Problem 35 - Algebra for College Students

Question

Solve each equation for \(y\). $$\frac{1}{2} x-\frac{1}{3} y=1$$

Step-by-Step Solution

Answer

$$y = -3 + \frac{3}{2} x$$

Step 1: Identify the given equation

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Step 2: Isolate terms involving y

Rewrite the equation to isolate terms involving \(y\) on one side. To achieve this, subtract \(\frac{1}{2} x\) from both sides of the equation: $$ -\frac{1}{3} y = 1 - \frac{1}{2} x $$.

Step 3: Solve for y

To solve for \(y\), multiply both sides of the equation by \(-3\): $$ y = -3 \times (1 - \frac{1}{2} x). $$ Simplify the expression: $$ y = -3 + \frac{3}{2} x. $$

Key Concepts

Isolate Variable

When solving linear equations, one key step is to isolate the variable you need to solve for. In this case, we want to isolate the variable \(y\). This means getting \(y\) by itself on one side of the equation. To start, identify terms in the equation that contain \(y\), and take steps to move other terms to the opposite side. For example, if you have \( \frac{1}{2} x - \frac{1}{3} y = 1 \), the first step is to move the \( \frac{1}{2} x \) term to the other side by subtracting it from both sides:

Linear Equation

A linear equation is an equation where each term is either a constant or the product of a constant and a single variable. These equations can be written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In the given exercise, our linear equation is \( \frac{1}{2} x - \frac{1}{3} y = 1 \). This means that we have two terms involving variables and constants, both raised to the first power, and it will graph as a straight line. Our goal is to simplify this equation to solve for the variable \(y\).

Algebraic Manipulation

Algebraic manipulation involves using operations to simplify and solve equations. It often includes actions like adding, subtracting, multiplying, or dividing both sides of the equation by the same number or expression. In our example, after subtracting \( \frac{1}{2} x \) from both sides of \( \frac{1}{2} x - \frac{1}{3} y = 1 \), we get: $$ - \frac{1}{3} y = 1 - \frac{1}{2} x. $$ To isolate \(y\), we multiply both sides by -3, the reciprocal of \( - \frac{1}{3} \). This action simplifies the equation to: $$ y = -3 \times (1 - \frac{1}{2} x). $$

Solving for y

Solving for \(y\) means we need to find an expression or value for \(y\) in terms of the other variable (or constants). To solve for \(y\), we simplify the right side of the equation from earlier: $$ y = -3 \times (1 - \frac{1}{2} x). $$ Distribute the -3 through the parentheses: $$ y = -3 + \frac{3}{2} x. $$ Now we have \(y\) expressed solely in terms of \(x\), which completes our solution. The final solution for the equation \( \frac{1}{2} x - \frac{1}{3} y = 1 \) is: $$ y = \frac{3}{2} x - 3. $$