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Chapter 2: Problem 92

Solve each formula for the specified variable. \(y=m x+b\) for \(x\)

Short Answer

Expert verified
Solve for x by isolating it: \(x = \frac{y - b}{m}\).

Step by step solution

01

- Isolate the term with the variable

Start by isolating the term that contains the variable we need to solve for. Subtract the constant term from both sides of the equation: y = mx + b y - b = mx
02

- Solve for the variable

Divide both sides by the coefficient of the variable to solve for x: y - b = mx \(x = \frac{y - b}{m}\)

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Key Concepts

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

Isolating Variables
Isolating variables is a key step in solving linear equations. It means rearranging the equation so that the variable you are solving for is alone on one side of the equation.

For example, in the equation \(y = mx + b\), you need to isolate \(x\). First, subtract \(b\) from both sides to get \(y - b = mx\). Now, \(x\) is still not isolated, but you have simplified the equation. This concept is crucial because isolated variables make it easier to understand how changes in one variable affect the others.
Rearranging Formulas
Rearranging formulas involves moving terms around in an equation so that you can solve for a specific variable. This usually involves a combination of addition, subtraction, multiplication, and division.

Let's take our example \(y = mx + b\). After isolating the term containing \(x\), we have \(y - b = mx\). To solve for \(x\), we need to get \(x\) by itself. This means dividing every term by the coefficient \(m\), which gives us \(x = \frac{y - b}{m}\). Notice how each operation is carefully applied to maintain the equality of the equation.
Step-by-Step Solution
A step-by-step solution helps break down complex problems into manageable parts. By following each step carefully, you make it easier to avoid mistakes and understand the logic behind the solution.

Here's how to solve \(y = mx + b\) for \(x\) in steps:
  • Step 1: Isolate the variable term - Start by subtracting \(b\) from both sides: \(y - b = mx\)
  • Step 2: Solve for the variable - Divide both sides by \(m\) to isolate \(x\): \(x = \frac{y - b}{m}\)

Each step is purposeful and leads you closer to the final solution. Be sure to double-check your work as you go along to catch any possible errors early.

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Most popular questions from this chapter

The world's largest taco contained approximately \(1 \mathrm{~kg}\) of onion for every \(6.6 \mathrm{~kg}\) of grilled steak. The total weight of these two ingredients was \(617.6 \mathrm{~kg}\). To the nearest tenth of a kilogram, how many kilograms of each ingredient were used to make the taco? (Data from Guinness World Records.

Solve each formula for the specified variable. \(C=\frac{5}{9}(F-32)\) for \(F\)

Solve each formula for the specified variable. \(\mathscr{A}=\frac{1}{2} h(b+B)\) for \(b\)

Solve each formula for the specified variable. \(y=m x+b\) for \(m\)

Beyonc茅 and Guns \(\mathrm{N}\) ' Roses had the two top-grossing concert tours for 2016 , together generating \(\$ 300\) million in ticket sales. Guns \(\mathrm{N}\) ' Roses took in \(\$ 38\) million less than Beyonc茅. How much did each tour generate? (Data from Pollstar.)
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