Sam's weekly delivery van cost is modeled by the linear equation $$C = 0.5m + 60$$, where $$C$$ is the weekly cost in dollars and $$m$$ is the number of miles he drives. This practical equation matches the slope-intercept form $$y = mx + b$$, replacing standard variables with descriptive ones that match the context.

**ⓐ Find $$C$$ when $$m = 0$$:**
$$C = 0.5(0) + 60 = 0 + 60 = 60$$
Sam's cost is \$60 per week when he drives $$0$$ miles.

**ⓑ Find $$C$$ when $$m = 250$$:**
$$C = 0.5(250) + 60 = 125 + 60 = 185$$
His cost is \$185 when he drives $$250$$ miles.

**ⓒ Interpret the slope and $$C$$-intercept:**
- The slope $$0.5$$ indicates that the weekly cost, $$C$$, increases by \$0.50 for each additional mile driven, $$m$$.
- The $$C$$-intercept is $$(0, 60)$$, meaning that when the number of miles driven is $$0$$, the initial weekly cost is \$60.

**ⓓ Graph the equation:**
Graphing this real-world application requires a scaled rectangular coordinate system to accommodate larger quantities. Start at the $$C$$-intercept $$(0, 60)$$. To make graphing the slope $$m = 0.5$$ easier, write it as a fraction $$\frac{0.5}{1}$$ and multiply the numerator and denominator by $$100$$ to get the equivalent fraction $$\frac{50}{100}$$. From the intercept, count a rise of $$50$$ and a run of $$100$$ to locate the next point at $$(100, 110)$$. Draw a straight line connecting these points to represent the cost function.

Claude

When evaluating linear equations that model real-world applications in slope–intercept form, two primary adjustments are frequently made to ground abstract mathematics into practical scenarios: 1. Descriptive Variables: Instead of relying exclusively on the generic variables $$x$$ and $$y$$, context-specific letters are often chosen to clearly identify the nature of the quantities being measured. Using $$C$$ for Celsius or $$h$$ for height naturally informs what the variables denote. 2. Scaled Axes: Because real-world environments routinely present sizable positive or dramatic negative limits, the standard boundaries of the rectangular coordinate system frequently fall short. The axes must logically be extended into larger scales to properly graph and encapsulate the true span of the application's collected data.

Applying Slope-Intercept Form to Real-World Data

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Interpreting the Slope and $$C$$-Intercept of $$C = 0.5m + 60$$

Loreen runs a calligraphy business. Her weekly cost is modeled by the linear equation $$C = 1.8n + 35$$, where $$C$$ is the weekly cost in dollars and $$n$$ is the number of wedding invitations she writes.

**ⓐ Find $$C$$ when $$n = 0$$:**
$$C = 1.8(0) + 35 = 0 + 35 = 35$$
Loreen's weekly cost is \$35 when she writes $$0$$ invitations.

**ⓑ Find $$C$$ when $$n = 75$$:**
$$C = 1.8(75) + 35 = 135 + 35 = 170$$
Her weekly cost is \$170 when she writes $$75$$ invitations.

**ⓒ Interpret the slope and $$C$$-intercept:**
- The slope $$1.8$$ means that the weekly cost increases by \$1.80 for every additional invitation written.
- The $$C$$-intercept is $$(0, 35)$$, meaning her weekly cost is \$35 even if $$0$$ invitations are written.

**ⓓ Graph the equation:**
To graph this equation, prepare a coordinate plane with a sufficiently large scale to fit the application's data. Start at the $$C$$-intercept $$(0, 35)$$. By utilizing the slope $$1.8$$ as an equivalent ratio (such as moving up $$18$$ units and right $$10$$ units, or multiplying to create larger integer steps), plot a second point. Draw a straight line passing through these points to visually represent the business cost model.

Interpreting the Slope and $$C$$-Intercept of $$C = 1.8n + 35$$

The Celsius-to-Fahrenheit conversion model, $$F = \frac{9}{5}C + 32$$, is an example of a real-world linear equation that utilizes descriptive variables (like $$F$$ and $$C$$) rather than the standard abstract variables $$y$$ and $$x$$. Comparing the given equation against the general slope–intercept form, $$y = mx + b$$, confirms it fits the pattern, with a slope of $$m = \frac{9}{5}$$ and an $$F$$-intercept of $$(0, 32)$$. ⓐ Find $$F$$ when $$C = 0$$: Substituting $$0$$ for $$C$$ yields $$F = \frac{9}{5}(0) + 32 = 32$$. ⓑ Find $$F$$ when $$C = 20$$: Substituting $$20$$ for $$C$$ gives $$F = \frac{9}{5}(20) + 32 = 36 + 32 = 68$$. ⓒ Interpret the slope and $$F$$-intercept: The slope of $$\frac{9}{5}$$ asserts that for every $$5$$-degree increase on the Celsius scale ($$C$$), the Fahrenheit temperature ($$F$$) rises by $$9$$ degrees. The $$F$$-intercept at $$(0, 32)$$ implies that when the Celsius measurement is precisely $$0^\circ$$, the temperature on the Fahrenheit scale is $$32^\circ$$. ⓓ Graph the equation: Because real-world applications frequently feature numbers well beyond standard integer math, the rectangular coordinate system must be expanded to a larger scale. First, mark the $$F$$-intercept at $$(0, 32)$$. From there, compute a rise of $$9$$ units and a run of $$5$$ units to map a second point at $$(5, 41)$$. Connecting these points with a straight line constructs the graphical relationship.

Interpreting the Slope and $$F$$-Intercept of $$F = \frac{9}{5}C + 32$$

The linear equation $$h = 2s + 50$$ models a woman's estimated height in inches, $$h$$, based on her shoe size, $$s$$. Since the equation is in slope-intercept form ($$y = mx + b$$), we can identify the slope as $$2$$ and the $$h$$-intercept as $$(0, 50)$$. In this context, the slope of $$2$$ indicates that the estimated height increases by $$2$$ inches for every $$1$$ unit increase in shoe size. The $$h$$-intercept of $$50$$ signifies that a hypothetical person wearing a women's shoe size of $$0$$ would have an estimated height of $$50$$ inches. Evaluating the equation for a shoe size of $$s = 8$$ yields a height of $$h = 2(8) + 50 = 66$$ inches. To graph this relationship, one can plot the $$h$$-intercept at $$(0, 50)$$ and use the slope ($$\frac{2}{1}$$) to count up $$2$$ units and right $$1$$ unit to find a second point.

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