Problem 15 - Introductory Algebra for College Students

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Do not graph the equation. \(7 x-9 y=0\)

Step-by-Step Solution

Answer

The \(x\)-intercept and the \(y\)-intercept of the equation \(7x - 9y = 0\) are both 0.

Step 1: Finding the x-intercept

Replace \(y\) with 0 in the equation to find the x-intercept. So we get \(7x - 9*0 = 0 \Rightarrow 7x = 0 \). Solving the equation for \(x\), we get \(x = 0\).

Step 2: Finding the y-intercept

Replace \(x\) with 0 in the equation to find the y-intercept. So we get \(7*0 - 9y = 0 \Rightarrow -9y = 0\). Solving the equation for \(y\), we get \(y = 0\).

Key Concepts

Understanding the x-intercept

The concept of the x-intercept is essential when analyzing linear equations. It refers to the point where a graph crosses the x-axis. Therefore, the y-coordinate of this point is always zero, which means we set \( y = 0 \) in the equation to find the x-intercept.
Let's consider the equation given in the exercise: \(7x - 9y = 0\). To find the x-intercept, we substitute 0 for \( y \) in the equation. This action simplifies the equation to \( 7x = 0 \), and solving for \(x\) gives \( x = 0 \).
Key points to remember about finding the x-intercept:

• Set \( y \) to zero in the equation.
• Solve for \( x \) to find the x-intercept.
• The x-intercept is a point \((x,0)\) on the graph.

Decoding the y-intercept

The y-intercept is another vital component, representing the point where a graph crosses the y-axis. At this intercept, the x-coordinate is zero. This means we substitute \( x = 0 \) in the equation to find the y-intercept.
From the given equation \(7x - 9y = 0\), by replacing \( x \) with 0, we have \(-9y = 0\). Solving this equation gives \( y = 0 \). So, the y-intercept for this linear equation is at the point \((0,y)\), specifically \((0,0)\).
Key points to consider when finding the y-intercept:

• Set \( x \) to zero in the equation.
• Solve for \( y \) to identify the y-intercept.
• The y-intercept occurs at a point \((0,y)\).

Basics of Linear Equations

Linear equations are equations of the first degree; they can be expressed in the form \(ax + by = c\). In these equations, every graph forms a straight line when plotted. This specific equation, \(7x - 9y = 0\), is a linear equation representing a line through the origin.
The steps to finding intercepts, as shown with the equation, are:

• Identify and rearrange the equation to make finding intercepts straightforward.
• Zero out variables to find specific intercept values.
• Linear equations typically have at most one x-intercept and one y-intercept.

Understanding and analyzing linear equations are crucial in identifying various intercepts, which helps in visualizing the equation's graph and its behavior on the coordinate plane.