Problem 20 - Algebra Form and Function

Question

Write an equation in point-slope form for the line. Through (-1,-8) and parallel to \(y=5 x-2\)

Step-by-Step Solution

Answer

Answer: The equation of the line is \(y + 8 = 5(x + 1)\).

Step 1: Identify the slope of the given line

In the equation \(y=5x-2\), it is already in the slope-intercept form, which is \(y=mx+b\), where m is the slope, and b is the y-intercept. In this case, the slope of the given line is 5.

Step 2: Write the point-slope form equation template

The point-slope form of an equation is given by \(y - y_1=m(x-x_1)\), where (x1, y1) is a point on the line, and m is the slope. As we want to find a line parallel to the given one, the slope will also be 5. So the equation will look like \(y - (-8) = 5(x - (-1))\)

Step 3: Simplify the equation

Now that we have our point-slope form equation, we can simplify it as follows: \(y + 8 = 5(x + 1)\) This is the equation of the line that passes through the point (-1, -8) and is parallel to the line \(y = 5x - 2\).

Key Concepts

slope-intercept form

The slope-intercept form of a line is one of the most commonly used ways to express the equation of a line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) indicates the y-intercept. This form easily showcases two key characteristics of a line:

• The slope \( m \) tells us how steep the line is. A larger absolute value of \( m \) means a steeper line.
• The y-intercept \( b \) gives the point where the line crosses the y-axis.

This form is especially useful as it allows quick understanding of the line's direction and position on a graph. For instance, in the equation \( y = 5x - 2 \), the slope is 5, suggesting that for each step increase in \( x \), \( y \) increases by 5. Meanwhile, it crosses the y-axis at -2. Understanding slope-intercept form helps greatly when comparing lines or determining parallel and perpendicular lines.

parallel lines

Parallel lines are a fundamental concept in geometry. They are lines in a plane that never intersect, no matter how far they extend. The critical feature of parallel lines is that they have the same slope. This means they incline at the same angle relative to the x-axis.

When you have two linear equations in slope-intercept form, for example, \( y = 5x - 2 \) and \( y = 5x + 7 \), both have a slope of 5, which indicates they are parallel. To determine if two lines are parallel, simply compare their slopes:

• If the slopes are equal, \( m_1 = m_2 \), the lines are parallel.
• If the slopes differ, the lines will eventually intersect and are not parallel.

In our exercise, since we were provided with a line equation \( y = 5x - 2 \), and required to find a parallel line passing through a specific point \((-1, -8)\), the key was using the same slope of 5 to maintain parallelism.

equation of a line

Finding the equation of a line involves determining a formula that represents all the points on the line. There are several forms for expressing this equation, but they typically rely on the slope and particular points on the line.

For lines passing through a specific point and having a known slope, like in our problem scenario, the point-slope form is a direct way to construct the equation: \[ y - y_1 = m(x - x_1) \]Here, \((x_1, y_1)\) is a known point, and \( m \) is the slope.

• Start by taking the known point and placing it in the equation opposite the corresponding variable, \( x_1, y_1 \).
• Insert the slope into the equation.
• Simplify if necessary to convey the relationship clearly.

For example, with a point \((-1, -8)\) and slope 5, our line's equation became \( y + 8 = 5(x + 1) \). Finding a line's equation is a fundamental skill in algebra, helping to understand not only how a line moves but also how it relates to other lines in a geometric space.