Problem 31 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((1,-6) ;\) parallel to the line \(x+2 y=6\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -\frac{1}{2}x - \frac{11}{2}\).

Step 1: Find the Slope of the Given Line

To find the slope of the line given by the equation \(x+2y=6\), rewrite it in slope-intercept form \(y=mx+b\). Start by solving for \(y\): subtract \(x\) from both sides to get \(2y=-x+6\). Then, divide each term by 2 to find \(y=-\frac{1}{2}x+3\), where the slope \(m\) is \(-\frac{1}{2}\).

Step 2: Use the Slope of a Parallel Line

Since parallel lines have the same slope, the slope of the line we need to find is also \(-\frac{1}{2}\).

Step 3: Use the Point-Slope Form

To find the equation of the line that passes through the point \((1, -6)\) and has a slope of \(-\frac{1}{2}\), use the point-slope form of the line equation, \(y-y_1=m(x-x_1)\). Here, \(m=-\frac{1}{2}\), \(x_1=1\), and \(y_1=-6\). Substitute these values into the formula: \(y + 6 = -\frac{1}{2}(x - 1)\).

Step 4: Simplify the Equation

Distribute the slope on the right side: \(-\frac{1}{2}(x - 1) = -\frac{1}{2}x + \frac{1}{2}\). The equation then becomes \(y + 6 = -\frac{1}{2}x + \frac{1}{2}\). To solve for \(y\), subtract 6 from both sides: \(y = -\frac{1}{2}x + \frac{1}{2} - 6\) which simplifies to \(y = -\frac{1}{2}x - \frac{11}{2}\).

Key Concepts

Point-Slope Form

The point-slope form of a line's equation is a practical tool for finding the equation of a line when you know a point on the line and its slope. It's expressed as \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) represents the coordinates of a specific point on the line, and \(m\) is the slope. This form helps you directly plug in the values for the point and the slope.
For example, if you have a point \((1, -6)\) and a slope \(-\frac{1}{2}\), you can substitute these values directly:

• \(y_1 = -6\) and \(x_1 = 1\)
• \(m = -\frac{1}{2}\)

Plugging these into the equation, we get: \( y + 6 = -\frac{1}{2}(x - 1) \). This enables us to quickly formulate the equation of the line.The point-slope form is beneficial when changes need to be made based on different points and slopes, keeping the process straightforward.

Slope-Intercept Form

The slope-intercept form is another way to express the equation of a line. It's written as \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept. The y-intercept \(b\) is where the line crosses the y-axis, which can be easily observed in this form.
In the relevant solution, when rewriting the line \(x + 2y = 6\) into this form, it helped identify the slope as \(-\frac{1}{2}\):

• Subtract \(x\) from both sides to get \(2y = -x + 6\)
• Divide each term by 2: \(y = -\frac{1}{2}x + 3\)

Using the slope-intercept form makes understanding and comparing lines easier. It shows the slope clearly and directly displays the intercept, simplifying graph analysis and computation steps. This form is especially useful when converting between different equation formats.

Parallel Lines

Parallel lines are fascinating because they never intersect; they always maintain the same distance apart. A defining feature of parallel lines is that they have the same slope. This concept helps in finding lines that are parallel to each other, as they should share the same angle of inclination.
For instance, to find a line parallel to \(x + 2y = 6\), rewrite it into the slope-intercept form \( y = -\frac{1}{2}x + 3 \). This shows that its slope is \(-\frac{1}{2}\). Therefore, any line parallel to it should also have a slope of \(-\frac{1}{2}\), such as the line through the point \((1, -6)\).

• Retain the slope \(-\frac{1}{2}\) from the given line.
• Use different y-intercept values to have different parallel lines passing through different coordinates.

Understanding parallel lines aids in robust line equation formulation, keeping track of how lines interact within a plane.