Problem 68 - College Algebra Essentials

Question

Use intercepts to graph the each equation. $$6 x-9 y-18=0$$

Step-by-Step Solution

Answer

The x-intercept of the graph is 3, and the y-intercept is -2. Therefore, plot the points (3, 0) and (0, -2) and then draw a straight line passing through these points.

Step 1: Find the x-intercept

To find the x-intercept, set \(y=0\) in the equation and solve for \(x\). In the given equation, if you set \(y=0\), you'll have \(6x - 9*0 -18 = 0\), which simplifies to \(6x = 18\). If you solve this for \(x\), you'll get \(x = 3\). Therefore, the x-intercept is \(3\).

Step 2: Find the y-intercept

To find the y-intercept, set \(x=0\) in the equation and solve for \(y\). In the given equation, if you set \(x=0\), you'll have \(6*0 - 9y -18 = 0\), which simplifies to \(-9y = 18\). If you solve this for \(y\), you'll get \(y = -2\). Therefore, the y-intercept is \(-2\).

Step 3: Plot the intercepts and draw the line

Now that you have the x-intercept \(3\) and the y-intercept \(-2\), plot these points on the coordinate plane. The x-intercept is a point on the x-axis and hence its y-coordinate is zero. So, the x-intercept corresponds to the point (3, 0) and the y-intercept corresponds to the point (0, -2). After plotting these points, draw a straight line passing through these two points. This line represents the graph of the equation.

Key Concepts

X-Intercept

The x-intercept of a line is the point at which the line crosses the x-axis on the coordinate plane. Understanding the x-intercept gives us a valuable starting point for graphing linear equations. To find the x-intercept, we set the value of \(y\) to zero and solve the equation for \(x\). This represents the scenario where the line meets the x-axis, and the y-coordinate of any point on the x-axis is, by definition, zero.

In our exercise, to find the x-intercept for the equation \(6x - 9y - 18 = 0\), we substitute \(y\) with 0 and simplify the equation to \(6x = 18\), which upon solving gives us \(x = 3\). This means the line intersects the x-axis at the point \((3, 0)\), leaving us with a straightforward visual cue when sketching the graph.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. To find it, we set the x-coordinate to zero. This reflects the point where the line will pass through the y-axis, which becomes helpful in plotting the line on a graph.

For the given problem \(6x - 9y - 18 = 0\), we set \(x\) equal to zero and solve for \(y\), giving us the equation \(-9y = 18\). Solving for \(y\) yields \(y = -2\), indicating that the y-intercept of the line is the point \((0, -2)\). With the y-intercept found, we have another critical point that assists us in graphing the linear equation accurately.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graph equations, plot points, and visualize geometric shapes. It consists of two number lines: the horizontal axis (x-axis) and the vertical axis (y-axis) that intersect at a point called the origin (0,0). Each point on this plane is identified by an ordered pair of numbers known as coordinates.

The x-coordinate shows a point's horizontal position, while the y-coordinate shows its vertical position. When graphing an equation, like our linear equation \(6x - 9y - 18 = 0\), the x-intercept and y-intercept provide two specific points that we can plot on this plane, which we can then connect to form the line that represents the equation.

Solving Linear Equations

Solving linear equations involves finding the values of the variables that make the equation true. Linear equations are of the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. The solutions to these equations can often be graphically represented as a straight line on the coordinate plane.

By setting one variable to zero and solving for the other, we find the intercepts, which are crucial points that anchor the line on the graph. Solving for the x-intercept gives us the point on the line where \(y=0\), and the y-intercept gives us the point where \(x=0\). Once these points are plotted, drawing a line through them graphically represents all possible solutions to the equation.