Problem 49 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Through \((1,2),\) parallel to \(y=5\)

Step-by-Step Solution

Answer

The equation of the line is \(y = 2\).

Step 1: Identifying the Slope

The line "parallel to \(y = 5\)" means it must have the same slope as \(y = 5\). The equation \(y = 5\) is a horizontal line, and the slope of any horizontal line is \(m = 0\).

Step 2: Using Point-Slope Form

To find the equation of a line parallel to \(y = 5\), we use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Here, \((1, 2)\) is the point, so substituting, we get: \(y - 2 = 0(x - 1)\).

Step 3: Simplifying the Equation

Simplify the equation from the previous step: \(y - 2 = 0\times(x - 1)\) simplifies to \(y - 2 = 0\).

Step 4: Writing in Slope-Intercept Form

Convert the equation \(y - 2 = 0\) to slope-intercept form \(y = mx + b\). Since \(-2\) is on one side, we add \(2\) to both sides to get \(y = 2\). This is already in slope-intercept form since \(m = 0\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a popular and standard way to represent a line. It is expressed as \( y = mx + b \), where:

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

This form is particularly useful because it provides a clear view of both the slope and y-intercept of a line. To convert equations into this form, we often rearrange equations or perform algebraic manipulations. For example, if given a linear equation like \( y - 2 = 0 \), we simply solve for \( y \) to find \( y = 2 \), which matches the slope-intercept form with \( m = 0 \) and \( b = 2 \). This process highlights just how versatile and straightforward the slope-intercept form can be.

Parallel Lines

Understanding parallel lines is crucial when dealing with equations and graphs. Parallel lines are lines in the same plane that do not intersect. One key characteristic of parallel lines is that they have the same slope.

• If given two linear equations, to check for parallelism, ensure their slopes \( m \) are equal.
• If two lines are parallel and we know the equation of one, we can find the equation of any other line parallel to it by simply maintaining the same slope.
• For example, lines parallel to \( y = 5 \) will all have a slope of \( m = 0 \) because \( y = 5 \) is a horizontal line.

When asked to find a line parallel to another, like in our exercise, the method involves keeping this slope consistent and applying it to a new equation composed with any given points, reinforcing the consistency of their non-intersecting paths.

Horizontal Lines

Horizontal lines are unique and easily recognizable features in algebra and the coordinate plane. They run left to right and are represented by an equation like \( y = c \), where \( c \) is a constant value.

• These lines have a defining slope of \( m = 0 \), meaning they have no incline.
• The y-coordinate remains constant across all x-values, resulting in no vertical change.
• In a graph, a horizontal line passes parallel to the x-axis.

The equation \( y = 5 \), for instance, signifies a horizontal line at the y-coordinate of 5. Any line parallel to it would also be a horizontal line, just at a different y-value, unless specified to pass through a particular point. Hence, when you see horizontal lines, remember their unchanging nature and straightforward representation in the mathematical world.