Problem 49 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the line. Graph the equation. Label the points where the line crosses the axes. $$ 2 x+4 y=16 $$

Step-by-Step Solution

Answer

The x-intercept is 8 and the y-intercept is 4

Step 1: Determine the x-intercept

The x-intercept is the value of x when y=0. We start by setting \(y = 0\) in the equation \(2x + 4y = 16\). After substituting \(y=0\) to the equation, we get \(2x = 16\). Then, solve for \(x\) by dividing by 2.

Step 2: Determine the y-intercept

The y-intercept is the value of \(y\) when \(x=0\). We start by setting \(x = 0\) in the equation \(2x + 4y = 16\). After substituting \(x=0\) to the equation, we get \(4y = 16\). Then, solve for \(y\) by dividing by 4.

Step 3: Plot the x and y intercepts

Plot the points where the line crosses the axes i.e., (x-intercept,0) and (0,y-intercept) on the graph, and link those two points with a straight line to represent the line for the given equation.

Key Concepts

x-intercept

An x-intercept is a point where a graph crosses the x-axis. This means that at the x-intercept, the y-coordinate is always zero. To find the x-intercept of a linear equation, you set the y value equal to zero and solve for x in the equation. This gives you the point \[(x, 0)\]For example, using the equation provided, \[2x + 4y = 16\]we set \( y = 0 \) and solve:\[2x + 4(0) = 16 \rightarrow 2x = 16 \rightarrow x = 8\]So, the x-intercept is at the point (8, 0). This is where the line will cross the x-axis. Finding the x-intercept is crucial as it provides one of the key points needed to sketch the line on a graph.

y-intercept

The y-intercept is a point where a graph crosses the y-axis. At the y-intercept, the x-coordinate is always zero. To determine the y-intercept from a linear equation, set x to zero and solve for y. This results in the point \[(0, y)\]Following this process with our equation, we have:\[2x + 4y = 16\]Set \( x = 0 \) and solve:\[2(0) + 4y = 16 \rightarrow 4y = 16 \rightarrow y = 4\]Thus, the y-intercept is at the point (0, 4). The y-intercept is vital because it provides the second necessary point to plot a straight line accurately. Including both the x and y intercepts gives a clear guide to understanding the line's trajectory.

Graphing Linear Equations

Graphing a linear equation involves plotting points and drawing the line that represents the equation. A straight line can be completely defined by two points, which in most cases will be the x-intercept and the y-intercept. Let's walk through the steps briefly:

• First, find the x-intercept by setting y = 0 and solve for x.
• Next, identify the y-intercept by setting x = 0 and solve for y.
• With these two points, plot them on the graph where the x-axis and y-axis intercepts occur.
• Draw a straight line through these points, extending it across the grid.

For example, with the intercepts (8, 0) and (0, 4) from our exercise, these points direct the line's path across the graph. The process of graphing lets you visualize the relationship that the linear equation represents, making it easier to analyze and understand. This graphical representation is especially beneficial for identifying how the line behaves as x and y values change, and how it may interact with other linear equations in a coordinate system.