Problem 63 - Elementary Algebra Within Reach

Question

In Exercises \(63-66\), sketch the graph of the equation. $$ 2 x+4 y=8 $$

Step-by-Step Solution

Answer

The x-intercept is at (4,0) and the y-intercept is at (0,2). The graph of the equation is a straight line passing through these points.

Step 1: Find the x-intercept

Set y to 0 and solve equation \(2x + 4y = 8\) for x. This gives the x-intercept: solve \(2x + 4(0) = 8\) for x, which results in x = 4. Therefore, the x-intercept is at point (4,0).

Step 2: Find the y-intercept

Set x to 0 and solve equation \(2x + 4y = 8\) for y. This gives the y-intercept: solve \(2(0) + 4y = 8\) for y, which results in y = 2. Therefore, the y-intercept is at point (0,2).

Step 3: Sketch the graph

Plot the points (4,0) and (0,2) on a coordinate system. Draw a straight line passing through these points. This line is the graph of the given equation.

Key Concepts

X-Intercept

Understanding the x-intercept of a linear equation is crucial to graphing the equation accurately. The x-intercept is a point where the line cuts the x-axis on the coordinate system. More specifically, it is the value of x when y is equal to zero. To find this intercept, you simply set y to zero in the equation and solve for x. For instance, in the equation 2x + 4y = 8, setting y to zero gives you 2x = 8. Solving for x, you'll find that x equals 4, indicating that the x-intercept is at the point (4,0). Plotting this point on a graph will help you begin to form the shape of the line.

Y-Intercept

In contrast to the x-intercept, the y-intercept refers to the point where the line meets the y-axis. To find the y-intercept, you set the x value to zero and solve for y. In the same equation 2x + 4y = 8, for instance, if we plug in x = 0 we get 4y = 8. By dividing both sides by 4, we determine that y equals 2, and thus the y-intercept is (0,2). It's important to note that every straight line on a coordinate system will cross the y-axis at one point, which is represented as (0,y). The y-intercept provides you with a second point to plot on your graph, creating a line when connected to the x-intercept.

Coordinate System

The coordinate system is the backbone of graphing where every point is defined by a pair of numbers corresponding to its position on the x (horizontal) and y (vertical) axes. This system allows for precise plotting and a clear visual representation of equations. Ideally laid out in a grid format, it consists of four quadrants with the x and y-axes dividing them. As you plot points on the graph, such as the x and y-intercepts found earlier, you are utilizing the coordinate system to visually express mathematical relationships. This transformation of algebraic expressions into geometrical figures aids in interpreting and solving equations.

Plotting Points

Plotting points is a fundamental skill in graphing linear equations. It involves placing dots on the coordinate system where each dot represents a solution to the equation. Start by identifying your points, like the x-intercept (4,0) and y-intercept (0,2) we found. Then, simply draw a dot where the x value corresponds to the horizontal position and the y value corresponds to the vertical position on the graph. Connecting these dots with a straight line forms the graph of the equation. Remember, the points must be accurate to ensure the integrity of the graph; even slight errors in plotting can result in incorrect interpretation of the line's slope and position.