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Chapter 8: Problem 157

Solve the equations for the variable. $$13 z+6.45=8 z+23.75$$

Short Answer

Expert verified
z = 3.46

Step by step solution

01

- Isolate terms with the variable

To isolate the variable term on one side, subtract \(8z\) from both sides of the equation: \[13z + 6.45 - 8z = 8z + 23.75 - 8z\]This simplifies to: \[5z + 6.45 = 23.75\]
02

- Isolate the constant term

Subtract \(6.45\) from both sides to isolate the term with the variable:\[5z + 6.45 - 6.45 = 23.75 - 6.45\]This simplifies to: \[5z = 17.30\]
03

- Solve for the variable

Divide both sides of the equation by 5 to solve for \(z\):\[\frac{5z}{5} = \frac{17.30}{5}\]So, \[z = 3.46\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating variable terms
When solving equations, the first step is often to isolate the variable term on one side of the equation. This means you want all the terms with the variable (in this case, 'z') on one side and the constant terms (just the numbers) on the other side.
Let's look at our example equation: \[13z + 6.45 = 8z + 23.75\]
The goal is to get 'z' by itself. To do this, we subtract \(8z\) from both sides. Subtraction of the same term on both sides keeps the equation balanced:\[13z + 6.45 - 8z = 8z + 23.75 - 8z\]
When we simplify this, we get:\[5z + 6.45 = 23.75\]
This process has successfully isolated the variable term \(5z\) on the left side of the equation.
Subtracting constants
After isolating variable terms, the next step is to focus on isolating the constant term present alongside the variable term. In our simplified equation, we have:\[5z + 6.45 = 23.75\]
To isolate the variable \(z\), we need to remove the constant term \(6.45\) from the left side. We do this by subtracting \(6.45\) from both sides of the equation:\[5z + 6.45 - 6.45 = 23.75 - 6.45\]
When simplified, this results in:\[5z = 17.30\]
Subtraction here helps to clear away the constant part, leaving us a simpler equation where the variable term \(z\) is more easily solved.
Dividing to solve for variable
Now that we have isolated the variable term, the final step is to solve for the variable itself. Our simplified equation is:\[5z = 17.30\]
To solve for \(z\), we need to divide by the coefficient of \(z\), which is 5. That's because right now \(z\) is multiplied by 5, and division is the inverse operation of multiplication. So, we divide both sides by 5:\[\frac{5z}{5} = \frac{17.30}{5}\]
This simplifies to:\[z = 3.46\]
Dividing helps to isolate the variable \(z\), giving us the solution for our equation. By following these steps鈥攊solating variable terms, subtracting constants, and then dividing鈥攚e can systematically solve for the variable in linear equations.

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