Problem 53 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation and check your proposed solution in. \(0.4(2 z+6)+0.1=0.5(2 z-3)\)

Step-by-Step Solution

Answer

The solution to the equation is \(z = 20\).

Step 1: Simplify both sides of the equation

Divide the numbers inside the brackets on both sides of the equation.\n We get: \[0.4 \cdot 2z + 0.4 \cdot 6 + 0.1 = 0.5 \cdot 2z - 0.5 \cdot 3\] which simplifies further to: \[0.8z + 2.4 + 0.1 = z - 1.5.\]

Step 2: Keep z on one side of the equation and constant terms on the other side

To isolate \(z\), move all terms involving \(z\) to one side (say the left side) and rest other terms to the right side.\n As a result, we obtain: \(0.8z - z = -1.5 - 2.4 - 0.1\). This simplifies to \(--0.2z = -4\).

Step 3: Solve for z

Now solve for \(z\) by dividing each side by -0.2.\n As a result, the equation turns out to be \(z = -4 / -0.2\), so the solution is \(z = 20\).

Step 4: Check the solution

Verify the obtained solution (z = 20) by substituting into the original equation.\n Substitution gives \(0.4(2 \cdot 20 + 6) + 0.1 = 0.5(2 \cdot 20 - 3)\).\n Working it out, it gives \( 9.8 = 9.8\) . This confirms that the solution is correct.

Key Concepts

Understanding Linear Equations

Linear equations are foundational concepts in algebra. They are equations of the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable. In linear equations, the highest power of the variable is 1. This means that they form a straight line when graphed on a coordinate plane. Linear equations can have one solution, no solution, or infinitely many solutions, depending on the relationship between the lines they form.
In the given problem, we have the linear equation: \[0.4(2z + 6) + 0.1 = 0.5(2z - 3)\]This equation involves the variable \(z\) and constants. The goal is to find the value of \(z\) that makes the equation true. Linear equations are essential in mathematics because they model real-world problems where relationships are consistent and proportional.

How to Solve Equations Effectively

Solving equations is about finding the value of the variable that satisfies the equation. Here are the general steps for solving a linear equation:

• Simplify each side: First, distribute any numbers outside parentheses and combine like terms on both sides of the equation.
• Isolate the variable: Rearrange the equation to get the variable on one side and constants on the other. Use operations like addition, subtraction, multiplication and division.
• Solve for the variable: Once the variable is isolated, perform the necessary calculation to find its value.

In the exercise, we applied these steps to solve for \(z\):

• Simplified the equation to \(0.8z + 2.5 = z - 1.5\).
• Rearranged to isolate \(z\), resulting in \(-0.2z = -4\).
• Solved for \(z\), giving us \(z = 20\).

Understanding these steps helps in solving similar problems efficiently and with confidence.

The Importance of Equation Verification

Equation verification is a vital step to ensure that the solution you have found is correct. It involves substituting the obtained solution back into the original equation and checking whether both sides are equal. This step is crucial because it confirms the accuracy of your work.
For our problem, we substituted \(z = 20\) into the original equation, leading to:\[0.4(2 \times 20 + 6) + 0.1 = 0.5(2 \times 20 - 3)\]Calculating both sides gave us \(9.8 = 9.8\). Since both sides match, the solution is verified as correct. This step prevents errors and boosts your confidence in the solution's correctness.
To master solving equations, always remember to verify your answers. It is a simple but powerful way to ensure your work is reliable and accurate.