Problem 62 - Algebra Form and Function

Question

Which line has the greater (a) Slope? (b) \(y\) -intercept? $$ y=\frac{1}{5} x, \quad y=1-6 x $$

Step-by-Step Solution

Answer

Answer: (a) The first line has a greater slope. (b) The second line has a greater y-intercept.

Step 1: Identify the slope and y-intercept of each line

The first line, y = (1/5)x, has a slope of 1/5 and y-intercept of 0 (since there is no constant term, it implies the constant term is 0). The second line, y = 1 - 6x, has a slope of -6 and y-intercept of 1. Now we compare the slopes and y-intercepts of both lines.

Step 2: Compare the slopes

The first line has a slope of 1/5 and the second line has a slope of -6. Since 1/5 > -6, the first line has a greater slope.

Step 3: Compare the y-intercepts

The first line has a y-intercept of 0 and the second line has a y-intercept of 1. Since 0 < 1, the second line has a greater y-intercept To sum up: (a) The first line has a greater slope. (b) The second line has a greater y-intercept.

Key Concepts

Understanding Slope

The slope of a line is a measure of its steepness and the direction it goes. Imagine a hill: a gentle hill has a small slope, while a steep hill has a large slope. In mathematics, the slope is often represented by the letter \(m\). For the line \(y = \frac{1}{5}x\), the slope is \(\frac{1}{5}\). This means for every step you take to the right on the graph, you go up by \(\frac{1}{5}\) of a step.
In contrast, the line \(y = 1 - 6x\) has a slope of \(-6\). Here, for every step to the right, you go down by 6 steps. This negative number indicates the line is decreasing, or going down as you move from left to right.
When comparing these slopes, a positive slope like \(\frac{1}{5}\) is greater than a negative slope like \(-6\).

• Positive Slope: Line goes upwards.
• Negative Slope: Line goes downwards.

Exploring the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. This is the value of \(y\) when \(x\) is 0. For \(y = \frac{1}{5}x\), the y-intercept is 0. This means the line starts at the origin point \(0,0\).
On the other hand, for \(y = 1 - 6x\), the y-intercept is 1. So, this line crosses the y-axis at point \(0,1\).
The y-intercept tells us where exactly on the y-axis the line lands, making it easier to draw graphs. Between these two equations, the line \(y = 1 - 6x\) has a greater y-intercept because it crosses the y-axis at 1, which is higher than 0.

• Value on Y-axis: Where the line touches the y-axis.
• Importance: Helps to start drawing the line on a graph.

Comparison of Linear Equations

When comparing linear equations, like \(y = \frac{1}{5}x\) and \(y = 1 - 6x\), we look at both slope and y-intercept.
The slope tells us the direction and angle of the line, while the y-intercept pinpoints the line's starting position on the y-axis. The first equation has a small, positive slope, indicating a gentle upward line starting from the origin. The second equation has a steep negative slope, which means the line is very steep and descends quickly from a y-intercept of 1.
By examining these parts, we can understand and compare different lines:

• Slopes: Indicate if lines rise or fall, and how steeply.
• Y-Intercepts: Show the initial crossing point on the y-axis.

Together, understanding these components helps you sketch and compare linear functions effectively.