Problem 71 - College Algebra Essentials

Question

Use intercepts to graph the each equation. $$8 x-2 y+12-0$$

Step-by-Step Solution

Answer

The x-intercept is \((-1.5, 0)\) and y-intercept is \((0,6)\). By connecting these points, the equation \(8x - 2y + 12 = 0\) is represented as a straight line on the graph.

Step 1: Finding the x-intercept

To find the x-intercept, set the parameter \(y\) equal to 0 in the equation and solve for \(x\). The equation becomes \(8x + 12 = 0\). The solution is \(x = -\frac{3}{2}\).

Step 2: Finding the y-intercept

To find the y-intercept, set the parameter \(x\) equal to 0 in the equation and solve for \(y\). The equation becomes \(-2y + 12 = 0\). The solution is \(y = 6\).

Step 3: Plotting the Graph

Start by plotting the x-intercept at \((-1.5,0)\) and then plot the y-intercept at \((0, 6)\). Connect these points with a straight line to represent the equation \(8x - 2y + 12 = 0\) on the graph.

Key Concepts

X-Intercept

The x-intercept is a key concept when dealing with linear equations and their graphs. It's the point where the graph of the equation crosses the x-axis. At this point, the value of y is always 0. For the given equation \( 8x - 2y + 12 = 0 \), you find the x-intercept by setting \( y = 0 \).

• Substitute \( y = 0 \) into the equation: \( 8x + 12 = 0 \).
• Solve for \( x \): \( 8x = -12 \) gives \( x = -\frac{3}{2} \).

This means the x-intercept is at the point \( (-1.5, 0) \). Understanding this concept is crucial because it gives you a specific point to start graphing a linear equation. It's one of the two critical points needed to graph a line using intercepts.

Y-Intercept

The y-intercept of a linear equation is where the graph crosses the y-axis. Here, the value of \( x \) is always 0, which simplifies finding this intercept.

• To find the y-intercept of \( 8x - 2y + 12 = 0 \), set \( x = 0 \).
• The equation simplifies to \( -2y + 12 = 0 \).
• Solve for \( y \): \( -2y = -12 \) gives \( y = 6 \).

Thus, the y-intercept is \( (0, 6) \). This point provides another anchor for drawing the graph. Together with the x-intercept, you can draw the line representing the equation, greatly simplifying the process of graphing.

Coordinate Plane

A coordinate plane is a two-dimensional surface formed by two perpendicular lines known as axes. The horizontal axis is called the x-axis, while the vertical axis is the y-axis. These axes intersect at the origin, denoted as \( (0,0) \).

• Each point on the plane can be specified by a pair of numbers, \( (x,y) \), known as coordinates.
• Coordinates indicate the position on the plane relative to the origin.

When you plot the x-intercept \( (-1.5, 0) \) and the y-intercept \( (0, 6) \) on the coordinate plane, you can connect these points to graph the linear equation. The coordinate plane is essential as it provides a visual representation of equations and solutions, allowing us to clearly see the relationships and intersections of lines.

Linear Equation Solutions

Linear equations, like \( 8x - 2y + 12 = 0 \), have solutions that can be plotted on a graph as a straight line. The solutions are the set of all points \( (x, y) \) that satisfy the equation.

• Each solution corresponds to a point on the line.
• The process of solving involves finding meaningful points such as intercepts, which can be graphed to identify the line.

In this exercise, the x-intercept and y-intercept are both solutions that help us establish the line visually. By connecting these intercepts, you draw the line that represents all potential solutions for the equation. Understanding how to find and use these solutions is crucial for graphing and interpreting linear equations accurately.