Problem 65 - Algebra Form and Function

Question

Which line has the greater (a) Slope? (b) \(y\) -intercept? $$ y+2=3(x-1), \quad y=6-50 x $$

Step-by-Step Solution

Answer

Answer: The line with the equation \(y=3x-5\) has the greater slope, and the line with the equation \(y=-50x+6\) has the greater y-intercept.

Step 1: Rewrite the first equation in slope-intercept form

To rewrite \(y+2=3(x-1)\) in slope-intercept form, we need to perform the following steps: 1. Distribute the 3 to \(x-1\): \(y+2=3x-3\) 2. Subtract 2 from both sides: \(y=3x-5\) Now, the first equation is written in slope-intercept form as \(y=3x-5\).

Step 2: Rewrite the second equation in slope-intercept form

To rewrite the second equation, \(y=6-50x\), in slope-intercept form, we need to rewrite it in the form \(y=mx+b\). Currently, it is written as \(y=-50x+6\), which is already in slope-intercept form.

Step 3: Identify the slopes and \(y\)-intercepts

Now that both equations are in slope-intercept form, we can compare their slopes and \(y\)-intercepts: 1. First equation: \(y=3x-5\), where the slope \(m_1=3\) and the \(y\)-intercept \(b_1=-5\) 2. Second equation: \(y=-50x+6\), where the slope \(m_2=-50\) and the \(y\)-intercept \(b_2=6\)

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

(a) Compare the slopes: \(m_1=3\) and \(m_2=-50\). Since 3 > -50, the line with the equation \(y=3x-5\) has the greater slope. (b) Compare the \(y\)-intercepts: \(b_1=-5\) and \(b_2=6\). Since 6 > -5, the line with the equation \(y=-50x+6\) has the greater \(y\)-intercept.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when working with linear equations. In this form, any linear equation can be expressed as \[y = mx + b\], where:

• \(m\) is the slope,
• \(b\) is the y-intercept.

This form reveals important information about lines:

• The slope \(m\) shows the steepness and direction of the line.
• The y-intercept \(b\) indicates where the line crosses the y-axis.

When a problem presents an equation not in slope-intercept form, such as \(y+2=3(x-1)\), rearranging it into this form helps uncover those details. This step involves expanding the equation and isolating \(y\) to match \[y = mx + b\].Through such transformations, the characteristics of the line become easier to analyze.

Linear Equations

Linear equations describe straight lines on a coordinate plane. They are called "linear" because their graph is a line. The structure of a linear equation allows you to predict the output \(y\) for any given input \(x\). Characteristics of linear equations include:

• A constant rate of change, which is the slope.
• The absence of variables raised to powers other than one.

Solving linear equations often involves converting them into a familiar format, like the slope-intercept form, to make comparisons or predictions. Every time you come across new equations, practice reformatting them into \[y = mx + b\]. This clarity enhances your understanding and aids in solving problems related to intersections, parallelism, and perpendicularity.

Y-Intercept

The y-intercept is the point where the graph of a line crosses the y-axis of a coordinate plane. In the context of the slope-intercept form \[y = mx + b\], it is represented by the constant \(b\). Key aspects of the y-intercept include:

• It signifies the starting value of \(y\) when \(x = 0\).
• Understanding the y-intercept helps in quickly sketching graphs of lines and determining their position relative to each other.
• When comparing linear equations, the y-intercept indicates which line is higher or lower at the y-axis.

For instance, in the equations given in the original exercise, \(y=-50x+6\) has a y-intercept of 6, which clearly illustrates that this line crosses the y-axis higher than the line \(y=3x-5\), having a y-intercept of -5. Recognizing these differences is vital for identifying shifts or trends described by linear equations.