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Chapter 3: Problem 50

In the following exercises, graph using the intercepts. $$ 3 x+y=3 $$

Short Answer

Expert verified
Plot (1, 0) and (0, 3) then draw a line through them.

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Starting with the equation 3x + y = 3, set y=0: 3x + 0 = 3. Thus, x = 1. So, the x-intercept is (1, 0).
02

- Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. Starting with the equation 3x + y = 3, set x=0: 3(0) + y = 3. Thus, y = 3. So, the y-intercept is (0, 3).
03

- Plot the intercepts

Plot the points (1, 0) and (0, 3) on a graph. These points are where the line crosses the x-axis and y-axis respectively.
04

- Draw the line

Draw a straight line through the points (1, 0) and (0, 3) to complete the graph of the equation 3x + y = 3.

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Key Concepts

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

x-intercept
The x-intercept is the point where a graph crosses the x-axis. To find it, we set y to 0 and solve for x in the equation. For the equation 3x + y = 3, we have:
Set y = 0: 3x + 0 = 3
Therefore, x = 1
So, the x-intercept is (1, 0). This tells us that the line meets the x-axis at the point (1, 0). Understanding x-intercepts helps in visualizing how the line behaves with respect to the x-axis, giving us a starting point to draw our graph.
y-intercept
The y-intercept is the point where a graph crosses the y-axis. To find it, we set x to 0 and solve for y in the equation. For the equation 3x + y = 3, we have:
Set x = 0: 3(0) + y = 3
Thus, y = 3
So, the y-intercept is (0, 3). This indicates that the line meets the y-axis at the point (0, 3). Finding the y-intercept helps us understand the initial vertical position of the line, aiding in the accurate graphing of the equation.
plotting points
After calculating the intercepts, it's time to plot these points on a graph. For the equation 3x + y = 3, the intercepts are (1, 0) and (0, 3).
1. Look for the point (1, 0) on the graph:
- Move 1 unit to the right along the x-axis.
- No vertical movement since the y-value is 0.
2. Locate the point (0, 3):
- Stay at the origin for the x-value 0.
- Move 3 units up along the y-axis.
Marking these points accurately ensures that the line reflecting the equation will be drawn correctly.
drawing lines
Drawing the line is the final step in graphing the linear equation. Using the plotted points (1, 0) and (0, 3), we draw a straight line through them. Here鈥檚 how:
1. Place a ruler or a straight edge on the graph to align with the points (1, 0) and (0, 3).
2. Draw a line passing through both points.
This line represents the linear equation 3x + y = 3. Drawing lines in this structured way ensures that the graph accurately depicts the relationship defined by the equation. Understanding this step consolidates the entire process of graphing linear equations using intercepts.

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