Question

Match the description with the linear model \(y=10\) or the linear model \(y=10 x .\) Graph the model. You rent a sailboard for \(\$ 10\) per hour.

Short answer

The model that best describes the scenario (rent a sailboard for \$10 per hour) is \(y=10x\). The graph of this model is a straight line passing through the origin with a slope of 10.

Step-by-step solution

Understand the scenario

In the given scenario, you are renting a sailboard for \$10 per hour. This implies that the total price you pay 'y' directly depends on the number of hours you rent 'x'. Therefore, this is an example of proportional relationship.

Identify the correct linear model

Now let's match the scenario with the given models. The first model \(y=10\) suggests that no matter what the value of 'x' (number of hours) is, the total cost 'y' stays constant at \$10. This does not match with our scenario. The second model \(y=10x\) means that for each hour of rent, you pay \$10, which perfectly matches our scenario. So, the correct model is \(y=10x\).

Graph the model

To graph the model \(y=10x\), consider the 'x' axis representing the number of hours and the 'y' axis representing the total cost. Now, plot a straight line through the origin (0,0) with a slope of 10. The slope indicates that for each additional hour (increase in x), the total cost (y) increases by \$10.

Key concepts

Understanding Proportional Relationships

A proportional relationship is a fundamental concept in algebra that describes a linear relationship between two quantities where one quantity is a constant multiple of the other. This relationship is represented by the equation of a line in the form of y = kx, where k is the constant of proportionality.

For instance, when you rent a sailboard for \(10 per hour, the total cost y depends on the number of hours x you rent it for. In essence, your cost increases uniformly as the hours increase, maintaining a constant ratio. This is an example of a proportional relationship since the total cost is directly proportional to the hours rented. The constant of proportionality in this case is \)10, indicating the cost per hour.

Graphing Linear Equations

Graphing linear equations is a way to visually represent the relationship between two variables. The graph of a linear equation is always a straight line. In the context of the exercise, the linear model y = 10x is graphed by first plotting the y-intercept, which in this case, is the origin (0,0) because there is no fixed starting cost - the cost is zero when the number of hours rented is zero.

From there, we use the slope to find other points on the line. Given that the slope is 10, we go up 10 units in the y direction (cost) for every 1 unit we go to the right in the x direction (hours). Plotting these points and drawing a straight line through them gives us the graph of the linear equation. This visual representation helps to quickly see how the total cost changes with the number of hours rented.

The Slope of a Line

The slope of a line is a measure of its steepness and direction. Mathematically, the slope is the ratio of the change in y (vertical change) to the change in x (horizontal change) between two distinct points on the line, often written as m. In the equation y = mx + b, m represents the slope.

If we take the previous example where the slope is 10, it means that for each additional hour rented (increase in x), the cost (y) increases by $10. A positive slope indicates that as x increases, y also increases. Conversely, a negative slope would imply that as x increases, y decreases. A steeper slope means a larger increase in y for a given increase in x, and this concept is critical for interpretation and understanding of linear relationships in various contexts.