Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the line described, giving it in slope-intercept form if possible. Through \((3,-2),\) parallel to \(2 x-y=5\)

Step-by-Step Solution

Answer

The equation of the line is \(y = 2x - 8\).

Step 1: Identify the slope of the given line

The given line is in the form of the linear equation \(2x - y = 5\). To find the slope, we convert this equation into slope-intercept form \(y = mx + b\), where \(m\) is the slope. Start by isolating \(y\):\[2x - y = 5\]Add \(y\) to both sides: \[2x = y + 5\]Subtract \(5\) from both sides: \[y = 2x - 5\]Thus, the slope \(m\) is \(2\).

Step 2: Use the slope for the parallel line

Lines that are parallel share the same slope, so the new line will have the same slope, \(m = 2\).

Step 3: Use point-slope form to find the equation

With the slope \(m = 2\) and the point \((3, -2)\), use the point-slope formula, \(y - y_1 = m(x-x_1)\), where \((x_1, y_1)\) is the point on the line. Substitute \(m = 2\), \(x_1 = 3\), and \(y_1 = -2\):\[y - (-2) = 2(x - 3)\]Simplifying: \[y + 2 = 2x - 6\]

Step 4: Simplify to slope-intercept form

To convert \(y + 2 = 2x - 6\) into the slope-intercept form \(y = mx + b\), solve for \(y\):\[y = 2x - 6 - 2\]Simplify the right side: \[y = 2x - 8\]Thus, the equation of the line is \(y = 2x - 8\).

Key Concepts

Linear Equations

Linear equations form the backbone of algebra and are usually expressed in the form \(Ax + By = C\). These equations represent straight lines when plotted on a graph. The two most common forms in which linear equations are expressed are the standard form \(Ax + By = C\) and the slope-intercept form \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept — the point where the line crosses the y-axis.
When converting a linear equation from standard form to slope-intercept form, focusing on isolating \(y\) on one side is key. This makes it easier to identify the slope \(m\) and further analyze the properties of the line.

• Standard form: Generally looks like \(Ax + By = C\)
• Slope-intercept form: Simplifies to \(y = mx + b\)

This simplification helps in understanding and graphing linear equations effectively.

Parallel Lines

Parallel lines are fascinating geometrical entities. In mathematics, lines are considered parallel if they maintain a constant distance from each other and never intersect, regardless of how far they extend. This occurs when two lines have exactly the same slope.
For example, if you have the equation of a line \(y = 2x - 5\) with a slope \(m = 2\), any line that is parallel to this will also have a slope of \(m = 2\). The y-intercept may vary, allowing for different vertical placements of the line, but the slope remains consistent.

• Parallel lines have identical slopes.
• The distance between them remains constant.
• They never meet or intersect.

Understanding parallel lines is crucial when solving problems that involve constructing or identifying lines that share the same slope.

Point-Slope Form

The point-slope form is a practical tool for deriving the equation of a line when you know the slope and a specific point it passes through. The formula is represented as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the given point and \(m\) is the slope.
This form is particularly useful when we need to write the equation of a line quickly, given a point and a slope. It's a straightforward process:

• Identify the slope \(m\).
• Note down the point \((x_1, y_1)\).
• Substitute these values into the point-slope formula.

After substituting, solving for \(y\) will help convert the equation into the slope-intercept form. For instance, given \(m = 2\) and the point \((3, -2)\), the point-slope formula becomes \(y + 2 = 2(x - 3)\), which can be further simplified to find the slope-intercept form of the equation. This conversion helps in visualizing the line on a graph effectively.