Problem 72 - Elementary Algebra

Question

Solve. $$ 70=50-12 y $$

Step-by-Step Solution

Answer

The solution is \( y = \frac{-5}{3} \).

Step 1: Isolate Term with Variable

Start by isolating the term with the variable on one side of the equation. Currently, the equation is set up as:\[ 70 = 50 - 12y \]To isolate the term \(-12y\), subtract 50 from both sides of the equation:\[ 70 - 50 = -12y \]which simplifies to:\[ 20 = -12y \]

Step 2: Solve for Variable

Now that we have \(20 = -12y\), we need to solve for \(y\) by isolating it. Do this by dividing both sides of the equation by \(-12\):\[ \frac{20}{-12} = y \]Simplify the fraction on the left:\[ y = \frac{-5}{3} \]

Key Concepts

Solving Equations

Equations are like balance scales, where each side of the equation must be equal. Solving equations involves finding the value of the unknown variable that makes both sides equal. In the given exercise, we see an equation: \( 70 = 50 - 12y \). The main goal is to find the value of \( y \) that satisfies this equation. To achieve this, we use a variety of techniques like simplifying expressions and performing operations to both sides of the equation.
Whenever we perform an operation, such as adding or subtracting a number, we do it to both sides to keep the equation balanced. This involves:

• Combining like terms
• Using inverse operations (like adding when subtracting is involved)
• Balancing the equation by doing the same operations on both sides

This logical process ensures we isolate the variable and eventually determine its numerical value.

Isolation of Variables

Isolating the variable means rearranging the equation so that the variable stands alone on one side of the equation. This is a critical step because it allows us to find the value directly associated with the variable. In our exercise, the equation starts as:\( 70 = 50 - 12y \). Our first move was to isolate the term with the variable (\(-12y\)) by subtracting 50 from both sides. This results in a simpler equation: \( 20 = -12y \). By moving terms around, we help "untangle" the variable from the rest of the numbers in the equation. The key actions include:

• Identifying the term containing the variable
• Performing operations to isolate this term (like addition, subtraction)
• Reassessing the equation after each operation to ensure the variable's isolation progresses

The final result should have the variable alone on one side, ready for the final solution steps.

Fractions in Equations

Fractions often appear when dividing terms during the isolation of a variable, and simplifying them correctly is vital. In the equation \( 20 = -12y \), we isolate \( y \) by dividing both sides by \(-12\). This results in the equation:\( \frac{20}{-12} = y \). Dividing a number by another can often lead to a fraction, which represents a number that isn't whole. Understanding basic fraction simplification rules helps make this manageable:

• Find the greatest common divisor (GCD) of the numerator and denominator
• Divide both by the GCD to simplify the fraction

In this case, the GCD of 20 and 12 is 4, so we simplify \( \frac{20}{-12} \) to \( \frac{-5}{3} \). Fractions provide exact values for solutions, making them a crucial aspect of solving algebraic equations.