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Chapter 2: Problem 91

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

Short Answer

Expert verified
\(m = \frac{y - b}{x}\)

Step by step solution

01

- Isolate the Term Involving the Variable

Start with the equation \( y = m x + b \). To solve for \( m \), you need to move all other terms to the other side of the equation. First, subtract \( b \) from both sides.\[ y - b = m x \]
02

- Solve for the Variable

To isolate \( m \), divide both sides of the equation by \( x \).\[ \frac{y - b}{x} = m \]

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Key Concepts

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

Solving for a Variable
In algebra, solving for a variable means finding the value of the specific variable in an equation. When the equation is provided, the goal is to manipulate it until the desired variable is isolated on one side. For instance, if we have the equation \(y = mx + b\) and we need to solve for \(m\), we would rearrange the equation:

1. Start with the given equation: \(y = mx + b\)
2. Subtract \(b\) from both sides: \(y - b = mx\)
3. Finally, divide both sides by \(x\) to isolate \(m\): \(\frac{y - b}{x} = m\).

These steps of manipulating the equation ensure that we can find the value of the variable we are interested in.
Isolating Terms
Isolating terms is a crucial part of solving for a variable. This means rearranging the equation so that the term involving the variable of interest stands alone on one side of the equation. In the given example, we isolated \(m\) by moving other terms to the opposite side:

1. Given the equation \(y = mx + b\), we needed to isolate the term involving \(m\)
2. This was achieved by subtracting the constant term \(b\) from both sides to get \(y - b = mx\)
3. Next, to isolate \(m\), we divided both sides by \(x\): \(\frac{y - b}{x} = m\).

By following these steps, we systematically make sure each operation brings us closer to having the variable on its own.
Linear Equations
Linear equations are expressions where each term is either a constant or product of a constant and a single variable. They typically have one or more variables and are of the first degree; meaning the variable is not raised to any power other than one. An example of a linear equation is \(y = mx + b\).

Linear equations come in various forms:
  • Slope-Intercept Form (e.g., \(y = mx + b\))
  • Standard Form (e.g., \(Ax + By = C\))

    • Solving linear equations may involve different steps depending on the context, but the goal is always to isolate the variable. Understanding these foundational concepts helps us tackle more complex algebra problems.

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    Most popular questions from this chapter

    Solve each problem. Anna saved \(\$ 1950,\) which was \(65 \%\) of the amount she needed for a used car. What was the total amount she needed for the car?

    Simple interest 1 in dollars is calculated using the formula \(\boldsymbol{I}=\boldsymbol{p r t}, \quad\) where p represents the principal, or amount, in dollars that is invested or borrowed, r represents the annual interest rate, expressed as a decimal, and \(t\) represents time, in years. In each exercise, find the value of the remaining variable in the simple interest formula. $$ I=\$ 33, r=2 \%, t=3 \mathrm{yr} $$

    Choose the letter of the correct response. Which equation does not require the use of the multiplication property of equality? A. \(3 x-5 x=6\) B. \(-\frac{1}{4} x=12\) C. \(5 x-4 x=7\) D. \(\frac{x}{3}=-2\)

    Solve each problem. (Assume that all items are equally priced.) If 6 gal of premium unleaded gasoline costs \(\$ 22.56\), how much would it cost to completely fill a 15-gal tank?

    \(x+\frac{2}{3}=-\frac{1}{6}\)
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