Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the line described, giving it in slope-intercept form if possible. Through \((-4,-7),\) parallel to \(x+y=5\)

Step-by-Step Solution

Answer

The equation is \( y = -x - 11 \).

Step 1: Identify the Slope of the Given Line

The given line is represented by the equation \( x + y = 5 \). First, rewrite this equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope. To do this, solve for \( y \):\[y = -x + 5\]From this, we see the slope \( m = -1 \).

Step 2: Use the Point-Slope Formula

Since the new line is parallel to the given line, it shares the same slope \( m = -1 \). Use the point-slope form of the equation \( y - y_1 = m(x - x_1) \), where \((x_1, y_1) = (-4, -7)\):\[y - (-7) = -1(x - (-4))\]Simplifying, we have:\[y + 7 = -1(x + 4)\]

Step 3: Simplify to Slope-Intercept Form

Distribute the slope \(-1\) and simplify to find the equation in slope-intercept form \( y = mx + b \):\[y + 7 = -1 \cdot x - 1 \cdot 4\]Simplifying, we get:\[y + 7 = -x - 4\]Subtract 7 from both sides to isolate \( y \):\[y = -x - 4 - 7\]So, \( y = -x - 11 \).

Key Concepts

Equation of a Line

An equation of a line is a mathematical expression that describes all the points along that line. It’s a way for us to *predict* on the graph where a line will be based on a rule, rather than having to draw or visualize it.
The most common form of the line's equation is the slope-intercept form:

• Slope-Intercept Form: The general formula for this form is \( y = mx + b \). Here, \( m \) represents the slope of the line, while \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

To convert an equation into slope-intercept form, you need to solve it for \( y \). For example, with an equation \( x + y = 5 \), solving for \( y \) would mean isolating \( y \) on one side:
Starting from:
\[ x + y = 5 \]
Subtract \( x \) from both sides yields:
\[ y = -x + 5 \]
This shows the slope \( m = -1 \) and y-intercept \( b = 5 \).
Understanding this form lets us easily graph lines and interpret relationships between variables.

Parallel Lines

Parallel lines are like two train tracks running side by side and never meeting. These lines have what's called an identical slope, which means they tilt or rise at the same angle, ensuring they never intersect.

• Identical Slopes: If two lines are described as parallel, they must share the same slope \( m \). For example, if the slope of one line is \( m = -1 \), a parallel line will also have a slope of \( m = -1 \).

Understanding parallel lines is vital when determining the equation of a line based on known conditions. For instance, if we know a line must be parallel to another given line, we simply use the same slope in our calculations, as was done in converting \( x + y = 5 \) to a line passing through a certain point in the problem above.
This makes constructing parallel lines straightforward, as the only variation lies in where the line intersects the y-axis (its y-intercept, different along parallel lines unless they are coincident lines).

Point-Slope Formula

The point-slope formula is a tool in algebra that helps us write the equation of a line when we know a point on the line and the slope of the line. By knowing just one point, \((x_1, y_1)\), and the slope \( m \), you can form the line's equation.

• Point-Slope Formula: This formula is represented as \( y - y_1 = m(x - x_1) \). This equation tells us how far \( y \) changes for a given change in \( x \) from our chosen point.

In practical use, such as in our original exercise, knowing the point \((-4, -7)\) and a slope of \( -1 \) allows the writing of the equation:
\[y - (-7) = -1(x - (-4))\]
Which we then simplify to:
\[y + 7 = -1(x + 4)\]
This is how we start building the line's equation. Converting or simplifying this further gives us the slope-intercept form, useful for graphing and interpreting relationships.
It provides a flexible framework for quickly rearranging and understanding line equations whenever a slope and a single point are known.