Problem 12 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given points. $$ (0,9) \text { and }(8,7) $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that is parallel to the given line and passes through the point \((2,4)\) is \(y-4 = -3(x-2)\).

Step 1: Determine the Slope of the Given Line

The given equation is in the form \(y = mx + c\), where \(m\) is the slope. So, the slope of the given line is -3.

Step 2: Using Point-Slope Form

The point-slope form of the line is \(y-y1=m(x-x1)\). Plugging the point \((2,4)\) and the slope -3 into this formula, we get \(y-4 = -3(x-2)\).

Step 3: Simplify the Equation of the Line

Expand the right side of the above equation and simplify to get the equation of the line in point-slope form.

Key Concepts

Equation of a Line

An equation of a line provides a mathematical description of a line on the coordinate plane. The most common forms for expressing a linear equation include the point-slope form, slope-intercept form, and standard form. In our exercise, we are particularly dealing with the point-slope form, which is written as:

• \(y - y_1 = m(x - x_1)\)

Here, \((x_1, y_1)\) represents a specific point on the line, and \(m\) is the slope of the line. This form is especially useful when you know one point on the line and the slope, as it directly incorporates these values into the equation.
By plugging in the values for the slope and coordinates of the point, you end up with a specific equation that represents the line. This makes it a straightforward way to articulate lines when both a point and the slope are known.

Parallel Lines

Parallel lines are lines that run alongside each other in the same plane and never intersect. This characteristic is deeply linked to the slopes of the lines. When two lines are parallel, they share the same slope but have different y-intercepts.

• If you have one line with a slope of \(m\), any other line that is parallel to it will also have a slope of \(m\).

In the original exercise, we looked at a line given by the equation \(y = -3x + 1\). This line has a slope of \(-3\). Any line parallel to this one will also have a slope of \(-3\). To find the equation of a line that is parallel to the given line, simply use this slope with another point that the new line passes through, as demonstrated with the point \((2,4)\). By maintaining the same slope, you ensure that the new line is parallel to the original line.

Slope-Intercept Form

The slope-intercept form is another common way to write the equation of a line, expressed as:

• \(y = mx + b\)

Here, \(m\) represents the slope of the line and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. This form is popular because it directly shows both the slope and y-intercept, making it easy to sketch the line.
After determining the equation of a line in point-slope form, you can easily convert it into slope-intercept form by simplifying the equation to isolate \(y\). This conversion helps in quickly understanding the line's behavior and interaction with the y-axis. In the exercise, the parallel line equation was originally \(y - 4 = -3(x - 2)\). By expanding and simplifying, it can be transformed into slope-intercept form for an easy read of both its slope and y-intercept.