Problem 35 - Understanding Elementary Algebra with Geometry

Question

Sets of values are given for variables having a linear relationship. In each case, write the slope-intercept form for the equation of the line corresponding to the given set of values and answer the accompanying question. $$\begin{array}{|l|c|c|c|} \hline x \text { (Number of clerks working) } & 6 & 8 & 9 \\ \hline y \text { (Number of minutes waiting time) } & 8 & 5 & 3.5 \\ \hline \end{array}$$ What would the waiting time be if 4 clerks are working?

Step-by-Step Solution

Answer

The waiting time would be 11 minutes if 4 clerks are working.

Step 1 - Identify points

Identify the given points from the table. The points are (6, 8), (8, 5), and (9, 3.5).

Step 2 - Calculate the slope (m)

Use the slope formula for two points \( (x_1, y_1) \text{ and } (x_2, y_2) \): \ m = \frac{y_2 - y_1}{x_2 - x_1} \.Calculate the slope for the points (6, 8) and (8, 5):\ m = \frac{5 - 8}{8 - 6} = \frac{-3}{2} = -1.5 \.

Step 3 - Find the y-intercept (b)

Use the slope-intercept form equation \( y = mx + b \) with one of the points to solve for b.Use the point (6, 8): 8 = -1.5(6) + b 8 = -9 + b b = 17.

Step 4 - Write the equation

Combine the slope and y-intercept to get the slope-intercept form of the equation. Thus, the equation is \[ y = -1.5x + 17 \].

Step 5 - Solve for the waiting time when x = 4

Substitute x = 4 into the equation \[ y = -1.5x + 17 \]. y = -1.5(4) + 17 y = -6 + 17 y = 11.

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is a special format that makes it easy to understand the relationship between the variables quickly. In this form, the equation is written as:
\[ y = mx + b \]
The letter \'m\ represents the slope of the line. This tells you how steep the line is. The letter \'b\ is the y-intercept. It shows where the line crosses the y-axis.
For example, if we have the equation \[ y = -1.5x + 17 \], the slope is -1.5 and the y-intercept is 17. Using these two pieces of information, we can plot the line and predict y values based on given x values.

calculate slope

Calculating the slope is a crucial step when working with linear equations in algebra. The formula to find the slope between two points \(x_1, y_1\) and \(x_2, y_2\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's use the points (6, 8) and (8, 5) from the exercise. Substituting in the values, we get:
\[ m = \frac{5 - 8}{8 - 6} = \frac{-3}{2} = -1.5 \]
The slope \(m\) of -1.5 tells us that for every unit increase in the x-value, the y-value decreases by 1.5. This negative value indicates the line slopes downwards.

find y-intercept

Once we have the slope, the next step is to find the y-intercept (b). We use any point from the given points along with the slope in the slope-intercept equation \(y = mx + b\).
Let's use the point (6, 8). We already know that \(m = -1.5\). Substituting these values into the equation:
8 = -1.5 * 6 + b
Simplifying this, we get:
8 = -9 + b
b = 17
This means the y-intercept is 17. The line crosses the y-axis at (0, 17).

substitute values

Finally, with the slope and y-intercept known, we can predict other values on the line by substituting x-values into the equation.
For example, to find the waiting time when 4 clerks are working (x = 4), we use the equation \[ y = -1.5x + 17 \] and substitute 4 for x:
\[ y = -1.5(4) + 17 \]
Simplifying, we get:
\[ y = -6 + 17 \]
\[ y = 11 \]
So, if 4 clerks are working, the waiting time would be 11 minutes. This substitution method helps predict outcomes for any variable input in linear equations.