Problem 49 - Introductory Algebra for College Students

Question

Solve each equation and check your proposed solution in Exercises. $$0.92 y+2=y-0.4$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 25\).

Step 1: Rearrange the equation

Rearrange the equation by bringing like terms together. This can be done by subtracting \(0.92y\) from both sides of the equation. After this step, the equation will be: \(2 = y - 0.4 - 0.92y\).

Step 2: Simplify

Now, simplify the right side of the equation. Combine \(y\) and \(-0.92y\) together, you will get \(0.08y = 2\).

Step 3: Isolate the variable y

To solve for \(y\), divide both sides of the equation \(0.08y = 2\) by \(0.08\). You will get \(y = 25\).

Step 4: Check the result

Substitute \(y = 25\) into the original equation \(2 = y - 0.4 - 0.92y\), you will get \(0.92 (25) + 2 = 25 - 0.4\). After calculation, the left side equals 25 and the right side also equals 25, which proves the solution is correct.

Key Concepts

Rearranging Equations

Rearranging equations is the first step when solving algebraic equations. The goal is to move terms around to get all the variable terms on one side of the equation and constants on the other. In the given problem, we start with the equation \(0.92y + 2 = y - 0.4\). To rearrange it, subtract \(0.92y\) from both sides to group the \(y\) terms together. Now, the equation looks like this: \(2 = y - 0.4 - 0.92y\). This step helps to set the stage for making further simplifications.

Simplifying Expressions

Simplifying expressions means combining like terms to make the equation easier to solve. For this equation, you have terms involving \(y\) on the right side: \(y - 0.92y\). By combining these, you find \(0.08y\). Now, the equation is rewritten as \(0.08y = 2\). Simplifying helps to both clarify and reduce the equation so that we can isolate the variable effectively in the next steps.

Isolating Variables

Isolating the variable means getting the variable alone on one side of the equation. It usually involves either adding, subtracting, multiplying, or dividing both sides by the same number. After simplifying to \(0.08y = 2\), we can isolate \(y\) by dividing both sides by \(0.08\), giving us the solution \(y = 25\). So, isolating the variable helps us find the actual value of the unknown in the equation.

Checking Solutions

Checking solutions is a crucial step to confirm whether the value found for the variable is correct. This involves substituting the value back into the original equation to see if both sides are equal. For the solution \(y = 25\), substitute it back into \(0.92y + 2 = y - 0.4\). Calculate: \(0.92(25) + 2\) equals \(23 + 2 = 25\) and verify that \(25 - 0.4\) also results in \(24.6\), proving both sides equal 24.6 and confirming the solution is correct. This ensures no mistakes were made during solving.