Problem 43 - Algebra for College Students

Question

For Problems \(31-44\), solve each equation for the indicated variable. $$ \frac{y-1}{x+6}=\frac{-2}{3} $$ for \(y\)

Step-by-Step Solution

Answer

\(y = \frac{-2x - 9}{3}\).

Step 1: Cross Multiply

To eliminate the fractions, we use cross multiplication. Multiply both sides of the equation by the denominators:\[(y - 1) imes 3 = (-2) imes (x + 6).\]

Step 2: Simplify Both Sides

Simplify both sides of the equation obtained from cross multiplication:\[3(y - 1) = -2(x + 6).\]

Step 3: Distribute Values

Distribute the constants on both sides of the equation:\[3y - 3 = -2x - 12.\]

Step 4: Isolate the Variable y

Add 3 to both sides to isolate the term with the variable y:\[3y = -2x - 12 + 3.\]Simplify the right side:\[3y = -2x - 9.\]

Step 5: Solve for y

Divide the entire equation by 3 to solve for y:\[y = \frac{-2x - 9}{3}.\]

Key Concepts

Cross Multiplication

When faced with equations involving fractions, cross multiplication is a helpful method to simplify the problem. In essence, cross multiplication is used to eliminate fractions by multiplying the numerator of each fraction by the denominator of the other. This results in an equation without fractions.
For example, consider the equation \[\frac{y-1}{x+6}=\frac{-2}{3}.\]By cross multiplying, you multiply both sides as follows:

• Numerator of the first fraction \((y-1)\) times the denominator of the second fraction \(3\).
• Numerator of the second fraction \(-2\) times the denominator of the first fraction \((x+6)\).

This leads to:\[3(y - 1) = -2(x + 6).\]Notice how this operation effectively clears the fractions and gives you a straightforward equation to work with. Cross multiplication is a smooth, fuss-free way to handle equations with fractions.

Variable Isolation

Variable isolation is the process of rearranging an equation to get the unknown variable by itself on one side of the equation. This is a fundamental technique in algebra, allowing you to solve for a specific variable.
In our example :\[3(y - 1) = -2(x + 6),\]we need to isolate \(y\). Start by simplifying and distributing any numbers outside brackets to eventually solve for \(y\):

• Expand on both sides: \(3y - 3 = -2x - 12\).
• Add \(3\) to both sides to move constants away from the \(y\) term, resulting in \(3y = -2x - 9\).
• Finally, divide everything by \(3\) to get \(y = \frac{-2x - 9}{3}\).

Variable isolation helps maintain focus on the specific variable you are interested in, gradually clearing away all other elements.

Distributive Property

The distributive property allows us to multiply a single term by each term within a set of parentheses. This is particularly useful when simplifying expressions or solving equations.
The property states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b+c) = ab + ac\) holds true.

• In our equation \(3(y - 1) = -2(x + 6)\), apply the distributive property to both sides.
• Left side: \(3 \times (y - 1)\) simplifies to \(3y - 3\).
• Right side: \(-2 \times (x + 6)\) simplifies to \(-2x - 12\).

Using the distributive property simplifies the terms in an equation, allowing easier manipulation and solvability. It's an essential tool anytime you encounter parentheses in algebraic equations.