Problem 73 - Calculus Volume 1

Question

Write the equation of the line satisfying the given conditions in slope- intercept form. \(x\) -intercept \(=5\) and \(y\) -intercept \(=-3\)

Step-by-Step Solution

Answer

The equation is \(y = \frac{3}{5}x - 3\).

Step 1: Understand intercepts

The x-intercept is where the line crosses the x-axis, which means the y-coordinate is 0. Thus, we have the point \((5, 0)\). The y-intercept is where the line crosses the y-axis, meaning the x-coordinate is 0. Thus, we have the point \((0, -3)\).

Step 2: Find the slope

To find the slope \(m\), use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) with the intercepts as points. Here, \((x_1, y_1) = (5, 0)\) and \((x_2, y_2) = (0, -3)\). Substitute these values into the formula: \[m = \frac{-3 - 0}{0 - 5} = \frac{-3}{-5} = \frac{3}{5}\]

Step 3: Use slope-intercept form

The slope-intercept form of the equation of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. From Step 2, the slope \(m = \frac{3}{5}\) and from Step 1, the y-intercept \(b = -3\). Substitute these values into the form: \[y = \frac{3}{5}x - 3\]

Step 4: Write the final equation

The equation in slope-intercept form, representing the line with the given intercepts, is \(y = \frac{3}{5}x - 3\).

Key Concepts

x-intercept

The x-intercept is a crucial point where a line crosses the x-axis. At this point, the value of the y-coordinate is always zero. Imagine a seesaw perfectly leveled on a playground; the center is the x-intercept. This makes it essential to understand and identify the x-intercept when dealing with linear equations. In our exercise, the x-intercept is given as 5. This means the line cuts through the x-axis at the point (5, 0). The x-intercept is a helpful clue in constructing the equation of a line because it tells us where the line starts horizontally.

• Visualize the line crossing the x-axis.
• Remember: at the x-intercept, the y-value is 0.

Understanding the x-intercept helps us move towards finding other crucial attributes of the line, such as the slope and the y-intercept.

y-intercept

The y-intercept is where the line crosses the y-axis, and at this point, the x-coordinate is always zero. Think of it as where the journey of the line begins vertically.In the context of the exercise, we are told that the y-intercept is -3. This indicates that the line crosses the y-axis at the point (0, -3).

• The y-intercept provides us with the 'b' value in the slope-intercept form equation, which is: \(y = mx + b\).
• In our case, \(b = -3\).

Knowing the y-intercept not only helps in plotting the line graphically but also in formulating the equation of the line directly once the slope is known. Always look for this intersection first when writing equations in slope-intercept form.

slope of a line

The slope of a line is a measure of its steepness and direction. It's like the incline of a hill - steep lines have higher slopes, while gentle lines have lower slopes.To calculate a line's slope, use two points on the line and the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Where:

• \((x_1, y_1)\) is the first point.
• \((x_2, y_2)\) is the second point.

In this problem, the points are the x-intercept (5, 0) and the y-intercept (0, -3). Using the formula:\[m = \frac{-3 - 0}{0 - 5} = \frac{-3}{-5} = \frac{3}{5}\]This means the slope of the line is \(\frac{3}{5}\), indicating a positive slope where the line rises to the right.The slope is the 'm' in the slope-intercept form \(y = mx + b\). Knowing the slope allows us to predict how the line moves and to write the equation accurately. Remember, a positive slope raises the line, and a negative slope lowers it.