(1) Because a line goes up by $\square$ when it goes to the right by 1, the slope is $\square$.
Answer (1)
The slope of a line is the change in y divided by the change in x . If a line goes up by y when it goes to the right by 1, then:
The change in y ( Δ y ) is y .
The change in x ( Δ x ) is 1.
The slope m is calculated as m = Δ x Δ y = 1 y = y .
Therefore, the slope is y .
Explanation
Understanding the Problem The problem states that a line goes up by a certain amount when it goes to the right by 1. We need to determine the slope of the line.
Defining Slope The slope of a line is defined as the change in y divided by the change in x . In mathematical terms, slope m = Δ x Δ y , where Δ y is the change in y and Δ x is the change in x .
Applying Given Information The problem states that when the line goes to the right by 1 (change in x = 1 ), the line goes up by a certain amount (change in y ). Let's call the amount the line goes up y . So, Δ x = 1 and Δ y = y .
Calculating the Slope Therefore, the slope of the line is m = 1 y = y . This means the slope of the line is equal to the amount the line goes up when it goes to the right by 1.
Final Answer Because a line goes up by y when it goes to the right by 1, the slope is y .
Examples
Understanding slope is crucial in many real-world applications. For example, when designing roads, engineers use the concept of slope to determine the steepness of the road. A steeper slope means a more difficult climb for vehicles. Similarly, in construction, the slope of a roof is important for water runoff. A higher slope allows water to drain more effectively, preventing leaks and damage to the building. In simple terms, slope helps us understand and control how things change over a certain distance, making our designs safer and more efficient.
