Problem 15 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(1.7 x=10.71\)

Step-by-Step Solution

Answer

The solution is \(x = 6.3\).

Step 1: Identify the Equation

The equation given is a simple linear equation: \(1.7x = 10.71\). Our goal is to find the value of \(x\) that satisfies this equation.

Step 2: Isolate the Variable

To isolate \(x\), divide both sides of the equation by 1.7: \[x = \frac{10.71}{1.7}\] This step simplifies to finding \(x\) by performing the division.

Step 3: Perform the Division

Calculate \(\frac{10.71}{1.7}\) to find the value of \(x\). Performing the division: \[x = 6.3\] Thus, the solution for the equation is \(x = 6.3\).

Step 4: Verify the Solution

To ensure the solution is correct, substitute \(x = 6.3\) back into the original equation and check:\[1.7 \times 6.3 = 10.71\]Calculate the left side: \(1.7 \times 6.3 = 10.71\), which matches the right side of the equation, confirming that \(x = 6.3\) is the correct solution.

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics. They form straight lines when graphed and have a standard format of \(ax + b = c\) where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable to solve for. In this exercise, the equation is simplified through operations to find the unknown value of \(x\).
Linear equations are straightforward because they only have one variable raised to the power of one. They are essential for understanding more complex algebraic principles.
Here is why linear equations are important:

• They model real-world problems and can help in decision making.
• They're foundational for understanding functions and graphing.
• They are an integral part of high school and college algebra curricula.

Division Operation

The division operation is a key step in solving linear equations. It helps in simplifying the equation and finding the value of the variable. In the original exercise, division is used to isolate \(x\).
When performing division in algebra:

• Ensure both sides of the equation are divided by the same non-zero number to maintain equality.
• Double-check calculations to avoid mistakes, especially with decimals.

Understanding division is essential because it changes the quantity by breaking it into parts. This skill is used to simplify terms and to find solutions in equations.
This brings clarity and precision in solving and checking results, as demonstrated in the exercise.

Checking Solutions

After solving an equation, it's crucial to verify the solution. This means plugging the solution back into the original equation to see if it holds true.
Reasons for checking solutions include:

• It confirms the accuracy of calculations.
• Identifies potential mistakes in earlier steps.
• It provides confidence in the final answer.

In our case, by substituting \(x = 6.3\) back into the equation \(1.7x = 10.71\), both sides equate perfectly. Therefore, the solution is correct and verified.

Variable Isolation

Variable isolation is about getting the variable alone on one side of the equation. This forms the core part of solving linear equations.
Here's how to effectively isolate variables:

• Use inverse operations like addition, subtraction, multiplication, or division.
• Maintain equilibrium by applying these operations to both sides of the equation.
• Reorganize and simplify until the variable stands alone.

Isolating the variable transforms the equation into an easily solvable form. It's a necessary skill to master for effectively tackling algebraic problems.
In this exercise, we isolate \(x\) by dividing both sides by 1.7, which leads directly to finding \(x = 6.3\), the correct solution.