Problem 33 - Contemporary Precalculus

Question

Find the equation of the line through the given points. $$(0,-5) and (-3,-2)$$

Step-by-Step Solution

Answer

Question: Given two points (0, -5) and (-3, -2), find the equation of the line passing through these points. Answer: The equation of the line is y = -x - 5.

Step 1: Find the Slope of the Line

Given points are \((0,-5)\) and \((-3,-2)\). To find the slope (m) of the line, we'll use the formula: $$m = \frac{y2-y1}{x2-x1}$$ In this case, \((x1, y1) = (0, -5)\) and \((x2, y2) = (-3, -2)\). Now, plugging in values, we get: $$m = \frac{-2 - (-5)}{-3 - 0}$$

Step 2: Simplify the Slope

Simplify the slope we found in Step 1: $$m = \frac{-2 + 5}{-3}$$ $$m = \frac{3}{-3}$$ $$m = -1$$

Step 3: Use the Point-Slope Form to Write the Equation

Now that we have found the slope, we can use the point-slope form to write the equation of the line. The point-slope form is: $$(y - y1) = m(x - x1)$$ Using the slope found in step 2, \(m = -1\), and one of the given points \((0, -5)\), we get: $$(y - (-5)) = -1(x - 0)$$

Step 4: Simplify the Equation

Finally, simplify the equation found in the previous step: $$y + 5 = -1x$$ $$y = -x - 5$$ Now we have found the equation of the line through the given points: $$y=-x-5$$

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep the line is, and it's often represented by the letter 'm'. When looking at the coordinates of two points on the line, you can calculate the slope using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
In simple terms, you subtract the y-coordinate of the first point from the y-coordinate of the second point (the rise), and divide it by the subtraction of the x-coordinate of the first point from the x-coordinate of the second point (the run). A positive slope means the line is inclined upwards, while a negative slope indicates it's inclined downwards. When the slope is zero, the line is horizontal, and if the slope is undefined, the line is vertical.

• Rise over run: Remembering this phrase can help you recall how to compute the slope.
• Positive vs. Negative Slope: A positive slope indicates an increasing line from left to right, while a negative slope indicates a decreasing line.
• Zero and Undefined Slopes: Horizontal lines have a slope of 0, and vertical lines have an undefined slope, often thought of as infinitely steep.

For the given points \( (0,-5) \) and \( (-3,-2) \) in our exercise, the slope calculation would yield a negative result, indicating a downward incline from left to right.

Point-Slope Form

When you have a point and a slope, the point-slope form is an efficient way to write the equation of a line. The general form is \( (y - y1) = m(x - x1) \) where \( m \) is the slope and \( (x1, y1) \) are the coordinates of a point on the line. In practical terms, this form emphasizes how the y-value on the line changes with respect to \( x \) and a specific point on the line.

• Why point-slope form: It's especially useful when you know one point on the line and the slope but don't have the line's y-intercept.
• Writing equations: This form allows you to write an equation quickly since it only requires plugging in two values: the slope and the coordinates of a single point.

Using the point-slope form, you can see the relationship between the line's vertical movement (changes in y) and its horizontal movement (changes in x). This is particularly important in understanding how the line behaves across the coordinate system. For the points mentioned in the exercise, once we have the slope, we select either point as \( (x1, y1) \) and proceed to formulate the equation.

Linear Equations

Linear equations are fundamental in algebra and represent lines on a Cartesian coordinate system. They have the standard form \( y = mx + b \) where \( m \) is the slope of the line and \( b \) is the y-intercept—the point where the line crosses the y-axis. The beauty of linear equations lies in their simplicity and the linearity — every x-value has one and only one y-value associated with it.

• Intercepts: The y-intercept is the starting point of the line on the y-axis when \( x=0 \). The x-intercept can also be found where the line crosses the x-axis and can be calculated by setting \( y=0 \) in the equation and solving for \( x \).
• Slope-Intercept Form: This is a specific type of linear equation where the equation is solved for \( y \) and is helpful for quickly graphing a line since you can pinpoint the y-intercept and use the slope to find another point.

Simplifying a point-slope form equation will often lead to the slope-intercept form, making it easier to draw the graph of the line. In the given exercise, after using the point-slope form with our found slope and a chosen point, we simplify to arrive at the equation in slope-intercept form: \( y = -x - 5 \) which tells us the line slopes downwards to the right with a y-intercept at \( -5 \).