Problem 40 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that is parallel to the given line and passes through the given point. $$ y=\frac{1}{2} x,(8,-10) $$

Step-by-Step Solution

Answer

The equation of the line, in slope-intercept form, that is parallel to the given line and passes through the given point is \(y=\frac{1}{2}x-14\).

Step 1: Identify the slope

The slope \(m\) of the line \(y=\frac{1}{2}x\) is \(\frac{1}{2}\). Because the line we are trying to find should be parallel to the given one, it will have the same slope, hence \(m=\frac{1}{2}\).

Step 2: Apply the point-slope form

The point-slope form of a line is \(y-y_1=m(x-x_1)\), where \((x_1, y_1)\) are coordinates of a particular point on the line. We’ll substitute \((x_1, y_1)\) with \((8, -10)\) and the value of \(m\) we got in step 1, we'll get: \(y-(-10) = \frac{1}{2}(x-8)\). This simplifies to: \(y+10=\frac{1}{2}x-4\).

Step 3: Convert to slope-intercept form

We can rewrite the equation of the line from step 2 to the slope-intercept form \(y = mx + b\), by subtracting 10 from both sides: \(y=\frac{1}{2}x-4-10 \). This simplifies to the final equation: \(y=\frac{1}{2}x-14\).

Key Concepts

Parallel Lines

Understanding the concept of parallel lines is crucial when working with linear equations. Parallel lines are two lines that never intersect and have the same slope. In mathematical terms, if two non-vertical lines are parallel, their slopes will be equal.

For example, if given the line equation \(y = \frac{1}{2}x\), any line parallel to it must also have a slope of \(\frac{1}{2}\). This property allows us to write equations for new lines that we know will never meet the original line. Therefore, in our problem, we use the original line's slope to determine the slope of the parallel line we are trying to find.

Point-Slope Form

Point-slope form is a way of writing the equation of a line given a point on the line and its slope. The form of the equation is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a specific point on the line and \(m\) is the slope.

This form is extremely useful when you want to write the equation of a line that passes through a specific point and has a certain slope. Additionally, this form can easily be manipulated into other forms of linear equations, such as slope-intercept form, which is often more convenient for graphing and analysis.

Slope of a Line

The slope of a line is a measure of its steepness and is usually denoted by \(m\). It is calculated as the ratio of the change in \(y\) (the vertical change) to the change in \(x\) (the horizontal change) between two distinct points on the line.

To find the slope, use the formula \(m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line. For a line to be parallel to another, its slope must be exactly equal to the slope of the other line.

Line Equation

The line equation in slope-intercept form is one of the most familiar forms, written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, the point at which the line crosses the y-axis.

The slope-intercept form makes it easy to graph a line and understand its behavior. Converting a point-slope form equation to slope-intercept form involves solving for \(y\) and simplifying, so that you can clearly identify the slope and the y-intercept. In our case, we converted the initial point-slope form equation to slope-intercept form by isolating \(y\) and simplifying the constants to find \(b\).