Problem 21 - Elementary Algebra

Question

Solve. $$ -110 y+25=15 y+310 $$

Step-by-Step Solution

Answer

\( y = -\frac{57}{25} \)

Step 1: Move Variable Terms to One Side

To isolate the variable \( y \), we first need to move all terms containing \( y \) to one side of the equation. Begin by subtracting \( 15y \) from both sides:\[-110y + 25 - 15y = 15y + 310 - 15y\]which simplifies to:\[-125y + 25 = 310\].

Step 2: Move Constant Terms to the Other Side

Next, we will isolate the term with \( y \) by moving the constant term to the other side. Subtract 25 from both sides:\[-125y + 25 - 25 = 310 - 25\]resulting in:\[-125y = 285\].

Step 3: Solve for the Variable

Now, solve for \( y \) by dividing each side of the equation by \(-125\):\[y = \frac{285}{-125}\]This simplifies to:\[y = -\frac{57}{25}\].

Step 4: Simplify the Fraction

Finally, simplify the fraction \(-\frac{57}{25}\). This fraction is already in its simplest form because 57 and 25 have no common factors other than 1, so:\[y = -\frac{57}{25}\].

Key Concepts

Variable Isolation

When faced with an algebraic equation like \(-110y + 25 = 15y + 310\), the goal is to find the value of the variable \(y\). The first major step in solving such equations is variable isolation. This means rearranging the equation so that the variable you want to solve for, in this case, \(y\), is on one side of the equation by itself.

• Start by identifying all terms in the equation that include \(y\).
• Move them to one side of the equation. In our example, subtract \(15y\) from both sides, which simplifies to \(-125y + 25 = 310\).

This process is vital as it directs the pathway to solving the equation, allowing for easier manipulation of the constants and the variable. Remember, whatever operation you do to one side, you must do to the other to maintain the balance of the equation.

Simplifying Fractions

In algebra, you often encounter fractions when isolating variables, just as in our example with \(y = \frac{285}{-125}\). Simplifying fractions is a crucial step in providing a clean, understandable result. It involves reducing the fraction to its simplest form.

• First, identify any common factors between the numerator and denominator.
• If no common factors exist, as in our case, the fraction is already simplified.

In our example, the fraction \(-\frac{57}{25}\) is in its simplest form because the greatest common divisor of 57 and 25 is 1. Hence, further simplification is not possible. Always double-check your fraction to ensure it's as simple as possible. A simplified fraction often makes it easier to interpret the final value.

Algebraic Manipulation

Algebraic manipulation is used to rearrange and simplify expressions and equations to find the value of unknowns. In solving the equation \(-110y + 25 = 15y + 310\), after isolating the variable and transferring terms, algebraic manipulation involves performing operations that simplify your equation. This process continues until you lock down the variable value.

• Begin with moving variable and constant terms appropriately, such as subtracting or adding terms from both sides.
• For our equation, after simplifying to \(-125y = 285\), you divide both sides by the coefficient of \(y\), which is \(-125\).
• The division leads you directly to the solution, \(y = -\frac{57}{25}\).

This course of action underscores the importance of understanding operational order and mathematical properties. Each step in your manipulation should follow logically, maintaining equation balance while approaching a simpler form.