Problem 34 - Algebra for College Students

Question

Solve each equation for \(y\). $$8 x-3 y=4$$

Step-by-Step Solution

Answer

y = \frac{8}{3}x - \frac{4}{3}

Step 1 - Isolate the term with y

Start with the equation: \[ 8x - 3y = 4 \] Subtract \(8x\) from both sides to isolate the \(-3y\) term. \[ -3y = 4 - 8x \]

Step 2 - Solve for y

Divide both sides of the equation by \(-3\) to solve for \(y\). \[ y = \frac{4 - 8x}{-3} \]

Step 3 - Simplify the equation

Simplify the expression on the right side: \[ y = \frac{4}{-3} - \frac{8x}{-3} \] Simplify further to: \[ y = -\frac{4}{3} + \frac{8}{3}x \]Thus, the solution is: \[ y = \frac{8}{3}x - \frac{4}{3} \]

Key Concepts

Isolating Variables

One of the most important steps in solving linear equations is to isolate the variable you are solving for. In our example, we need to solve for y in the equation 8x - 3y = 4.
To isolate y, the goal is to get y on one side of the equation by itself. Start by removing the other terms from the side containing y. You do this by performing inverse operations.
In this case, subtract 8x from both sides to move the 8x term:
8x - 3y - 8x = 4 - 8x
This simplifies to:
-3y = 4 - 8x
Remember to always perform the same operation on both sides of the equation to maintain equality.

Simplifying Expressions

Simplifying expressions helps write the equation in the simplest form possible. This not only makes it more readable but also easier to solve.
In our example, after isolating y, we got -3y = 4 - 8x. To solve for y, we need to divide both sides by -3:
-3y / -3 = (4 - 8x) / -3
This simplifies to:
y = (4 / -3) + (-8x / -3)
The negative signs cancel out, resultng:
y = -4/3 + 8/3x
Notice how each term was simplified individually for better clarity. Simplifying each term ensures that the equation is easy to understand and work with for further algebraic operations.

Steps in Algebra

Solving linear equations involves a series of logical and ordered steps. Let’s outline the steps involved in solving 8x - 3y = 4 for y to better understand the process:
1. **Isolate the term with y:**
Start with the equation: 8x - 3y = 4.
Subtract 8x from both sides: -3y = 4 - 8x.
2. **Solve for y:**
Divide both sides by -3: y = (4 - 8x) / -3.
3. **Simplify the equation:**
Simplify the expression: y = -4/3 + 8/3x.
It’s helpful to practice these steps on different equations to get comfortable with the process. Solving linear equations is a fundamental skill in algebra, and mastering it will make other mathematical concepts easier to understand.