of a car was \(m\) dollars. It then depreciated by \(x \%\). Later, it appreciated by \(y \%\) to \(n\) dollars. If there are no other changes in the price, is \(m>n\) ? (1) \(y=\frac{x}{1-\frac{x}{100}}\) (2) \(y=\frac{3 x}{4}\)

A researcher collected \(n\) readings from a scientific experiment. Is the standard deviation of the measurements less than 5 ? (1) The mean of the positive difference between each measurement and the mean is 5 . (2) The variance of the set of \(n\) measurements is 25 .
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