Problem 56 - Precalculus with Limits

Question

In Exercises 51-64, find the slope-intercept form of the equation of the line that passes through the given point and has the indicated slope \(m\). Sketch the line. \((8, 2)\), \(m = \frac{1}{4}\)

Step-by-Step Solution

Answer

The slope-intercept form of the equation of the line that passes through the point (8, 2) and has a slope of \(1/4\) is \(y = \frac{1}{4}x\).

Step 1: Substitute the known values

Firstly, substitute the given point \((8,2)\) and the slope \(m = \frac{1}{4}\) into the slope intercept equation \(y = mx + b\). That gives the equation \(2 = \frac{1}{4} * 8 + b\).

Step 2: Solve for \(b\)

Secondly, solve this equation for \(b\). The equation becomes \(2 - \frac{1}{4} * 8 = b\), which simplifies to \(2 - 2 = b\). So, \(b = 0\).

Step 3: Write the final equation

Substitute the solved value for \(b\) into the equation. So, the slope-intercept form of the equation of the line is \(y = \frac{1}{4}x + 0\). Since adding 0 doesn't change the equation, it can be simplified to \(y = \frac{1}{4}x\).

Step 4: Sketch the line

Lastly, a rough sketch of the line can be drawn. The line crosses through the point (8,2) and has a slope of 1/4, which means for each 1 unit increase in \(x\), \(y\) increases by 1/4 unit.

Key Concepts

Understanding Linear Equations

Linear equations are foundational in mathematics and describe a straight line when graphed on a coordinate plane. They are generally expressed in the form: \(y = mx + b\), where:

• \(y\) is the dependent variable, often seen on the vertical axis.
• \(x\) is the independent variable, plotted on the horizontal axis.
• \(m\) represents the slope, describing the line's steepness.
• \(b\) is the y-intercept, indicating where the line crosses the y-axis.

Linear equations model relationships where variables change at a constant rate. They are powerful tools in predicting and understanding phenomena in various areas like economics and science. Recognizing linear equations enables us to describe and visualize linear relationships between two variables effectively.

Exploring Point-Slope Form

Point-slope form is another way to write linear equations, particularly useful when we need to create an equation of a line from a point and its slope. The point-slope form is written as: \[ y - y_1 = m(x - x_1) \] Here:

• \((x_1, y_1)\) is a known point on the line.
• \(m\) is the slope of that line.

By rearranging the formula, you can convert it into the more familiar slope-intercept form \(y = mx + b\). This conversion is helpful when you need the line equation in a simpler format for graphing or analysis. Understanding point-slope form enhances your ability to quickly define line equations from minimal information.

Graphing Lines and Understanding Slope-Intercept Form

Graphing lines might sound complicated, but understanding the slope-intercept form makes it straightforward. The slope-intercept form \(y = mx + b\) instantly tells you two crucial details:

• The slope \(m\), indicating how steep the line is, or how y changes with x.
• The intercept \(b\), showing the y-value where the line intersects the y-axis.

When graphing:
Start at the y-intercept \(b\), then use the slope \(m\) to determine the direction and angle of the line. For instance, a slope \(m = \frac{1}{4}\) means for every four units you move horizontally (on the x-axis), you will move up one unit vertically (on the y-axis).
Graphing allows us to visualize data and predictions easily, making it an essential skill in various areas from engineering to economics.