Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the equation, if possible. $$y=\frac{3 x-1}{4 x}$$

Step-by-Step Solution

Answer

The x-intercept is \(\frac{1}{3}\) and there is no y-intercept.

Step 1: Find the x-intercept

Set \(y = 0\) and solve the equation for \(x\). \(0 = \frac{3x-1}{4x}\)As a rule, if \(\frac{a}{b} = 0\), then \(a = 0\). Apply this rule to get:\(3x - 1 = 0\)Then solving for \(x\) gives:\(x = \frac{1}{3}\)

Step 2: Find the y-intercept

Set \(x = 0\) and solve the equation for \(y\). \(y = \frac{3(0)-1}{4(0)}\)Substituting \(0\) into the equation gives:\(y = \frac{-1}{0}\)This is undefined, hence the graph of the equation does not intersect the y-axis, so there is no y-intercept.

Key Concepts

Solving Equations

To find the intercepts of an equation, we must solve it by substituting particular values for either variable. This approach stems from knowing specific points where the equation crosses either the x-axis or y-axis. For x-intercepts, replace

• \(y = 0\) and solve the resulting equation for \(x\).
• For y-intercepts, set \(x = 0\). Then solve for \(y\).

By solving \[y = \frac{3x - 1}{4x}\]for both of these conditions, you can discover where the equation's graph crosses the axes if it does. This method of solving equations helps identify the intercepts clearly and efficiently.

Graph of the Equation

Visualizing equations helps us understand intercepts. For the given equation, \[y = \frac{3x - 1}{4x},\]the graph represents the function in a coordinate system. The x and y-intercepts are specific points:

• The x-intercept occurs where the graph crosses the x-axis (when \(y = 0\)).
• If possible, the y-intercept occurs where the graph crosses the y-axis \((x = 0)\).

Plotting the graph can sometimes illustrate the behavior of a function, making it easy to see where it does or doesn't cross axes. Understanding the shape can provide insights into more complex aspects like asymptotes or undefined behavior.

Undefined

In mathematics, division by zero is undefined. This appears when solving for the y-intercept in equations where the denominator can be zero. For example, \[y = \frac{3(0)-1}{4(0)}\]results in \[\frac{-1}{0},\]which cannot be calculated because division by zero isn't possible. When encountering division by zero, it implies the expression has no real value and doesn't intersect where expected, such as the y-axis in this nuance.

Intercepts in Algebra

Intercepts are fundamental concepts in algebra, reflecting points where graphs meet the axes:

• The x-intercept shows where the graph crosses the x-axis. It represents the values of \(x\) when \(y = 0\) and can often be found by directly solving the equation.
• The y-intercept reveals where the graph intersects the y-axis. It's computed by finding the value of \(y\) when \(x = 0\), unless it results in undefined expression.

Properly understanding intercepts aids in interpreting graphs, analyzing functions, and solving more complicated algebraic tasks. They provide a quick snapshot of a function's behavior and are essential in applications spanning diverse fields.