Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an equation of the line that passes through the points. Use the slope- intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible). $$(1,0.6),(-2,-0.6)$$

Step-by-Step Solution

Answer

The equation of the line that passes through the given points is \(y = 0.4x + 0.2\).

Step 1: Calculate the Slope

The slope of a line passing through the points \((x1, y1)\) and \((x2, y2)\) is given by the formula \[m = \frac{y2 - y1}{x2 - x1}\]. Substituting (1, 0.6) for \((x1, y1)\) and (-2, -0.6) for \((x2, y2)\) into the formula gives us: \[m = \frac{-0.6 - 0.6}{-2 - 1} = 0.4\]

Step 2: Substitute the Slope and a Point into the Equation

Next, substitute the calculated slope and one of the given points into the slope-intercept form \(y = mx + b\). Using the point (1, 0.6), we get the equation \(0.6 = 0.4(1) + b\). Solve this equation for \(b\) to find the y-intercept.

Step 3: Solve for the y-Intercept

Solving the equation \(0.6 = 0.4(1) + b\) for \(b\) entails subtracting 0.4 from both sides of the equation, resulting in \(b = 0.6 - 0.4 = 0.2\). This is the y-intercept.

Step 4: Write the Equation of the Line

Lastly, substitute the calculated slope \(m = 0.4\) and y-intercept \(b = 0.2\) into the slope-intercept form \(y = mx + b\) to get the final equation of the line: \(y = 0.4x + 0.2\).

Key Concepts

Slope Calculation

Understanding how to calculate the slope is essential when determining the equation of a line. The slope, often denoted as \(m\), indicates how steep a line is and the direction it goes. To find the slope of a line passing through two points, use the formula: \[m = \frac{y2 - y1}{x2 - x1}\\]Here, \((x1, y1)\) and \((x2, y2)\) are the coordinates of the two points. The difference between the \(y\)-coordinates (\(y2 - y1\)) shows the vertical change, while the difference between the \(x\)-coordinates (\(x2 - x1\)) shows the horizontal change.

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.
• Zero slope indicates a horizontal line.
• An undefined (division by zero) slope represents a vertical line.

Applying this to our points, \[(x1, y1) = (1, 0.6)\]and\[(x2, y2) = (-2, -0.6)\],the slope \(m\) is calculated as \\[m = \frac{-0.6 - 0.6}{-2 - 1} = 0.4\].[br] This positive slope suggests the line increases as it moves from left to right.

Equation of a Line

The equation of a line in slope-intercept form is a simple way to represent straight lines. It's written as:\[y = mx + b\].In this equation:

• \(m\) is the slope of the line, showing how steep it is.
• \(b\) is the y-intercept, where the line crosses the y-axis.

To find the equation of a line that passes through a specific point, you can substitute the known slope \(m\) and one of the points into this equation. For example, using the point \((1, 0.6)\) and the previously calculated slope \(m = 0.4\), substitute into the slope-intercept formula:\[0.6 = 0.4(1) + b\].[br] Solving for \(b\), which is the y-intercept, helps complete the equation of the line.

Y-Intercept

The y-intercept is a crucial part of a line's equation. It represents the point where the line crosses the y-axis. In the slope-intercept form equation \(y = mx + b\), \(b\) is the y-intercept. To find it, you can re-arrange the equation after substituting a known point and the slope. Solving for the y-intercept involves isolating \(b\) on one side of the equation. Using our example, starting with:\[0.6 = 0.4(1) + b\],we simply subtract \(0.4\) from both sides: \[b = 0.6 - 0.4 = 0.2\].[br] This calculation gives the y-intercept, meaning the line crosses the y-axis at \((0, 0.2)\), completing the line's equation to be \(y = 0.4x + 0.2\).Key points to remember about y-intercepts:

• Always expressed as a coordinate (0, b).
• Adjusts the line vertically on the graph.

Graphing Utility

A graphing utility is a helpful tool to visualize the equation of a line. Once you have the slope and y-intercept, you can plot the line easily. These tools allow for:

• Viewing precise graphs instantly.
• Seeing intersections and slopes clearly.
• Making quick adjustments to equations and immediate visualization.

For the equation \(y = 0.4x + 0.2\), enter it into a graphing tool to see how it visually represents the calculation. This line will cross the y-axis at \((0, 0.2)\) and has a positive incline, consistent with the slope \(0.4\) calculated earlier. Graphing utilities give a clear visual understanding, making learning faster and often more intuitive.