Problem 37 - Algebra 1

Question

Write an equation in point-slope form of the line that passes through the given point and has the given slope. $$ (8,-1), m=0 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the point (8, -1) and has the slope 0 is \( y = -1 \).

Step 1: Identify The Given Point and Slope

The task gives the point as (8,-1) and the slope 'm' as 0. In that case, the x-coordinate \( x_1 = 8 \), the y-coordinate \( y_1 = -1 \), and the slope \( m = 0 \).

Step 2: Substitute the Values

Now substitute these values into the point-slope form equation \( y - y_1 = m(x - x_1) \). Replacing the values, we get \( y - (-1) = 0(x - 8) \).

Step 3: Simplify the Equation

We then simplify the equation. Zero times any number is zero, so \( 0( x - 8) = 0 \). The equation becomes \( y - (-1) = 0 \) which further simplifies to \( y + 1 = 0 \). The final equation is \( y = -1 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most commonly used representations of a linear equation. It's written as \( y = mx + b \), where:

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

This form highlights both the slope and the position of the line on the y-axis, making it very helpful for quickly sketching or understanding a linear equation's behavior.
To convert an equation from any form, like the standard or point-slope form, into the slope-intercept form, isolate \( y \) and reformulate in the \( y = mx + b \) structure.
In our exercise, after simplifying the point-slope form equation, we ended up with \( y = -1 \). This means the line is horizontal and does not have a slope component; hence it's of the form \( y = mx + b \) with \( m = 0 \) and \( b = -1 \). This indicates a straight line crossing the y-axis at \(-1\).

Linear Equations

Linear equations are equations that form a straight line when graphed on a coordinate plane. These equations can be expressed in different formats, including point-slope form, slope-intercept form, and standard form. The essence of a linear equation is that it has no exponents higher than one on the variables.
At the heart of linear equations is the concept of a slope \( m \), which determines the tilt of the line. A zero slope, as in our exercise, means the line is perfectly flat and horizontal.
The beauty of working with linear equations is their predictability—they graph as straight lines and can easily be interpreted. Linear equations describe relationships with constant change. This simplification makes them very useful for modeling various scenarios in mathematics and the real world.

Graphing Linear Equations

Graphing linear equations is a fundamental skill in math. It involves plotting points on a coordinate grid to form a straight line. Begin by identifying key features of the line, such as its slope and intercepts.
When graphing, you can use different starting points:

• If working with the slope-intercept form, start at the y-intercept \( b \) and then use the slope \( m \) to determine your next points by rising or falling (changing y) relative to the run (changing x).
• For a point-slope form equation, begin with the given point, then use the slope to find additional points on the line.

In our exercise, we determined the equation \( y = -1 \). This is a horizontal line plot where every point on the line has a y-coordinate of \(-1\). No matter the x-value, the line remains flat, highlighting the absence of vertical change.