Problem 64 - Introductory Algebra for College Students

Question

Graph each equation in a rectangular coordinate system. $$2 x-3 y=6$$

Step-by-Step Solution

Answer

The y-intercept when \(x = 0\) is \(-2\). The x-intercept, when \(y = 0\), is \(3\). Thus, these points are \((0, -2)\) and \((3, 0)\). Plot these two points and draw the line passing through them to graph the equation.

Step 1: Calculate the y-intercept

The y-intercept is found by setting \(x = 0\) in the equation and solving for \(y\). So, put \(x = 0\) into \(2x - 3y = 6\).

Step 2: Calculate the x-intercept

The x-intercept is found by setting \(y = 0\) in the equation and solving for \(x\). So, put \(y = 0\) into \(2x - 3y = 6\).

Step 3: Plot the points and draw the line

Plot the y-intercept and the x-intercept on the rectangular coordinate system and draw a straight line passing these two points. This line represents the graph of the equation.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also called the Cartesian coordinate system, is a two-dimensional plane that is fundamental to graphing equations. It consists of two perpendicular axes: the horizontal axis, known as the x-axis, and the vertical axis, known as the y-axis. These axes intersect at a point called the origin, which has coordinates (0, 0). Each point on the plane is defined by an ordered pair of numbers (x, y), where 'x' represents the horizontal position and 'y' represents the vertical position.

When graphing linear equations, the rectangular coordinate system allows us to visualize the relationship between two variables. For instance, if we have the equation of a line like in this exercise, graphing it on this system helps us understand how changes in one variable affect the other. It's important to become familiar with plotting points on this grid, as it is a foundational skill for more advanced mathematics.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It occurs when the value of x is zero in the equation. To find the y-intercept, you substitute x = 0 in the linear equation and solve for y. For example, in the given equation \(2x - 3y = 6\), set \(x = 0\) to find the y-intercept:

• Substitute: \(2(0) - 3y = 6\)
• Simplify: \(-3y = 6\)
• Solve for y: \(y = -2\)

Thus, the y-intercept is the point \((0, -2)\). This point provides one of the crucial positions needed to sketch the line on the graph, offering a starting place for how the line intersects the y-axis.

X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. To determine this, you let y equal zero in the equation and solve for x. This method is similar to finding the y-intercept but focuses on where the line meets the horizontal axis. To illustrate, let's find the x-intercept for the equation \(2x - 3y = 6\):

• Substitute: \(2x - 3(0) = 6\)
• Simplify: \(2x = 6\)
• Solve for x: \(x = 3\)

Hence, the x-intercept is the point \((3, 0)\). Understanding the x-intercept is vital because it tells you the exact spot where the line hits the x-axis, helping to shape the complete picture of the line's path within the rectangular coordinate system.