Problem 18 - Algebra and Trigonometry

Question

\(9- 46\) 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 is \(z = -70\).

Step 1: Eliminate Fractions

First, we need to eliminate the fractions in the equation. Multiply every term by 10, which is the least common multiple of the denominators 5 and 10:\[ 10 \times \frac{z}{5} = 10 \times \frac{3}{10} z + 10 \times 7 \]This gives us:\[ 2z = 3z + 70 \]

Step 2: Isolate Variable Terms

Next, we want to isolate the variable terms on one side by subtracting \(3z\) from both sides:\[ 2z - 3z = 70 \]This simplifies to:\[ -z = 70 \]

Step 3: Solve for z

Finally, solve for \(z\) by dividing both sides by -1:\[ z = -70 \]

Key Concepts

Solving Equations

Solving linear equations is a fundamental skill in algebra. It involves finding the value of the variable that makes the equation true. When dealing with linear equations, each step you take will help you simplify the problem until the variable is isolated. Linear equations can be recognized by their highest degree of the variable being one. They often come in the standard form \( ax + b = c \), where you aim to find the value of \( x \). For any equation, the goal is to find a solution that satisfies the equation, meaning that both sides are equal when the value is substituted back in.
Here's a general approach to solve linear equations:

• Begin by simplifying each side of the equation (if necessary) by combining like terms.
• Use the properties of equality to isolate the variable.
• If fractions are present, consider eliminating them to make calculations easier.

Approaching equations with these steps in mind will help you systematically find solutions.

Isolating Variables

Isolating variables is a crucial part of solving equations. The aim here is to get the variable by itself on one side of the equation, usually the left side, to simplify solving. The process often requires performing operations on both sides of the equation to maintain balance and ensure the equation still holds true.To isolate variables, you can:

• Add or subtract terms to move constants or other variables to the opposite side of the equation.
• Use multiplication or division to get the variable to stand alone.

In our given solution, the term \( 3z \) was subtracted from \( 2z \) on both sides, which simplified to \(-z = 70\). This simplification helped in further isolating \( z \) by itself. Isolating helps you see the problem more clearly and reduces equations to their simplest form.

Eliminating Fractions

Eliminating fractions from equations can make them easier to solve. Fractions often add complexity due to additional steps and calculations, but removing them can simplify the process of finding solutions. In our case, the equation started with fractions:\( \frac{z}{5} = \frac{3}{10}z + 7 \).
To eliminate fractions:

• Identify the denominators present in the equation.
• Find the least common multiple (LCM) of these denominators.
• Multiply every term in the equation by this LCM to clear the fractions.

In the solution, multiplying every term by \( 10 \), the least common multiple of \( 5 \) and \( 10 \), transformed the equation to \( 2z = 3z + 70 \). This step removes fractions and provides a cleaner equation to work with, allowing you to isolate the variable more efficiently.