Problem 37 - Algebra Form and Function

Question

Are the lines parallel? $$ y=2+3(x+5) ; y=2+4(x+5) $$

Step-by-Step Solution

Answer

Answer: No, the lines are not parallel, as their slopes are not equal.

Step 1: Write both equations in slope-intercept form

To rewrite both equations in the slope-intercept form, we need to distribute the constants across the parentheses. For the first equation: $$ y = 2 + 3(x + 5) \\ y = 2 + 3x + 15 $$ Now combine the constants: $$ y = 3x + 17 $$ For the second equation: $$ y = 2 + 4(x + 5) \\ y = 2 + 4x + 20 $$ Now combine the constants: $$ y = 4x + 22 $$ Now we have both equations in slope-intercept form: $$ y = 3x + 17 \\ y = 4x + 22 $$

Step 2: Compare the slopes

Comparing the slopes of both lines, we can see that the first line has a slope of \(3\), while the second line has a slope of \(4\). Since the slopes are not equal, the lines are not parallel.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a widely used and very intuitive way to represent lines in the coordinate plane. This form is written as:\[ y = mx + b \]where:

• \(m\) is called the slope of the line, representing how steep the line is.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

When you see a line equation in this form, it gives you a quick snapshot of the line's characteristics. You easily understand how the line behaves by just looking at its slope and y-intercept. This form simplifies many mathematical concepts because it directly shows how the line tilts (its slope) and where it starts on the y-axis (its y-intercept). Recognizing this form becomes critical when working on problems involving linear equations.

Line Equations

Line equations are mathematical expressions used to describe a straight line on a graph. They help us determine specific points on the line using variables. The most common form is the slope-intercept form, but lines can also be represented using other forms such as the point-slope form and the standard form. For example, the initial equations \(y=2+3(x+5)\) and \(y=2+4(x+5)\) are in a form that requires expansion before being compared or analyzed further. By expanding, we convert these into a more straightforward form, namely the slope-intercept form.Converting line equations to slope-intercept form involves:

• Distributing any constants across parentheses.
• Combining like terms, typically constant numbers, to clean up the equation.

This conversion makes these equations easier to understand and compare and is particularly helpful for identifying the linea properties such as slope, thereby simplifying the comparison of different lines.

Slope Comparison

Slope comparison is a fundamental aspect when determining if two lines are parallel, perpendicular, or neither. It involves looking specifically at the slope value, which indicates the inclination of a line. For lines to be parallel, their slopes must be equal.For instance, if we have lines with equations \(y = 3x + 17\) and \(y = 4x + 22\), the slopes given by the coefficients of \(x\) (\(3\) and \(4\)) clearly show these lines are not parallel because they are not equal.Here’s why slope matters:

• Parallel lines: Same slope, never meet.
• Perpendicular lines: Slopes are negative reciprocals.
• Intersecting lines: Different slopes.

Thus, slope comparison allows us not only to classify the relationship between lines but also to predict their interactions in the space they share. In our scenario, the differing slopes confirm that these two lines are not parallel.