Problem 18 - Precalculus Mathematics for Calculus

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. $$\frac{z}{5}=\frac{3}{10} z+7$$

Step-by-Step Solution

Answer

The solution to the equation is \( z = -70 \).

Step 1: Clear the Fraction

To eliminate the fractions in the equation, multiply every term by 10, which is the least common denominator. This will simplify the equation:\( 10 \times \frac{z}{5} = 10 \times \frac{3}{10}z + 10 \times 7 \).After multiplying, the equation becomes: \( 2z = 3z + 70 \).

Step 2: Isolate the Variable

To solve for \( z \), first get all terms containing \( z \) on one side by subtracting \( 3z \) from both sides:\( 2z - 3z = 70 \).Simplifying, we get:\( -z = 70 \).

Step 3: Solve for the Variable

Now, to solve for \( z \), multiply both sides by -1 to get:\( z = -70 \).

Key Concepts

Fractions in Linear Equations

Fractions often appear in linear equations, which can make them seem more complex at first glance. However, understanding how to handle them is crucial for solving these types of equations efficiently.
In the equation \( \frac{z}{5} = \frac{3}{10}z + 7 \), we encounter fractions on both sides. These fractions can complicate calculations, but there are straightforward ways to deal with them<:br>

• Identify the least common denominator (LCD), which is the smallest number that all denominators can divide without leaving a remainder.
• Multiply every term by this LCD to clear the fractions. This step simplifies the equation into a standard linear form without fractions.

After clearing the fractions in our example by multiplying by 10, we transform the equation into \( 2z = 3z + 70 \). This simplification makes it much easier to approach subsequent solving steps.

Solving Linear Equations

Linear equations are the foundation of algebra. They can be solved with a series of straightforward steps that can make even fraction-laden equations clear.
The objective is to solve for the variable, which in this case is \( z \). Starting with the equation \( 2z = 3z + 70 \) after clearing the fractions, follow these steps:

• Identify all the terms that contain the variable and those that are constant. In our example, \( 2z \) and \( 3z \) are variable terms, while 70 is a constant.
• Bring all terms containing the variable to one side of the equation. This often involves addition or subtraction of terms.

The process transforms the equation into \( 2z - 3z = 70 \), which simplifies to \( -z = 70 \). Each of these steps takes you closer to isolating the variable, which is crucial for finding its value.

Variable Isolation Techniques

Isolating the variable is the final step in solving linear equations, which enables you to find its value.
For the equation \( -z = 70 \), isolating \( z \) requires getting \( z \) on its own on one side of the equation.
Here's how you can do it step-by-step:

• Multiply both sides of the equation by -1 to change \( -z \) into \( z \). This results in the equation \( z = -70 \).
• Further check your solution by substituting \( z = -70 \) back into the original equation to ensure it satisfies all the initial conditions.

Variable isolation is about using simple arithmetic to manipulate the equation so that the unknown variable is alone and its value can easily be read. Understanding this process is essential for solving not just simple, but also more complex equations later. By mastering these steps, you gain a valuable toolset for tackling a variety of mathematical problems.