Problem 1 - Algebra for College Students

Question

Solve each equation for \(y\). $$\frac{y-5}{x+3}=-\frac{4}{3}$$

Step-by-Step Solution

Answer

y = -\frac{4}{3}x + 1

Step 1 - Understand the Equation

We are given the equation \(\frac{y-5}{x+3} = -\frac{4}{3}\). We aim to solve for the variable \(y\).

Step 2 - Clear the Fraction

Multiply both sides of the equation by \(x + 3\) to clear the fraction. This gives us \((y - 5) = -\frac{4}{3} \times (x + 3)\).

Step 3 - Distribute the Constant

Distribute \(-\frac{4}{3}\) on the right-hand side of the equation: \(y - 5 = -\frac{4}{3}x - 4\).

Step 4 - Isolate the Variable \(y\)

Add 5 to both sides to isolate \(y\): \(y = -\frac{4}{3}x - 4 + 5\).

Step 5 - Simplify

Combine the constant terms on the right-hand side: \(y = -\frac{4}{3}x + 1\).

Key Concepts

linear equations

A linear equation is an algebraic expression where the highest power of the variable is one. This means that the variable is not squared or raised to any higher power. Linear equations often look like this: \ ax + by = c \ where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.

• They graph to a straight line.
• They have constant rates of change.

It's important to recognize linear equations because they are easier to solve than non-linear equations.

fractions

Fractions represent parts of a whole. They are written as \(\frac{numerator}{denominator}\). When working with fractions in equations:

• Clear the fraction by multiplying every term by the denominator.
• Simplify the resulting equation.

In our exercise, we have the fraction: \ \frac{y-5}{x+3} = -\frac{4}{3} \ To solve it:

1. We multiply both sides by \(x + 3\) to eliminate the denominator.
2. This leaves us with a simpler equation to solve.

variables manipulation

Manipulating variables is about rearranging equations to isolate the variable you are solving for. This involves:

• Addition and subtraction to move terms from one side to the other.
• Multiplication and division to simplify or clear coefficients.

For the exercise given:

1. We distribute \(-\frac{4}{3}\) across \(x + 3\).
2. Then, add or subtract terms to get \(y\) alone.

This manipulation helps reveal the value of the variable we are interested in.

isolation of variables

Isolating variables means getting a single variable by itself on one side of the equation. This is critical in solving for that variable:
\ \frac{y - 5}{x + 3} = -\frac{4}{3} \

• First, we clear the fraction by multiplying both sides by \(x + 3\).
• Next, distribute \(-\frac{4}{3}\): \(y - 5 = -\frac{4}{3}x - 4\).
• Finally, add 5 to both sides: \(y = -\frac{4}{3}x + 1\).

Now, \(y\) is isolated, and the problem is solved.