Question

Label any intercepts and sketch a graph of the plane. $$ x+2 y=4 $$

Short answer

The graph of the plane for equation \(x + 2y = 4\) will intersect with the x-axis at point (4,0) and with the y-axis at point (0,2). The slope of the plane is -1/2.

Step-by-step solution

Rearrange equation

The first step is to solve the given equation for y in terms of x. This will give a clearer picture of the slope and y-intercept of the plane. The equation \(x + 2y = 4\) can be rearranged to \(y = \frac{4 - x}{2}\). So, the slope of the plane is -1/2 and y-intercept is 2.

Calculate intercepts

Next, calculate the x and y intercepts. The x-intercept is found when y=0, which gives \(x = 4\). The y-intercept is found when x=0, which also gives \(y = 2\). So, the intercepts are at points (4,0) and (0,2).

Sketch the graph

Finally, sketch the graph of the plan using the intercepts and the slope. The plane will cross the x-axis at point (4,0) and the y-axis at point (0,2). The slope of the plane is -1/2, which means the plane will decline at a rate of 1 unit of y for every 2 units of x.

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for graphing linear equations. It is expressed as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) indicates the y-intercept. The beauty of this form lies in its ability to provide a clear picture of how the graph will look with just two pieces of information: the slope and the y-intercept. For the equation \( x + 2y = 4 \), converting it to slope-intercept form reveals that \( m = -\frac{1}{2} \) and \( b = 2 \). This indicates the line will tilt downwards as it moves from left to right (since the slope is negative) and will cross the y-axis at the point \( (0, 2) \).

X-Intercept

The x-intercept of a line is the point where it crosses the x-axis. At this point, the value of \( y \) is zero. To find the x-intercept from an equation, you would set \( y \) to zero and solve for \( x \). Taking our equation in slope-intercept form \( y = -\frac{1}{2}x + 2 \), when \( y = 0 \), \( x \) would be 4. Hence, the graph of this equation will intersect the x-axis at the point \( (4, 0) \). Knowing the x-intercept is an important step in graphing the line, as it gives a precise starting point from which to draw the line on the grid when sketching its graph.

Y-Intercept

Conversely, the y-intercept is found where the line crosses the y-axis, and this occurs when \( x = 0 \). For our example, setting \( x \) to zero in the slope-intercept form yields the point \( (0, 2) \) for the y-intercept. This is where you'll begin to plot the line on a graph. The y-intercept is not only a starting point for drawing the line but also provides insight into the line's vertical position on the graph. Since the y-intercept in our example is 2, it means the line will begin two units above the origin on the y-axis.

Sketching Graphs

With these intercepts in hand, sketching the graph of a linear equation becomes much easier. Begin by plotting the y-intercept on the graph at \( (0, 2) \). Next, plot the x-intercept at \( (4, 0) \). Now, draw a straight line connecting these two points. The slope of \( -\frac{1}{2} \) informs you that for every two units you move horizontally to the right, you must move one unit down. Continue to apply this pattern to draw your line in both positive and negative directions. By accurately plotting the intercepts and following the slope, you will create an exact representation of the equation on the graph. This method ensures a clear and visual understanding of the relationship described by the linear equation.