Problem 67 - Elementary and Intermediate Algebra

Question

Complete the formula. t= ___ s+___ \(\begin{array}{|c|c|}\hline s & {t} \\ \hline 18 & {55} \\ \hline 33 & {100} \\\ \hline 47 & {142} \\ \hline\end{array}\)

Step-by-Step Solution

Answer

The formula is \(t = 3s + 1\).

Step 1: Identify the Pattern

Examine the table and observe the relationship between the values of \(s\) and \(t\). Find the increment or decrement involving the given values for each pair in the table. This will help us recognize if it's a linear pattern.

Step 2: Determine the Slope

Use the formula for the slope (\(m\)) in a linear equation: \(m = \frac{(t_2 - t_1)}{(s_2 - s_1)}\). Select two points from the table to calculate this. For example, using (18, 55) and (33, 100), \(m = \frac{(100 - 55)}{(33 - 18)} = \frac{45}{15} = 3\). This confirms a consistent rate of change for every unit increase in \(s\).

Step 3: Calculate the Y-Intercept

With the slope known, use the point-slope form \((t = ms + c)\) where \(c\) is the y-intercept. Use one of the table pairs to solve for \(c\): \(55 = 3(18) + c\). Solving for \(c\), \(55 = 54 + c\), gives \(c = 1\).

Step 4: Confirm the Equation

Verify the linear equation \(t = 3s + 1\) using another pair of values to ensure consistency throughout the table. Substitute \(s = 33\) into this equation: \(t = 3(33) + 1 = 100\). This matches the spreadsheet data at \(s = 33\). Therefore, the formula for the relationship holds true for all given values.

Key Concepts

Slope Calculation

Understanding the concept of slope is essential when dealing with linear equations. The slope determines how steep the line is, and indicates the rate at which one variable changes in relation to another. In our task, we're looking to find the slope in a table comparing values of \(s\) and \(t\). To find the slope, use the formula

• \(m = \frac{(t_2 - t_1)}{(s_2 - s_1)}\)

Choose two points from your data. For example, take (18, 55) and (33, 100). Plug these values into the formula:

• \(m = \frac{(100 - 55)}{(33 - 18)} = \frac{45}{15} = 3\)

This calculation tells us that for every 1-unit increase in \(s\), \(t\) increases by 3 units. This consistent increment confirms that the relationship between \(s\) and \(t\) is linear. Slope calculations give us a clear insight into the behavior of linear equations.

Y-Intercept

The y-intercept is where the line will cross the y-axis when \(s = 0\). It's an essential part of the linear equation, allowing us to pinpoint the starting value of \(t\) when the independent variable \(s\) is zero. After determining the slope, the task is to find the y-intercept \(c\) in the equation \(t = ms + c\). By using one of the known points, such as (18, 55), and the slope calculated previously (which is 3), we are able to solve for \(c\):

• \(55 = 3(18) + c\)

This simplifies to:

• \(55 = 54 + c\)

so:

• \(c = 1\)

This means that when \(s = 0\), \(t\) would be 1, identifying our line's starting point on the y-axis. Recognizing the y-intercept helps us fully understand the nature of our linear equation.

Point-Slope Form

The point-slope form is an incredibly useful tool in linear algebra, especially when you know a point on the line and the slope. This form is expressed as:

• \(t - t_1 = m(s - s_1)\)

Here, \((s_1, t_1)\) represents a point on the line, and \(m\) is the slope. This form will come in handy when you need to write the equation of a line but only have a point and the slope.However, in this task, after initially using the point-slope form to organize our thinking, we convert it into the slope-intercept form \(t = ms + c\) to easily identify both the slope and the y-intercept together. For instance, using the slope \(m = 3\) and point (18, 55), this would verify the equation:

• \(t - 55 = 3(s - 18)\)

By simplifying and finding \(t\), you can confidently write the equation in slope-intercept form, showing the full relationship between the variables. Understanding point-slope form helps bridge the gap between different arithmetical operations needed to provide deeper insights into linear equations.