Problem 36 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Parallel to the line \(y=-\frac{3}{4}(x-100)-99\) passing through \((1,3)\)

Step-by-Step Solution

Answer

The slope-intercept form is \( y = -\frac{3}{4}x + \frac{15}{4} \).

Step 1: Identify the slope

To find the slope-intercept form, we first need to identify the slope of the given line. The line equation is given in point-slope form: \( y = -\frac{3}{4}(x-100) - 99 \). The slope here is \( m = -\frac{3}{4} \).

Step 2: Use the slope for the parallel line

Since parallel lines have the same slope, the line passing through the point \((1, 3)\) will have the slope \( m = -\frac{3}{4} \).

Step 3: Use point-slope form

We use the point-slope form of the line equation, which is: \( y - y_1 = m(x - x_1) \). Substituting \( m = -\frac{3}{4} \), \(x_1 = 1\), and \(y_1 = 3\), we get: \[ y - 3 = -\frac{3}{4}(x - 1) \].

Step 4: Simplify to slope-intercept form

The slope-intercept form is \( y = mx + b \). Simplify the equation: \( y - 3 = -\frac{3}{4}x + \frac{3}{4} \). Add 3 to both sides: \( y = -\frac{3}{4}x + \frac{3}{4} + 3 \). Simplify further: \[ y = -\frac{3}{4}x + \frac{3}{4} + \frac{12}{4} = -\frac{3}{4}x + \frac{15}{4} \].

Key Concepts

Parallel Lines

Parallel lines in geometry have a fascinating property. They never cross each other and are always the same distance apart. A key to understanding parallel lines in algebra is their slopes. If two lines are parallel, they have the same slope - in other words, they move at the same angle.

When we talk about a line being parallel to another, we're often asked to find the equation of this new parallel line. Our starting point is always the slope of the original line; they share this exact value.

• If the slope of our original line is \(m = -\frac{3}{4}\), any line parallel to it will also have a slope of \(-\frac{3}{4}\).
• Simply put, parallel lines stay equidistant because of their identical slopes.

Understanding this concept is crucial for solving problems that involve creating new lines parallel to given ones. Remember, it's all about maintaining that same slope!

Point-Slope Form

The point-slope form is an essential tool in algebra, especially when dealing with lines. This form helps express the equation of a line when we know a point on the line and its slope.

The point-slope formula looks like this:

• \( y - y_1 = m(x - x_1) \)

Here's how it works:

• \(x_1\) and \(y_1\) are the coordinates of the given point on the line.
• \(m\) represents the slope of the line.

Using the formula means plugging in these values. For example, if a line passes through the point (1, 3) and has a slope of \(-\frac{3}{4}\), we plug these into the formula:

\[ y - 3 = -\frac{3}{4}(x - 1) \]
This point-slope form is especially helpful when moving to other forms of linear equations, like the slope-intercept form. It provides a clear, straightforward path from information given to equation of a line.

Linear Equations

Linear equations are foundational in algebra, representing lines on a graph. They are equations of the form \(y = mx + b\), known as the slope-intercept form, where every component has a significant role.

• \(m\) is the slope: It describes how steep the line is and its direction (ascending or descending).
• \(b\) is the y-intercept: This is where the line crosses the y-axis, giving a starting point on the graph.

When transforming an equation from point-slope form to slope-intercept form, we're converting our information into this universal representation.

Using the original equation \( y - 3 = -\frac{3}{4}(x - 1) \), our goal is to simplify and solve it to reach the slope-intercept form:

Adding 3 to both sides, we get:

\[ y = -\frac{3}{4}x + \frac{3}{4} + 3 \]

Simplifying further, the final linear equation in slope-intercept form is:

\[ y = -\frac{3}{4}x + \frac{15}{4} \]
This provides a clean, visually straightforward way to depict lines, making it easy to understand their placement on a graph.