Problem 36 - Algebra for College Students

Question

Solve each equation for \(y\). $$\frac{2}{3} x-\frac{2}{5} y=2$$

Step-by-Step Solution

Answer

y = -5 + \frac{5}{3}x

Step 1: Isolate the term with y

Start by moving the term involving y to one side of the equation. Subtract \(\frac{2}{3}x\) from both sides: \[ -\frac{2}{5}y = 2 - \frac{2}{3}x \]

Step 2: Simplify the right side

Combine the terms on the right side by finding a common denominator. The common denominator of 3 and 1 is 3: \[ -\frac{2}{5}y = 2 - \frac{2}{3}x \] \Express 2 as \(\frac{6}{3}\): \[ -\frac{2}{5}y = \frac{6}{3} - \frac{2}{3}x \]

Step 3: Further simplify the equation

Simplify the right side further by performing the subtraction: \[ -\frac{2}{5}y = \frac{6-2x}{3} \] \Thus, \[ -\frac{2}{5}y = \frac{6-2x}{3} \]

Step 4: Solve for y

Multiply both sides of the equation by the reciprocal of \(\frac{-2}{5}\) to isolate y: \[ y = \left( \frac{6-2x}{3} \right) \times \left( \frac{-5}{2} \right) \] \Simplify this equation: \[ y = \frac{-5(6-2x)}{6} \] \Distribute the -5 and simplify: \[ y = \frac{-30 + 10x}{6} \] \Divide each term by 6: \[ y = -5 + \frac{5}{3}x \]

Step 5: Final Answer

Thus, \(y = -5 + \frac{5}{3}x\)

Key Concepts

Linear Equations

Linear equations are mathematical statements involving variables raised to the power of one. These equations form straight lines when graphically represented. They typically look like: \( ax + by = c \) or \( y = mx + b \).
The one we are working on is:
\( \frac{2}{3} x -\frac{2}{5} y = 2 \).
To solve these, you often need to rearrange terms and perform basic operations to isolate one variable.

Isolating Variables

Isolating variables means manipulating the equation so that one variable stands alone on one side of the equation. In our situation, we want to isolate \( y \). To do this:

• First, move the term involving \( y \) to one side.
• Then, move all other terms to the opposite side by either addition, subtraction, division, or multiplication.

For example, to isolate \( y \) in \( \frac{2}{3} x - \frac{2}{5} y = 2 \), we first subtract \( \frac{2}{3} x \):
\( -\frac{2}{5}y = 2 - \frac{2}{3}x \).
This move gets the term with \( y \) by itself on one side.

Algebraic Manipulation

Algebraic manipulation involves changing the form of an equation using algebraic properties without changing its solution. Common manipulations include:

• Adding or subtracting the same value on both sides.
• Multiplying or dividing both sides by the same non-zero value.

For the given problem, first simplify the right side to a common denominator: \( \frac{6}{3} - \frac{2}{3}x \).
Then perform subtraction: \( \frac{6-2x}{3} \).
To isolate \( y \), multiply by the reciprocal of \( -\frac{2}{5} \):
\( y = \left( \frac{6-2x}{3} \right) \times \left( \frac{-5}{2} \right) \), leading to \( y = \frac{-30 + 10x}{6} \).

Common Denominators

Finding common denominators is essential when you need to combine fractions. It lets you work with comparable terms. A common denominator is a shared multiple of the denominators in the fractions.

• Identify the least common denominator (LCD).
• Convert all fractions to equivalent fractions with the same LCD.

In this problem:
The common denominator for 3 and 1 is 3:
Express 2 as \( \frac{6}{3} \).
This approach ensures you can combine and manage terms uniformly:
\( -\frac{2}{5}y = \frac{6}{3} - \frac{2}{3}x \), simplifying to \( \frac{6-2x}{3} \).