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Chapter 1: Problem 28

Graph each line by hand. Give the \(x\) - and y-intercepts. $$x+y=4$$

Short Answer

Expert verified
The x-intercept is (4, 0) and the y-intercept is (0, 4). Plot these points and draw the line.

Step by step solution

01

Rearrange for y-intercept

To find the y-intercept, rearrange the given equation to make y the subject. Start with the equation: \(x+y=4\), and solve for y: \(y=4-x\). At the y-intercept, set \(x=0\). Substitute \(x=0\) into the equation: \(y=4-0=4\). Thus, the y-intercept is \( (0, 4) \).
02

Rearrange for x-intercept

To find the x-intercept, set \(y=0\) in the original equation: \(x+y=4\). Substitute \(y=0\) into the equation: \(x+0=4\), which simplifies to \(x=4\). Thus, the x-intercept is \((4, 0)\).
03

Plot the intercepts

Using the intercepts found, plot the points \( (0, 4) \) and \( (4, 0) \) on a graph. The y-intercept at \( (0, 4) \) is where the line crosses the y-axis, and the x-intercept at \( (4, 0) \) is where the line crosses the x-axis.
04

Draw the line

Connect the plotted points \( (0, 4) \) and \( (4, 0) \) with a straight edge. Extend the line across the graph, ensuring that it passes through both intercepts, to accurately represent the line described by the equation \(x+y=4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the x-intercept
The x-intercept of a linear equation is a crucial point where the graph crosses the x-axis. To find this intercept for any equation, you simply set the value of y to zero and solve for x.

In the equation provided, \(x + y = 4\), we identify the x-intercept by substituting \(y = 0\) into the equation. This simplifies to \(x + 0 = 4\), which further reduces to \(x = 4\).

Therefore, the x-intercept is the point \((4, 0)\). Here, the graph touches or crosses the x-axis at the coordinate where x equals 4, and y is zero. This point is fundamental when plotting the line, as it guides us in shaping how the line should stretch across the graph.
Discovering the y-intercept
The y-intercept is another essential component of graphing linear equations. It is the point where the line crosses the y-axis. To discover the y-intercept, you set \(x = 0\) in the equation and solve for y.

For our equation \(x + y = 4\), setting \(x = 0\) leads to \(0 + y = 4\). Therefore, \(y = 4\).

Thus, the y-intercept is at the point \((0, 4)\). This point indicates where the graph intersects the y-axis, and is a vital reference point when drawing the line on the graph, setting the exact path it should take vertically.
The Art of Plotting Points
Plotting points on a graph is a straightforward yet essential skill in graphing linear equations accurately. It involves marking specific coordinates, like intercepts, which serve as a guide for sketching the line.

For the given equation \(x + y = 4\), we have identified two key points: the x-intercept \((4, 0)\) and the y-intercept \((0, 4)\).

To plot these points:
  • Start with the x-intercept \((4, 0)\), place a point at where \(x = 4\) on the x-axis and \(y = 0\).
  • Then, proceed to the y-intercept \((0, 4)\), marking a point where \(x = 0\) and \(y = 4\) on the y-axis.
Once these points are plotted, connecting them with a straight line will draw a clear representation of the equation.

This process not only visualizes the relationship described by the equation but also confirms the linear path determined by both intercepts.

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Most popular questions from this chapter

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