Problem 100 - Intermediate Algebra

Question

Solve each equation. $$ \frac{7}{3} y+1=0 $$

Step-by-Step Solution

Answer

The solution is \( y = -\frac{3}{7} \).

Step 1: Isolate the Variable

First, we need to isolate the variable \( y \). Start by subtracting 1 from both sides of the equation to undo the addition: \[\frac{7}{3}y + 1 - 1 = 0 - 1\]This simplifies to:\[\frac{7}{3}y = -1\]

Step 2: Solve for \( y \)

To solve for \( y \), multiply both sides of the equation by the reciprocal of \( \frac{7}{3} \), which is \( \frac{3}{7} \). This will cancel out the fraction on the left side:\[\left(\frac{3}{7}\right) \times \frac{7}{3}y = -1 \times \frac{3}{7}\]The left side simplifies to \( y \), and the right side simplifies to:\[y = -\frac{3}{7}\]

Key Concepts

Fraction Operations

Handling fractions might seem tricky at first, but with a clear process, it becomes a lot easier. When dealing with equations, fractions can represent the coefficients of variables, as we saw with \(\frac{7}{3} y\) in this exercise.

Here are some basic operations you can perform on fractions:

• **Adding/Subtracting Fractions:** To add or subtract fractions, ensure they have a common denominator. If not, find equivalent fractions with the same denominator before performing the addition or subtraction.
• **Multiplying Fractions:** Multiply the numerators together and the denominators together. For example, \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).
• **Dividing Fractions:** Inverse the second fraction and multiply. This is where reciprocal multiplication comes in, a concept we'll explore in detail soon.

Taking these steps systematically helps simplify fractions and makes equation-solving more manageable.

Isolating Variables

Isolating the variable is a crucial step in solving any algebraic equation. Think of it as peeling away layers of operations surrounding the variable you want to solve for. This process reveals the 'core', or the value of the variable, just like removing layers from an onion. In our exercise, we had the equation \(\frac{7}{3} y + 1 = 0\).

To isolate \(y\), we started by removing the addition of 1. We do this by performing the inverse operation, which is subtraction. Here's how:

• Subtract 1 from both sides of the equation: \(\frac{7}{3} y + 1 - 1 = 0 - 1\).
• This simplifies to \(\frac{7}{3} y = -1\), where all the terms have been simplified around \(y\).

Remember, whatever operation you do to one side of the equation, you must also do to the other side. This keeps the equation balanced.

Reciprocal Multiplication

Reciprocal multiplication is pivotal when solving equations with fractions, especially when isolating a variable. A reciprocal is essentially flipping the numerator and the denominator of a fraction. For example, the reciprocal of \(\frac{7}{3}\) is \(\frac{3}{7}\).

After isolating \(y\) in \(\frac{7}{3} y = -1\), we multiplied both sides of the equation by the reciprocal of \(\frac{7}{3}\). Here's why:

• Multiplying by the reciprocal \(\frac{3}{7}\) cancels out the fraction \(\frac{7}{3}\) on the left side of the equation because \(\left(\frac{3}{7}\right) \times \left(\frac{7}{3}\right) = 1\).
• This leaves us with \(1 \times y = y\) on one side of the equation.
• The right side of the equation becomes: \(-1 \times \frac{3}{7} = -\frac{3}{7}\).

Reciprocal multiplication is a powerful tool because it simplifies equations by eliminating fractions, making it easier to discern the variable's value \(y = -\frac{3}{7}\).