Problem 66 - Algebra Form and Function

Question

Which line has the greater (a) Slope? (b) \(y\) -intercept? $$ y-3=-4(x+2), \quad-2 x+5 y=-3 $$

Step-by-Step Solution

Answer

Answer: The second line has the greater slope and y-intercept.

Step 1: Convert the first equation into slope-intercept form

We have the equation \(y-3=-4(x+2)\). We will now solve this equation for \(y\): $$ y-3=-4(x+2) \\ y-3=-4x-8 \\ y=-4x-5 $$ The equation of the first line in slope-intercept form is \(y=-4x-5\).

Step 2: Convert the second equation into slope-intercept form

We have the equation \(-2x+5y=-3\). We will now solve this equation for \(y\): $$ -2x + 5y =-3 \\ 5y=2x-3 \\ y=\frac{2}{5}x - \frac{3}{5} $$ The equation of the second line in slope-intercept form is \(y=\frac{2}{5}x-\frac{3}{5}\).

Step 3: Compare the slopes

The slope of the first line is \(m_1 = -4\), and the slope of the second line is \(m_2 = \frac{2}{5}\). Since \(-4 < \frac{2}{5}\), the second line has the greater slope.

Step 4: Compare the y-intercepts

The y-intercept of the first line is \(b_1 = -5\), and the y-intercept of the second line is \(b_2 = -\frac{3}{5}\). Since \(-5 < -\frac{3}{5}\), the second line has the greater y-intercept.

Step 5: Conclusion

The second line has the greater slope and y-intercept.

Key Concepts

Slope Comparison

The slope of a line provides a measure of how steep the line is. It's represented by the letter \(m\) in the slope-intercept form of a linear equation, which is \(y = mx + b\). This form clearly separates the slope and the \(y\)-intercept.
To compare slopes, we examine the numbers attached to \(x\) in each equation once they're in slope-intercept form. In our two equations, we find:

• The first line has a slope of \(-4\)
• The second line has a slope of \(\frac{2}{5}\)

To determine which slope is greater, we need to consider their values on the number line. Here, \(-4\) is much smaller than \(\frac{2}{5}\), indicating the second line is less steep in terms of negative descent and has a greater slope than the first.
This is because a larger slope value means the line rises more quickly as you move from left to right.

Y-Intercept Comparison

The \(y\)-intercept of a line is where the line crosses the \(y\)-axis. This value is represented by \(b\) in the equation \(y = mx + b\).
For our equations:

• The first line has a \(y\)-intercept of \(-5\)
• The second line's \(y\)-intercept is \(-\frac{3}{5}\)

When comparing \(y\)-intercepts, you look at their positions on the number line. Larger \(y\)-intercepts are closer to positive values. Here, \(-\frac{3}{5}\) is greater than \(-5\) because it's closer to zero.
This comparison tells us that the second line crosses the \(y\)-axis higher up than the first line, indicating a larger \(y\)-intercept.

Linear Equations

Linear equations are expressions that describe straight lines in a graph. They are typically written in the form \(y = mx + b\), known as the slope-intercept form.
This straightforward format shows:

• \(m\), the slope, which tells us how steep the line is
• \(b\), the \(y\)-intercept, which indicates where the line crosses the \(y\)-axis

To convert any equation into slope-intercept form, manipulate the equation until \(y\) is isolated on one side. This form makes it easy to read off the values for the slope and \(y\)-intercept, facilitating quick comparisons between different lines.
Understanding this format helps in analyzing and graphing lines effortlessly. It's a fundamental concept for studying and solving various real-world problems involving linear relationships.