Problem 86 - Introductory Algebra for College Students

Question

Solve each equation .Use a calculator to help with the arithmetic. Check your solution using the calculator. 6\. \(3.7 x-19.46=-9.988\)

Step-by-Step Solution

Answer

The solution to the equation \(3.7 x - 19.46 = -9.988\) is \(x = 2.56\).

Step 1: Add 19.46 to both sides

The equation is \(3.7 x - 19.46 = -9.988\). To isolate x, the first step is to get rid of 19.46 on the left side of the equation. This can be done by adding 19.46 to both sides of the equation, yielding the new equation: \(3.7 x = -9.988 + 19.46\).

Step 2: Solve for x

Calculate \( -9.988 + 19.46 \) with a calculator, which results in \( 9.472 \). So, the equation now is: \( 3.7 x = 9.472 \). To solve for x, divide both sides of the equation by 3.7. This gives: \( x = 9.472 / 3.7 \).

Step 3: Calculate x

Use a calculator to find the value of \( x = 9.472 / 3.7 \), which gives \( x = 2.56 \).

Step 4: Check your solution

Substitute \( x = 2.56 \) into the original equation \(3.7 x - 19.46 = -9.988\). Evaluation will confirm if the solution is correct.

Key Concepts

Calculator Use in Algebra

Using a calculator in algebra can make solving complex equations much simpler. Calculators help perform arithmetic operations quickly and accurately. In algebra, a calculator assists in:

• Adding, subtracting, multiplying, and dividing numbers.
• Ensuring precision, especially when dealing with decimals and fractions.

When you face an equation like \(3.7x - 19.46 = -9.988\), a calculator is handy for determining results such as \(-9.988 + 19.46\) or \(9.472 \div 3.7\). Particularly when working with decimals, it’s crucial to rely on a calculator to reduce errors and confirm that your solution is as precise as possible. Nevertheless, always ensure that you understand the algebraic steps to solve the equation manually before relying on technology.

Arithmetic Operations

Arithmetic operations play a critical role in solving linear equations. These include addition, subtraction, multiplication, and division. Here's how they work:

• **Addition** and **Subtraction**: These operations are often used to isolate the variable. For instance, in the equation \(3.7x - 19.46 = -9.988\), adding 19.46 to each side helps remove it from the left side, simplifying the equation.
• **Multiplication** and **Division**: Utilize these to solve for the variable once it’s isolated. In our equation, you divide both sides by 3.7 to solve for \(x\).

These operations are fundamental and are used in sequence to systematically work through an equation. Understanding when and how to apply each operation is key to efficiently and correctly solving algebraic equations.

Checking Solutions in Algebra

Once you find a solution in algebra, it's important to verify that the solution satisfies the original equation. This is known as checking your work. In this example, the solution \(x = 2.56\) should be substituted back into the original equation, \(3.7x - 19.46 = -9.988\). Using a calculator can help quickly compute:

• Multiply 3.7 by 2.56 to get 9.472
• Then subtract 19.46 from 9.472, which should result in -9.988

If the left side of the equation equals the right side after the substitution, your solution is correct. This step confirms that no arithmetic errors were made and strengthens your understanding of solving equations.