Problem 57 - Algebra 2

Question

Explain why the graph of \(x+3 y=0\) has only one intercept.

Step-by-Step Solution

Answer

The line only has one intercept at the origin (0,0) where both x and y-intercepts coincide.

Step 1: Understanding the Equation of a Line

The equation given is a linear equation of the form \(x + 3y = 0\). Linear equations generally describe straight lines on a coordinate plane.

Step 2: Finding Intercepts

Intercepts are the points where the line crosses the axes. To find the x-intercept, set \(y = 0\) and solve for \(x\), and to find the y-intercept, set \(x = 0\) and solve for \(y\).

Step 3: Calculating the x-intercept

For the x-intercept, set \(y = 0\) in the equation, resulting in \(x + 3(0) = 0\), thereby \(x = 0\). Thus, the x-intercept is at the origin (0,0).

Step 4: Calculating the y-intercept

For the y-intercept, set \(x = 0\), which gives \(0 + 3y = 0\), thus \(y = 0\). This means the y-intercept is also at the origin (0,0).

Step 5: Analyzing Coinciding Intercepts

Notice that both the x-intercept and y-intercept occur at the same point, the origin (0,0). This means there is only one intercept point on this line.

Key Concepts

Intercepts

Intercepts are crucial in understanding how a line interacts with the axes on a coordinate plane. They denote the points where the line meets either the x-axis or the y-axis. An x-intercept occurs where the line crosses the x-axis. Here, the value of the coordinate \(y\) is zero. Similarly, a y-intercept exists where the line crosses the y-axis, with the coordinate \(x\) being zero.
For the equation \(x + 3y = 0\), the intercepts are found by substituting values and solving for the other variable. Often, you will find different points for x and y-intercepts, but it's possible for both to be the same point as in this case where the intercept occurs at the origin (0,0).
Such coinciding intercepts indicate the line passes through the origin, simplifying the interpretation of the line's graph.

Coordinate Plane

The coordinate plane is a two-dimensional space defined by the intersection of two perpendicular lines, known as the x-axis and y-axis. This grid allows us to precisely locate points using coordinates, denoted as \((x, y)\).

• The horizontal line is the x-axis.
• The vertical line is the y-axis.
• The point where they cross is called the origin.

When graphing linear equations, the coordinate plane is a valuable tool. Each solution to a linear equation represents a point on this plane. By connecting points of solutions, you depict the complete line described by the equation.

Origin

The origin is a fundamental concept on the coordinate plane. It serves as the reference point from which all other points are measured. Located at the intersection of the x and y axes, the coordinates of the origin are \((0,0)\).
The significance of the origin is highlighted in linear equations like \(x + 3y = 0\), where both intercepts coincide at the origin. In this scenario, the origin is the sole point of intersection between the line and both axes, illustrating its pivotal role in understanding graph behavior.
Recognizing that both intercepts occur at the origin simplifies the interpretation of how the line interacts with the coordinate plane.

Solve for Intercepts

Solving for intercepts involves a simple algebraic process applied to an equation to identify where a line meets the x and y axes. By setting respective variables to zero, you isolate the value of the remaining variable.
To solve the x-intercept of \(x + 3y = 0\), set \(y = 0\):

• \(x + 3(0) = 0\)
• \(x = 0\)

Similarly, to find the y-intercept, set \(x = 0\):

• \(0 + 3y = 0\)
• \(y = 0\)

Hence, both intercepts land at the origin, clearly demonstrating that solving for intercepts is an effective way to understand the interactions between a line and the axes in a linear equation.