Problem 34 - Algebra for College Students

Question

For Problems \(31-44\), solve each equation for the indicated variable. $$ \frac{7}{y-3}=\frac{3}{x+1} \text { for } y $$

Step-by-Step Solution

Answer

The solution for \(y\) is \(y = \frac{7x + 16}{3}.\)

Step 1: Cross Multiply

To eliminate the fractions, multiply both sides of the equation by \((y-3)(x+1)\). This gives: \[ 7(x+1) = 3(y-3). \]

Step 2: Distribute Constants

Expand the equation by distributing the constants: \[ 7x + 7 = 3y - 9. \]

Step 3: Isolate Terms Involving y

To isolate terms with \(y\), add 9 to both sides: \[ 7x + 16 = 3y. \]

Step 4: Solve for y

Divide each term by 3 to solve for \(y\): \[ y = \frac{7x + 16}{3}. \]

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to eliminate fractions in an equation, making it easier to solve. This is especially useful when dealing with proportions or equations where two fractions are set equal to each other.

• In the original equation, we had \( \frac{7}{y-3}=\frac{3}{x+1} \). Here, the numerator of one fraction is multiplied by the denominator of the other.
• This helps eliminate the fractional form, rendering the equation into a standard one, where traditional algebraic methods can be applied.

By cross-multiplying, we transform the equation into \( 7(x+1) = 3(y-3) \). This step is crucial as it sets the stage for further manipulation and simplification of the equation.

Equation Solving

Solving an equation involves performing operations to find the value of the unknown variable. After cross-multiplying in our example, we applied the distributive property:

• First, multiply each term inside the parentheses by the constants: \( 7(x+1) = 7x + 7 \) and \( 3(y-3) = 3y - 9 \).
• The subsequent equation \( 7x + 7 = 3y - 9 \) requires balancing both sides to simplify and prepare for isolating the variable.

Next, add 9 to both sides to move constants around and keep the equation balanced: \( 7x + 7 + 9 = 3y \). Thus, it simplifies to \( 7x + 16 = 3y \). With this in form, we can easily see how to isolate the variable.

Variable Isolation

The process of variable isolation is critical to solving for a specific variable in an equation. It involves rearranging the equation such that the variable you are solving for ends up alone on one side of the equation.

• From the equation \( 7x + 16 = 3y \), our goal is to isolate \( y \).
• This is achieved through division: \( y = \frac{7x + 16}{3} \) allows \( y \) to be the subject of the formula, providing a clear solution in terms of \( x \).

Variable isolation makes your final answer easy to interpret and substitutes the values for immediate use in further calculations or real-world applications. By keeping \( y \) isolated, one can easily adjust the equation to reflect changes in \( x \), showcasing useful flexibility.