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Chapter 4: Problem 64

In Exercises \(61-64\), solve for \(y\) in terms of \(x\). $$ 3 x+4 y-5=0 $$

Short Answer

Expert verified
The solution of the given equation for \(y\) in terms of \(x\) is \(y = -\frac{3}{4}x + \frac{5}{4}\).

Step by step solution

01

Isolate \(y\) term.

To start solving the problem for \(y\), the first step is to isolate the term containing \(y\). We can achieve this by moving all other terms to the right side of the equation. Subtract \(3x\) and 5 from both sides to obtain: \[4y = -3x + 5\]
02

Solve for \(y\)

With the \(y\) term isolated, we can now solve for \(y\) by dividing the entire equation by the coefficient of \(y\), which is 4. This gives us the final equation: \[y = -\frac{3}{4}x + \frac{5}{4}\]

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Key Concepts

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

Isolating Variables
One of the most important steps in solving linear equations is isolating the variable you are solving for. In this exercise, our goal is to solve for \( y \) in terms of \( x \). To begin this process, you must focus on getting the \( y \) term alone on one side of the equation.

Here's how:
  • Identify the term involving \( y \) and separate it from other terms. In our equation, \( 3x + 4y - 5 = 0 \), \( 4y \) is the term we need to isolate.
  • Subtract any other terms from both sides of the equation to achieve isolation. Here, we subtract \( 3x \) and \(-5\) to move them to the other side, resulting in \( 4y = -3x + 5 \).
Remember, when isolating the variable, we perform the same operation on both sides of the equation to maintain balance. This keeps the equation equivalent, meaning both sides hold the same value.
Equation Manipulation
Equation manipulation is the art of changing the form of an equation to isolate the desired variable or simplify expressions. It involves using mathematical operations such as addition, subtraction, multiplication, and division judiciously.

In our example, once we've isolated the \( y \) term to \( 4y = -3x + 5 \), the next step is further simplify to solve for \( y \). We do this by performing the following manipulations:
  • Divide the entire equation by the coefficient of \( y \), which is 4. This reduces the \( y \) term to just \( y \), leading us to \( y = -\frac{3}{4}x + \frac{5}{4} \).
Each mathematical operation, like dividing here, must be applied to both sides of the equation. This keeps the equation balanced and correct. The goal is to cleanly express the isolated variable with as few surrounding terms as possible.
Algebraic Expressions
Algebraic expressions form the backbone of solving linear equations. They allow us to express mathematical ideas clearly and concisely. An equation like \( 3x + 4y - 5 = 0 \) consists of various algebraic expressions.

Understanding these components enables us to manipulate and rearrange them as needed:
  • Coefficients, such as 3 and 4 in our example, tell us how many times the variable is used.
  • Constants like \(-5\) are separate from the variable and standalone as numbers within the equation.
Knowing how to work with these elements is key. It unlocks the ability to transform the equation, enabling the effective isolation and manipulation necessary to find the solution. Grasping the structure of algebraic expressions helps in visualizing the relationship between variables, paving the way for a seamless solving process.

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Most popular questions from this chapter

In Exercises 71-76, write an equation of the line that passes through the points. $$ (0,0),(-3,6) $$

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