Problem 53 - Elementary Algebra

Question

For the following problems, solve the literal equations for the indicated variable. When directed, find the value of that variable for the given values of the other variables. Solve \(I=\frac{E}{R}\) for \(R\). Find the value of \(R\) when \(I=0.005\) and \(E=0.0035\).

Step-by-Step Solution

Answer

Answer: 0.7

Step 1: Solve the given equation for R

To solve the equation \(I = \frac{E}{R}\) for \(R\), we will need to isolate the variable \(R\) on one side of the equation. In this case, we will multiply both sides of the equation by \(R\) and then divide by \(I\).

Step 2: Multiply both sides of the equation by R

Multiply both sides of the equation by \(R\): \(R \cdot I = \frac{E}{R} \cdot R\) \(RI = E\)

Step 3: Divide both sides of the equation by I

Now divide both sides of the equation by \(I\): \(\frac{RI}{I} = \frac{E}{I}\) \(R = \frac{E}{I}\) Now we have the equation solved for variable \(R\).

Step 4: Plug in the given values for I and E into the equation

Now, we will plug in the given values for \(I\) and \(E\): \(I = 0.005\) and \(E = 0.0035\). \(R = \frac{0.0035}{0.005}\)

Step 5: Calculate the value of R

Now we just need to divide the numbers to find the value of \(R\): \(R = \frac{0.0035}{0.005} = 0.7\) The value of \(R\) when \(I = 0.005\) and \(E = 0.0035\) is \(0.7\).

Key Concepts

Variable Isolation in Literal Equations

When working with literal equations, our primary goal is to solve for one specific variable, called isolation. It involves rearranging the equation so one variable stands alone on one side. This process allows us to express that chosen variable in terms of the othervariables present in the equation.

In the example equation, \(I = \frac{E}{R}\), isolating \(R\) involves a sequence of steps:

• Multiply both sides of the equation by \(R\) to get rid of the denominator \(R\). This results in \(RI = E\).
• Next, divide both sides by \(I\) to solve for \(R\). This gives us \(R = \frac{E}{I}\).

Through these algebraic manipulations, we effectively isolate the variable \(R\) so it can be expressed in terms of the other two variables \(E\) and \(I\). This operation is essential in making the variable you need the subject of the formula.

Understanding the Substitution Method

Once the variable is isolated, the substitution method can be utilized. This method involves substituting known values into the equation, to find the solution.Understanding how substitution works is incredibly useful for determining the value of an isolated variable for specific instances. After isolating the variable \(R\) in \(R = \frac{E}{I}\), we substitute the given values:

• Set \(I = 0.005\) and \(E = 0.0035\).
• Substituting into the equation gives: \(R = \frac{0.0035}{0.005}\).

This method is particularly helpful in real-world scenarios where you're given specific measurements or quantities and need to find unknowns quickly using pre-arranged formulas.

Substitution not only allows for checking if the isolated formula works but also provides a systematic approach to problem-solving in math.

Fraction Division Defined

Fraction division plays a key role in handling calculations when solving equations like our original literal equation. It involves dividing one fraction by another and is a critical step when substituting specific values.To divide fractions, you'll follow these simple steps:

• Write the division of the fractions as a multiplication by the reciprocal of the divisor. For \(R = \frac{0.0035}{0.005}\), it becomes \(R = 0.0035 \times \frac{1}{0.005}\).
• Multiply the numerators together and the denominators together to complete the calculation.

Dividing \(0.0035\) by \(0.005\) results in \(0.7\).

Understanding fraction division is essential when dealing with literal equations, as it simplifies the calculation process, ensuring you arrive at the correct result efficiently.