Problem 2 - Algebra for College Students

Question

Solve each equation for \(y\). $$\frac{y+1}{x-9}=\frac{3}{4}$$

Step-by-Step Solution

Answer

y = \frac{3x - 31}{4}

Step 1: Cross Multiply

To eliminate the fractions, cross-multiply the equation. Multiply the numerator of the left fraction by the denominator of the right fraction, and the numerator of the right fraction by the denominator of the left fraction. This gives:display porn movies content:\[4(y+1) = 3(x-9)\]

Step 2: Distribute

Distribute the 4 on the left side and the 3 on the right side:\[4y + 4 = 3x - 27\]

Step 3: Isolate the y-Term

Subtract 4 from both sides of the equation to isolate the term with y:\[4y = 3x - 31\]

Step 4: Solve for y

Divide both sides by 4 to solve for y:\[y = \frac{3x - 31}{4}\]

Key Concepts

cross-multiplication

Cross-multiplication is a method used to eliminate fractions in an equation by multiplying across the equal sign. This technique helps to simplify the equation and make it easier to solve.
For the equation \(\frac{y+1}{x-9}=\frac{3}{4}\), we cross-multiply by multiplying the numerator of one fraction by the denominator of the other fraction:

\[4(y+1) = 3(x-9)\].
This results in an equation with no fractions, making it easier to proceed to the next steps. Cross-multiplication is particularly useful when dealing with proportions and fractional equations.

distributive property

The distributive property helps to multiply a single term across a sum or difference inside parentheses. It states that \(a(b + c) = ab + ac\).
In our problem, we apply the distributive property to both sides of the equation:

\[4(y+1) = 4y + 4\]
and
\[3(x-9) = 3x - 27\]
This step is crucial because it expands the terms, removing the parentheses and making it possible to combine like terms or isolate variables in later steps.

isolating variables

Isolating a variable means getting it alone on one side of the equation. Here, we need to isolate \y\. Starting with the equation \4y + 4 = 3x - 27\, we need to remove the constant term from the left side:

Subtract 4 from both sides:
\[4y = 3x - 31\].
Now, \y\ is in a simpler form, only with a constant multiplied to it.
In equations, isolating the variable helps in finding the solution by making the variable's coefficient 1 through further manipulation or operations.

fraction manipulation

Fraction manipulation involves performing operations to simplify or solve fractions. After isolating the variable \y\, our equation \[4y = 3x - 31\] needs one final step:

Divide both sides by 4 to solve for \y\:
\[y = \frac{3x - 31}{4}\].
This step ensures \y\ is alone and the equation is simplified. Fraction manipulation is key in solving equations involving fractions, ensuring all terms are correctly handled and simplified.