Problem 40 - Algebra Form and Function

Question

Are the lines parallel? $$ y=7+4(x-2) ; y=8+2(2 x+3) $$

Step-by-Step Solution

Answer

Answer: Yes, the lines are parallel because their slopes are equal (m1 = m2 = 4).

Step 1: Rewrite the given equations in slope-intercept form (y = mx + b)

In our given equations, $$ y = 7 + 4(x - 2) \\ y = 8 + 2(2x + 3) $$ We want to rewrite these equations in the form y = mx + b. For the first equation: $$y = 7 + 4(x - 2)$$ $$y = 7 + 4x - 8$$ $$y = 4x - 1$$ For the second equation: $$y = 8 + 2(2x + 3)$$ $$y = 8 + 4x + 6$$ $$y = 4x + 14$$ Now we have the equations in slope-intercept form: $$ y = 4x - 1 \\ y = 4x + 14 $$

Step 2: Compare the slopes of the two lines

Now we can compare the slopes (m) of these two lines. For the first line: $$y = 4x - 1$$, m1 = 4 For the second line: $$y = 4x + 14$$, m2 = 4 We can see that the slopes of both lines are equal, m1 = m2.

Step 3: Conclude if the lines are parallel or not

Since the slopes of the two lines are equal (m1 = m2), we can conclude that the lines are parallel.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a useful way to express a linear equation of a line. This form is given by the equation \(y = mx + b\), where \(m\) represents the slope, and \(b\) represents the y-intercept. The slope \(m\) indicates how steep the line is and in which direction it is heading, up or down. The y-intercept \(b\) is the point where the line crosses the y-axis.

In the exercise above, the given equations were initially:

• \(y = 7 + 4(x - 2)\)
• \(y = 8 + 2(2x + 3)\)

These are not in the slope-intercept form at first glance. By expanding and simplifying, we rewrite them as:

• \(y = 4x - 1\)
• \(y = 4x + 14\)

Both equations now clearly show the slope \(m = 4\) and each has its own y-intercept. This form makes it straightforward to identify the slope and determine any line's behavior.

Slope Comparison

When comparing lines to see if they are parallel, the key feature to evaluate is their slope. Parallel lines have identical slopes and never intersect. This means the coefficient of \(x\) in their linear equations, essentially the \(m\) in \(y = mx + b\), must be the same for both lines to be parallel.

In our problem, upon rewriting the equations in slope-intercept form:

• First Line: \(y = 4x - 1\)
• Second Line: \(y = 4x + 14\)

The slopes \(m\) for both lines are equal to 4. Because these slopes are identical, we can tell that the lines are parallel. The equal slopes demonstrate that although these lines may differ in where they cross the y-axis, they will never touch each other.

Linear Equations

Linear equations describe lines in two-dimensional space and demonstrate a constant rate of change. The most commonly used form is the slope-intercept form \(y = mx + b\). Linear equations are significant because they simplify calculations by offering a straightforward way to visualize relationships and trends.

In this context, understanding how to convert and handle linear equations is crucial for recognizing properties such as parallelism. The transformation we performed in our exercise involved expressing the equations as linear functions. This helped us directly see the slopes and determine the relationship between the lines.

By mastering linear equations, we can effortlessly identify and categorize different types of lines, such as horizontal and vertical lines, which have particular characteristics. This ability is not only helpful in academic exercises but also in real-world applications like graphing data and optimizing functions.