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Chapter 2: Problem 189

Solve for the formula for \(y\). Solve the formula \(4 x+3 y=7\) for \(y\)

Short Answer

Expert verified
y = \frac{7 - 4x}{3}

Step by step solution

01

Isolate the term with y

Start by isolating the term that contains y. In the equation \(4x + 3y = 7\), subtract 4x from both sides so that we can move the term with y to the left side.\[3y = 7 - 4x\]
02

Solve for y

Next, we need to solve for y by isolating it. To do this, divide both sides of the equation by 3:\[y = \frac{7 - 4x}{3}\]

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Key Concepts

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

isolation of variables
Understanding how to isolate variables is a fundamental skill when solving equations. The goal is to get the desired variable by itself on one side of the equation.

Consider the equation provided: \[4x + 3y = 7\]

Here, the variable to isolate is \(y\). We start by removing any other terms on the same side as \(y\). To do this:
  • Subtract \(4x\) from both sides:
    • \[3y = 7 - 4x\]
    This step allows us to simplify the equation to focus on \(y\). The isolated variable is now clearly separated, making the problem easier to solve.
linear equations in two variables
Linear equations in two variables involve relationships between two different variables. These are usually written as \(ax + by = c\), where \(a\), \(b\) and \(c\) are constants.

Our equation \[4x + 3y = 7\]
is an example of such an equation. It represents a line when graphed on a coordinate plane. The main goal is to express one variable in terms of the other, making it easier to either solve or graph the equation.

If you are given a point or another equation, it becomes possible to find specific values for both variables by substitution or elimination.
However, for this exercise, we focus on expressing \(y\) in terms of \(x\).
algebraic manipulation
Performing algebraic manipulations involves using operations like addition, subtraction, multiplication, and division to transform an equation into a more desirable form.
In this case:
\[3y = 7 - 4x\]

To isolate \(y\), we need to get rid of the coefficient before it:
  • Divide both sides by 3:
    • \[y = \frac{7 - 4x}{3}\]
    This operation is crucial because it allows us to completely isolate \(y\), showing it in terms of \(x\).

    These steps demonstrate how algebraic manipulation simplifies equations and helps in isolating variables effectively.

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Most popular questions from this chapter

Solve. For each inequality, also graph the solution and write the solution in interval notation. $$ |3 x-4| \geq 2 $$

Solve each inequality, graph the solution on the number line, and write the solution in interval notation. $$ 3 x-2>4 \text { or } 5 x-3 \leq 7 $$

Solve each inequality. Graph the solution and write the solution in interval notation. $$ |x-7| \geq 1 $$

Solve. Five student government officers want to go to the state convention. It will cost them \(\$ 110\) for registration, \(\$ 375\) for transportation and food, and \(\$ 42\) per person for the hotel. There is \(\$ 450\) budgeted for the convention in the student government savings account. They can earn the rest of the money they need by having a car wash. If they charge \(\$ 5\) per car, how many cars must they wash in order to have enough money to pay for the trip?

Solve. $$ 3|x+2|-5=4 $$
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