Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the line described, giving it in slope-intercept form if possible. Through \((-1,4),\) parallel to \(x+3 y=5\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -\frac{1}{3}x + \frac{11}{3}\).

Step 1: Identify the Given Equation's Form

The given line is represented by the equation \(x + 3y = 5\). This is in standard form. We need to convert it to slope-intercept form to identify its slope.

Step 2: Convert to Slope-Intercept Form

We'll rearrange the equation \(x + 3y = 5\) into slope-intercept form \(y = mx + b\).Subtract \(x\) from both sides to get \(3y = -x + 5\).Now, divide every term by 3 to solve for \(y\): \[y = -\frac{1}{3}x + \frac{5}{3}\]This means the slope of the given line is \(-\frac{1}{3}\).

Step 3: Determine the Slope of the Parallel Line

Since parallel lines have the same slope, the line we need to find will also have the slope \(-\frac{1}{3}\).

Step 4: Use the Point-Slope Formula

With the slope known, use the point-slope form of a line \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is the given point:Substitute the point \((-1, 4)\) and the slope \(m = -\frac{1}{3}\):\[y - 4 = -\frac{1}{3}(x + 1)\].

Step 5: Simplify to Slope-Intercept Form

Expand and simplify the equation from the point-slope form:\[y - 4 = -\frac{1}{3} x - \frac{1}{3}\]Add 4 to both sides to solve for \(y\):\[y = -\frac{1}{3} x + \frac{11}{3}\]Now the equation is in slope-intercept form, \(y = mx + b\).

Key Concepts

Parallel Lines

Parallel lines are lines that never intersect, no matter how far they are extended. This unique property stems from having the same slope, meaning they "slant" the same way. In mathematical terms, if two lines are parallel, the coefficient of the variable x in their equations are the same, when expressed in slope-intercept form \(y = mx + b\).
Understanding parallel lines is crucial for geometry and algebra, as it helps us find equations of lines that run alongside a given line.
For example, if a line in slope-intercept form has the equation \(y = -\frac{1}{3}x + b\), any line parallel to this one will also have a slope of -\frac{1}{3}\. Even if you only know the starting point, the parallel slope gives you direction.

Standard Form

The standard form of a line is another way to express a linear equation as \(Ax + By = C\), where A, B, and C are integers. This format is often preferred for certain calculations and is especially helpful for quickly identifying the x and y-intercepts.
For instance, consider the line given by the equation \(x + 3y = 5\).
Here, A is 1, B is 3, and C is 5.

• To convert this into slope-intercept form (which is useful for identifying the slope), manipulate the equation to solve for y.
• This conversion involves shifting the x-term to the other side and then dividing every term by the coefficient of y.

By converting \(x + 3y = 5\) into slope-intercept form, we can easily identify the slope and proceed with finding parallel lines.

Point-Slope Form

The point-slope form is a valuable formula used to find the equation of a line when you know its slope and one point on the line. It is designed as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the given point.
This formula simplifies the process of crafting equations, especially when dealing with parallel or perpendicular lines.

• To use it effectively, substitute the known slope and coordinates of the point into the formula.
• In our example, with the slope \(-\frac{1}{3}\) and point \((-1, 4)\), the equation forms as: \y - 4 = -\frac{1}{3}(x + 1)\.
• Expanding this helps to convert it to other forms like slope-intercept.

This versatile tool can bridge between the point-focused and equation-focused understanding of a line.

Slope of a Line

The slope of a line is a number that describes both the direction and steepness of the line. It is typically represented by \(m\) in the slope-intercept form equation \(y = mx + b\). Understanding slope is key to grasping how lines are related, especially in concepts like parallel and perpendicular lines.

• If two lines have equal slopes, they are parallel.
• If their slopes multiply to -1, they are perpendicular.

For the given line \(x + 3y = 5\), once converted to slope-intercept form \(y = -\frac{1}{3}x + \frac{5}{3}\), it is evident that the slope is -\frac{1}{3}\. This information was vital to find a parallel line through a particular point, guiding us to the final equation efficiently.