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Chapter 5: Problem 65

Sketch the graph of the line whose \(x\) -intercept is 3 and whose \(y\) -intercept is 5

Short Answer

Expert verified
The equation of the line is \(y = -\frac{5}{3}x + 5\).

Step by step solution

01

Identify the Intercepts

The problem gives the x-intercept as 3 and the y-intercept as 5. These intercepts are points on the graph where the line crosses the x-axis and y-axis, respectively.
02

Write the Points

The x-intercept of 3 corresponds to the point (3, 0). The y-intercept of 5 corresponds to the point (0, 5).
03

Determine the Slope

Use the formula for slope, \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the points (3, 0) and (0, 5), we get: \[ \text{slope} = \frac{5 - 0}{0 - 3} = \frac{5}{-3} = -\frac{5}{3} \]
04

Write the Equation of the Line

Using the slope-intercept form of the equation of a line, \(y = mx + b\), with slope \(m = -\frac{5}{3}\) and y-intercept \(b = 5\): \[ y = -\frac{5}{3}x + 5 \]
05

Sketch the Graph

Plot the points (3, 0) and (0, 5) on a graph. Draw a line through these points. The line will have a downward slope, reflecting the negative slope of -\(\frac{5}{3}\).

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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 of a line is the point where the line crosses the x-axis. At this point, the y-coordinate is always 0.
For the given exercise, the x-intercept is 3. This means the line crosses the x-axis at the point (3, 0).
To find the x-intercept of any linear equation, set y to 0 and solve for x.
For example, in the equation from the exercise, \(y = -\frac{5}{3}x + 5\), set y to 0 and solve for x:
\[0 = -\frac{5}{3}x + 5\]
Solving for x gives the x-intercept.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. Here, the x-coordinate is always 0.
In the given problem, the y-intercept is 5. So, the line crosses the y-axis at (0, 5).
To find the y-intercept of a linear equation, set x to 0 and solve for y.
In the exercise鈥檚 equation \(y = -\frac{5}{3}x + 5\), set x to 0:
\[y = -\frac{5}{3}(0) + 5\]
This confirms the y-intercept of 5.
slope
The slope of a line measures its steepness and direction. It is represented by the letter m.
In the slope formula \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\), substitute the given points (3, 0) and (0, 5).
The calculation is:
\[ \text{slope} = \frac{5 - 0}{0 - 3} = -\frac{5}{3} \]
The negative value means the line slopes downward from left to right.
slope-intercept form
The slope-intercept form of a linear equation is useful for graphing lines quickly. It is written as \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept.
From the given exercise, we found the slope \(-\frac{5}{3}\) and the y-intercept 5.
Substituting these into the slope-intercept form gives us:
\[ y = -\frac{5}{3}x + 5 \]
This equation can now be used to graph the line and helps easily identify the slope and y-intercept directly.

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