Problem 36 - Algebra Form and Function

Question

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

Step-by-Step Solution

Answer

Answer: Yes, the lines are parallel because their slopes are the same (4).

Step 1: Rewrite given equations in slope-intercept form

For the given lines, we need to rewrite them in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. The first equation is: $$y = 5 + 4(x - 2)$$ To put it in slope-intercept form: y = 5 + 4x - 8 y = 4x - 3 The second equation is already in slope-intercept form: $$y = 2 + 4x$$ which can also be written as $$y = 4x + 2$$

Step 2: Identify the slopes

Now that the equations are in slope-intercept form, we can identify the slopes. Remember that the slope is the coefficient of x. For the first equation (y = 4x - 3), the slope (m1) is 4. For the second equation (y = 4x + 2), the slope (m2) is also 4.

Step 3: Compare the slopes

Once we have identified the slopes, we can check if they are the same. In this case, the slopes are: m1 = 4 m2 = 4 Since both slopes are the same, the lines are indeed parallel.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a very common way to express linear equations. It is written as \(y = mx + b\), where \(m\) stands for the slope of the line, and \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.
This form is extremely useful because it allows you to identify two key characteristics of a line directly from the equation:

• **Slope**: This tells you how steep the line is. It's the number in front of \(x\), and it shows the rise over run, or the change in y for each unit change in x.
• **Y-Intercept**: This is the starting point of the line on the y-axis. It's the constant term in the equation.

To use the slope-intercept form, you often need to manipulate the given equation. For example, given \(y = 5 + 4(x - 2)\), you first distribute 4, and then simplify to get \(y = 4x - 3\). Now, you have it in the slope-intercept form, making it easy to read the line's slope and y-intercept.

Slope Comparison

When examining if lines are parallel, the slopes of those lines are particularly significant. Parallel lines share an important feature: they have identical slopes. If two or more equations in slope-intercept form, like \(y = 4x - 3\) and \(y = 4x + 2\), display the same coefficient for \(x\), their slopes are equal.
For linear equations, this means:

• If \(m_1\) and \(m_2\) are slopes of two lines, the lines are parallel if \(m_1 = m_2\).
• If the slopes differ, the lines are not parallel.

In the example given, both equations reduce to a slope of \(m_1 = 4\) and \(m_2 = 4\). Since these slopes are equal, the lines represented by these equations are parallel. This characteristic of lines makes slope comparison a quick and effective method for identifying parallel lines in algebra.

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed on a coordinate plane. They're called 'linear' because they yield straight-line graphs.
They often appear in forms such as:

• Slope-intercept form: \(y = mx + b\)
• Standard form: \(Ax + By = C\), where A, B, and C are constants.

Linear equations are ubiquitous in algebra and real-life applications. They can represent various scenarios, such as constant speed, direct proportionality, and more. Understanding how to transform and analyze these equations provides insights into patterns and relationships in problem-solving.
Parallel lines, as discussed using slope comparison, are a concept that falls within the realm of linear equations. When working with these equations, identifying the form (like slope-intercept) and characteristics (slope and y-intercept) is vital.