Problem 33 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. $$ \text { Passing through }\left(\frac{1}{2}, \frac{3}{4}\right) \text { and }\left(\frac{1}{5}, \frac{2}{3}\right) $$

Step-by-Step Solution

Answer

The slope-intercept form is \( y = \frac{5}{18}x + \frac{8}{9} \).

Step 1: Formula for Slope

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \[ m = \frac{y_2-y_1}{x_2-x_1} \].

Step 2: Substitute Values into Slope Formula

For the points \( \left( \frac{1}{2}, \frac{3}{4} \right) \) and \( \left( \frac{1}{5}, \frac{2}{3} \right) \), substitute the coordinates into the slope formula to get: \[ m = \frac{\frac{2}{3} - \frac{3}{4}}{\frac{1}{5} - \frac{1}{2}} \].

Step 3: Simplify the Numerator and Denominator

First, find a common denominator for the fractions involved. For the numerator: \[ \frac{2}{3} - \frac{3}{4} = \frac{8}{12} - \frac{9}{12} = \frac{-1}{12} \].For the denominator: \[ \frac{1}{5} - \frac{1}{2} = \frac{2}{10} - \frac{5}{10} = \frac{-3}{10} \].

Step 4: Calculate the Slope

Now divide the simplified numerator by the simplified denominator:\[ m = \frac{\frac{-1}{12}}{\frac{-3}{10}} = \frac{-1}{12} \times \frac{-10}{3} = \frac{10}{36} = \frac{5}{18} \].

Step 5: Use Point-Slope Form to Find Equation

Use the point-slope form \( y - y_1 = m(x - x_1) \) to find the equation, substituting \( m = \frac{5}{18} \) and one of the points, say \( \left(\frac{1}{2}, \frac{3}{4}\right) \): \[ y - \frac{3}{4} = \frac{5}{18} \left(x - \frac{1}{2} \right) \].

Step 6: Simplify to Slope-Intercept Form

Expand and simplify the equation to get it into \( y = mx + b \) form: Multiply out \( \frac{5}{18}(x - \frac{1}{2}) \) to get \[ y - \frac{3}{4} = \frac{5}{18}x - \frac{5}{36} \].Add \( \frac{3}{4} \) to both sides:\[ y = \frac{5}{18}x + \frac{3}{4} + \frac{5}{36} \].Simplify the constants on the right to get\[ y = \frac{5}{18}x + \frac{27}{36} + \frac{5}{36} = \frac{5}{18}x + \frac{32}{36} \].Reduce \( \frac{32}{36} \) to get \( \frac{8}{9} \).Thus, the equation is \( y = \frac{5}{18}x + \frac{8}{9} \).

Key Concepts

Slope Calculation

Calculating the slope of a line is an important step in finding the equation of a line in slope-intercept form. The slope indicates the steepness or direction of a line. To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, use the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, you subtract the \(y\)-coordinates and the \(x\)-coordinates of the two points, then divide the first difference by the second. This formula gives you the "rise" over the "run".

For example, given the points \(\left( \frac{1}{2}, \frac{3}{4} \right)\) and \(\left( \frac{1}{5}, \frac{2}{3} \right)\), insert the values into the slope formula:

• \[ m = \frac{\frac{2}{3} - \frac{3}{4}}{\frac{1}{5} - \frac{1}{2}} \]

Remember to simplify the fractions involved to find the slope, which allows you to move forward with creating the line equation.

Point-Slope Form

The point-slope form is very useful for writing the equation of a line when you know one point on the line and the slope. The equation is structured as follows:

• \[ y - y_1 = m(x - x_1) \]

In this formula, \(m\) is the slope of the line, and \(x_1, y_1\) are the coordinates of the given point.

To apply this to our example, use \(m = \frac{5}{18}\) and the point \(\left(\frac{1}{2}, \frac{3}{4}\right)\). Plug these values into the point-slope form:

• \[ y - \frac{3}{4} = \frac{5}{18}(x - \frac{1}{2}) \]

Rearranging this equation into slope-intercept form \(y = mx + b\) lets you better view the line's properties, such as its slope and \(y\)-intercept.

Simplifying Fractions

Simplifying fractions is a crucial skill in algebra, notably in finding the slope of a line, as it ensures calculations are as simple as possible. When simplifying fractions:

• Find a common denominator for adding or subtracting fractions.
• Reduce the fraction to its lowest terms.

In our slope calculation, we started with this fraction:

• \[ \frac{\frac{-1}{12}}{\frac{-3}{10}} \]

Instead of dividing directly, multiply by the reciprocal:

• \[ \frac{-1}{12} \times \frac{-10}{3} = \frac{10}{36} \]

Finally, simplify \(\frac{10}{36}\) to \(\frac{5}{18}\). Each fraction in the steps of finding your equation needs simplifying, particularly when converting from point-slope to slope-intercept form.