Problem 22 - Algebra and Trigonometry

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: Eliminate Fractions

The equation given is \( \frac{z}{5} = \frac{3}{10}z + 7 \). In order to make the equation easier to solve, we will eliminate the fractions by finding a common denominator, which is 10. Multiply every term in the equation by 10 to get rid of the fractions: \( 10 \times \frac{z}{5} = 10 \times \frac{3}{10}z + 10 \times 7 \). This simplifies to: \( 2z = 3z + 70 \).

Step 2: Simplify the Equation

Now we have the equation: \( 2z = 3z + 70 \). The goal is to isolate \(z\) on one side. Begin by subtracting \(3z\) from both sides to move all the \(z\) terms to one side: \( 2z - 3z = 70 \). This simplifies to: \( -z = 70 \).

Step 3: Solve for z

We now have \( -z = 70 \). To solve for \( z \), multiply both sides by -1 to get rid of the negative sign: \( z = -70 \).

Key Concepts

Solving Linear Equations

Linear equations are mathematical expressions where the variable is raised to the power of one. Solving linear equations involves finding the value of the variable that makes the equation true. In the equation \( \frac{z}{5} = \frac{3}{10}z + 7 \), we want to find the value of \( z \) that satisfies this condition.
Linear equations like these are common in algebra and often serve as the foundation for more complex math problems. The process typically includes steps such as simplifying the equation, using inverse operations, and sometimes manipulating expressions to isolate the variable.
When you solve linear equations, the goal is always to have the variable by itself on one side of the equation. This is achieved through a series of logical and systematic operations. Let's break down some of these steps to clarify the process.

Elimination of Fractions

Eliminating fractions makes it easier to work with linear equations. Fractions can often complicate calculations, so getting rid of them is helpful. In the exercise, the equation \( \frac{z}{5} = \frac{3}{10}z + 7 \) involves fractions on both sides.
The first step in eliminating fractions is to find a common denominator. For this equation, the common denominator is 10. By multiplying every term in the equation by 10, the fractions are effectively removed:

• \( 10 \times \frac{z}{5} = 2z \)
• \( 10 \times \frac{3}{10}z = 3z \)
• \( 10 \times 7 = 70 \)

This simplification transforms the original fraction-laden equation into \( 2z = 3z + 70 \), which is much easier to handle.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying equations to solve for a variable. Once the fractions have been eliminated, algebraic manipulation becomes crucial. In our exercise, we get to a simplified equation: \( 2z = 3z + 70 \).
Now, we need to get all terms involving \( z \) on one side and constant terms on the other. This involves:

• Subtracting \( 3z \) from both sides: \( 2z - 3z = 70 \)
• This simplifies to: \( -z = 70 \)

We've successfully isolated \( z \) to a more manageable form, yet with a negative sign. This forms the bridge toward the final solution.

Equivalent Equations

An equivalent equation is formed when you apply operations that don't change the solution of the original equation. For example, when fractions are eliminated or terms are subtracted from each side, the equation remains equivalent to the original.
In our case, \( 2z = 3z + 70 \) and \( -z = 70 \) are equivalent to the initial equation \( \frac{z}{5} = \frac{3}{10}z + 7 \). These transformations do not alter the root solution of the equation, which is the value of \( z \).
To find this solution, we finally address the equation \( -z = 70 \). By multiplying both sides by -1, we get the equivalent equation \( z = -70 \). This operation gives us the same root as that of the original equation, showing consistency through the steps of solving and manipulating.