Problem 63 - Algebra Form and Function

Question

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

Step-by-Step Solution

Answer

The first line has a greater slope and a greater y-intercept. The slope of the first line is 1/2, while the slope of the second line is -1. The y-intercept of the first line is -3/4, while the y-intercept of the second line is -2.

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

Divide both sides of the equation by \(4\) to get the equation in the form of \(y = mx + b\): $$ \frac{1}{4} (2x) = \frac{1}{4} (4y + 3) \\ \Rightarrow \frac{1}{2} x = y + \frac{3}{4} \\ \Rightarrow y = \frac{1}{2} x - \frac{3}{4} $$ Now, the first line equation is in the slope-intercept form, \(y = mx + b\), with slope \(m_1 = \frac{1}{2}\) and \(y\)-intercept \(b_1 = -\frac{3}{4}\).

Step 2: Determine the slope and \(y\)-intercept of the second line equation

The second line equation is already in slope-intercept form, \(y = -x - 2\). So, the slope for the second line is \(m_2 = -1\) and the \(y\)-intercept is \(b_2 = -2\).

Step 3: Compare the slopes

We have the slopes of both lines as \(m_1 = \frac{1}{2}\) and \(m_2 = -1\). Since \(\frac{1}{2} > -1\), the first line has a greater slope.

Step 4: Compare the \(y\)-intercepts

We have the \(y\)-intercepts of both lines as \(b_1 = -\frac{3}{4}\) and \(b_2 = -2\). Since \(-\frac{3}{4} > -2\), the first line has a greater \(y\)-intercept. So, the first line has both a greater slope and a greater \(y\)-intercept.

Key Concepts

Slope

The slope of a line in a linear equation provides valuable information about the line's steepness and direction on a coordinate plane. Mathematically, the slope is represented by the letter "m" in the slope-intercept form of a linear equation, which is written as: \[ y = mx + b \]The slope measures how much the line rises or falls as you move from one point to another:- A positive slope means the line inclines upwards as it moves from left to right.- A negative slope indicates that the line declines downwards as it moves from left to right.- A larger numerical value of the slope shows a steeper line.- A slope of zero signifies a perfectly horizontal line.In our example, the first line has a slope of \( \frac{1}{2} \), indicating a gentle upward rise. The second line has a slope of \(-1\), meaning it declines steeply downward. Comparing these, \( \frac{1}{2} \) is larger than \(-1\), so the first line has the greater slope.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis on a graph. It is represented by the letter "b" in the slope-intercept form of a linear equation. This intercept gives us a starting point or a fixed location on the y-axis for the line. The y-intercept indicates the value of y when x is zero. In the general equation \( y = mx + b \), "b" tells you exactly where the line penetrates the y-axis. For our specific equations:- The first equation \( y = \frac{1}{2}x - \frac{3}{4} \) has a y-intercept of \(-\frac{3}{4}\).- The second equation \( y = -x - 2 \) has a y-intercept of \(-2\).The y-intercept can tell us a lot about the position of a line on a graph. Since \(-\frac{3}{4}\) is greater than \(-2\), the first line crosses the y-axis higher up than the second line.

Linear Equations

Linear equations describe straight lines when plotted on a graph. A linear equation is any equation that can be written in the standard form \( ax + by = c \) or the slope-intercept form \( y = mx + b \). The slope-intercept form is particularly useful because it immediately gives both the slope and y-intercept:- "m" represents the slope which shows the line's direction and steepness.- "b" is the y-intercept giving the point where the line crosses the y-axis.Working with linear equations allows students to easily graph and understand the behavior of lines on a coordinate plane. In our case:- The first line \( 2x = 4y + 3 \) was converted into the slope-intercept form \( y = \frac{1}{2}x - \frac{3}{4} \).- The second line \( y = -x - 2 \) is already in slope-intercept form.Understanding linear equations helps identify relationships between variables and can model real-world scenarios, making it an essential skill in math.