Problem 69 - Precalculus Essentials

Question

Use intercepts to graph equation. $$2 x+3 y+6=0$$

Step-by-Step Solution

Answer

The x-intercept of the given equation is (-3,0) and y-intercept is (0,-2). Plot these points on a Cartesian plane and draw the line which passes through these points to graph the given equation.

Step 1: Finding the x-intercept

In order to find the x-intercept, set \(y=0\) in the equation and solve for x. Thus, the equation becomes \(2x + 3(0) + 6 = 0 \rightarrow 2x + 0 + 6 = 0 \rightarrow 2x = -6\). Solving this equation for 'x' we get \(x = -6/2 = -3\). Thus, the x coordinate of the x-intercept is -3. So the x-intercept is (-3, 0)

Step 2: Finding the y-intercept

In order to find the y-intercept, set \(x=0\) in the equation and solve for y. The equation becomes \(2(0) + 3y + 6 = 0 \rightarrow 0 + 3y + 6 = 0 \rightarrow 3y= -6 \). Solving this equation for 'y' we get \(y = -6/3 = -2\). Thus, the y coordinate of the y-intercept is -2. So the y-intercept is (0, -2)

Step 3: Plotting the graph

Using the x and y intercepts obtained in previous steps, plot these points on the Cartesian plane. Draw a line passing through these points. This line represents the equation \(2x + 3y + 6 = 0\)

Key Concepts

Understanding Intercepts

When graphing linear equations, intercepts are key points on the graph that intersect the axes. The intercepts include both x-intercept and y-intercept. They provide a quick way to graph a line without needing to calculate the slope or any other points.

• The x-intercept is where the line crosses the x-axis. At this point, the value of y is always zero.
• The y-intercept is where the line crosses the y-axis. At this point, the value of x is always zero.

Since linear equations like \(2x + 3y + 6 = 0\) form straight lines, knowing these intercepts allows us to accurately draw the graph. Using intercepts is a simple way to visualize and understand the relationship a line has with the axes.

The X-Intercept

The x-intercept of a line is where the graph intersects the x-axis. At this point, the y-coordinate is zero, which means the line is horizontal to the x-axis at this spot.

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). From our example:
\[2x + 3(0) + 6 = 0\]
This simplifies to:
\[2x + 6 = 0\]
Solving for \(x\), you get:
\[2x = -6\]
\[x = -3\]
Therefore, the x-intercept is \((-3, 0)\).

Understanding the x-intercept helps in graphing because it tells you exactly where the line crosses the x-axis, allowing you to pinpoint one part of the line's path through the grid.

The Y-Intercept

The y-intercept is the point where the line crosses the y-axis, which means the x-coordinate is zero at this intersection point.

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). Using our example, you start with:
\[2(0) + 3y + 6 = 0\]
This simplifies to:
\[3y + 6 = 0\]
Solving for \(y\), you get:
\[3y = -6\]
\[y = -2\]
Thus, the y-intercept is \((0, -2)\).

Knowing the y-intercept is just as important as knowing the x-intercept when graphing a line. Together, they allow for an accurate understanding of where the line cuts through the axes, making it possible to draw the line on a graph.