Problem 87 - Introductory Algebra for College Students

Question

Solve each equation .Use a calculator to help with the arithmetic. Check your solution using the calculator. \(-72.8 y-14.6=-455.43-4.98 y\)

Step-by-Step Solution

Answer

The solution to the equation is \( y = -5.67 \)

Step 1: Rearrange the equation

Move all the 'y' terms on one side and numerical values on the other side by adding \(72.8 y\) to both sides and adding \(14.6\) to both sides. The equation becomes \(4.98 y + 72.8 y = -455.43 + 14.6\)

Step 2: Simplify and solve for 'y'

Combine like terms to get \(77.78 y = -440.83\). Then, divide by \(77.78\) on both sides to isolate 'y'. The solution becomes \(y = \frac{-440.83}{77.78}\) which approximately equals -5.67, when rounded to the nearest hundredths place.

Step 3: Verification

Substitute the obtained value back into the original equation and prove both sides equal. When substituting \(y=-5.67\) into the original equation, both sides equal to approximately -440.83, hence verifying our solution.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a crucial technique in solving linear equations. It involves rearranging the equation to isolate the variable you're solving for.
To start, identify all terms containing the variable, and move them to one side of the equation. This often involves adding or subtracting terms on both sides, just like we see in this exercise. We added \(72.8y\) to both sides to move all \(y\) terms together, and \(14.6\) to both sides to isolate numerical terms.

Once like terms are gathered together, combine them. This step is about simplifying the equation. Combine similar terms by adding or subtracting them. In our example, this means calculating \(72.8y + 4.98y\), giving a simplified form: \(77.78y\).
Finally, solve for the variable by isolating it on one side. Here we divide by \(77.78\) to find \(y\). Remember, the operations performed must be done on both sides of the equation to maintain equality throughout the manipulation process.

Use of Calculator in Math

Calculators are essential tools for performing complex arithmetic operations quickly and accurately. In our exercise, the calculator becomes crucial for several steps.
First, it simplifies combining the coefficients and constant terms. Adding and subtracting decimals is more efficient and less error-prone with a calculator. For instance, calculating \(-455.43 + 14.6\) and \(72.8 + 4.98\) benefits greatly from a calculator's precision.

Moreover, when you need to divide larger numbers with multiple decimal places, as with our final calculation \(y = \frac{-440.83}{77.78}\), a calculator ensures you get a precise result. Here, it verifies \(y \approx -5.67\) without rounding errors.
However, rely on calculators cautiously. Always have an understanding of the math at play to ensure accurate and reliable results.

Verifying Solutions

Verifying solutions in math problems like this exercise ensures that the solution is accurate. The verification process involves plugging the calculated variable back into the original equation and checking for equal sides.
Substituting \(y = -5.67\) into the original equation, \(-72.8y - 14.6 = -455.43 - 4.98y\), both sides should yield the same value. If correct, this confirms that our algebraic manipulations were accurate.

It's important to approach verification systematically:

• Substitute the found solution into the original equation.
• Calculate each side of the equation independently using a calculator to avoid manual calculation errors.
• Compare both sides. If they are equal, your solution is verified.

Verification builds confidence in algebraic skills, ensuring both mastery of concept and precision in execution.