Question

Which has the greater rate of increase: Line A with the equation \(y=5 x-7\), or Line \(B\) between the points \((-1,-4)\) and \((1,7)\) ?

Short answer

Line B has a greater rate of increase.

Step-by-step solution

Identify the slope of Line A

The equation of Line A is given as \(y = 5x - 7\). This equation is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope. Here, the slope \(m\) is 5.

Calculate the slope of Line B

Line B is defined by the points \((-1, -4)\) and \((1, 7)\). The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the points into the formula:\[ m = \frac{7 - (-4)}{1 - (-1)} = \frac{7 + 4}{1 + 1} = \frac{11}{2} = 5.5 \]

Compare the slopes

The slope of Line A is 5, and the slope of Line B is 5.5. Since 5.5 is greater than 5, Line B has a greater rate of increase.

Key concepts

Slope Calculation

The slope of a line is a measure of how steeply it rises or falls as you move along the line. Slope is very important in understanding the behavior of linear equations. For any line represented by two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is calculated using the formula: \[\frac{y_2 - y_1}{x_2 - x_1}\]

This formula tells you how much the y-coordinate (vertical change) changes for a given change in the x-coordinate (horizontal change). It's a simple ratio: rise over run.

When solving problems involving slopes, it's essential to make sure that the coordinates are correctly substituted into the formula. For instance, consider Line B that is determined by the points \((-1, -4)\) and \(1, 7 \). Plugging these into the slope formula, we have: \[ m = \frac{7 - (-4)}{1 - (-1)} = \frac{7 + 4}{1 + 1} = \frac{11}{2} = 5.5 \]

Thus, the slope of Line B is 5.5, which means for every unit increase in x, y increases by 5.5 units.

Linear Equations

A linear equation is any equation that can be written in the form \(y = mx + b\), known as the slope-intercept form.

Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is where the line crosses the y-axis. This form is particularly useful because it directly tells you both the slope and the y-intercept.

For example, the equation of Line A is given as \(y = 5x - 7\). In this form, we can identify that the slope \(m\) is 5 and the y-intercept \(b\) is -7. That means when \(x = 0\), \(y = -7\).

Being able to recognize and transform equations into the slope-intercept form makes graphing and understanding linear relationships straightforward. Since the slope \(m\) dictates the steepness, a larger \(m\) value means a steeper line. In our exercise, Line A has a slope of 5, indicating that for every unit increase in x, y increases by 5 units.

Rate of Change

The rate of change is another key concept directly linked to slope. It tells us how one quantity changes in relation to another. In the context of lines, it is the slope itself.

For Line A, given by \(y = 5x - 7\), the rate of change is 5. This means that for each unit the x-value increases, the y-value increases by 5 units.

For Line B, the rate of change is calculated from the points \((-1, -4)\) and \(1, 7 \). Using the slope formula, we found this rate to be 5.5. Therefore, for every unit increase in x, the y-value increases by 5.5 units.

Comparing the two slopes, 5.5 is greater than 5. This means Line B has a higher rate of change than Line A. Understanding which line has a greater rate of change helps to compare their growth rates effectively.

Remember that higher rates of change indicate faster increases or decreases, which is crucial in various real-life applications such as finance, physics, and everyday decisions.