Problem 66 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-1,3)\) and parallel to the line whose equation is \(3 x-2 y-5=0\)

Step-by-Step Solution

Answer

The line passing through the point (-1,3) and parallel to the line with equation \(3x - 2y - 5 = 0\) is given by the equation \(y = 1.5x + 4.5\).

Step 1: Identify the Slope of the Given Line

We first have to rearrange the given equation \(3x - 2y - 5 = 0\) to the slope-intercept form \(y = mx + b\). This gives us \(y = 1.5x + 2.5\). Thus, the slope of the given line, \(m\), is 1.5.

Step 2: Write the Equation in Point-Slope Form

Given that the line passes through (-1,3) and is parallel to the line above, it will have the same slope. Therefore, the point-slope form of the line is \(y - y_1 = m(x - x_1)\), which gives \(y - 3 = 1.5 (x + 1)\) after substituting the slope and point values.

Step 3: Write the Equation in Slope-Intercept Form

The final step is to rearrange the point-slope formula into the slope-intercept form \(y = mx + b\). Distributing on the right side and adding 3 to both sides gives us \(y = 1.5x + 4.5\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for students grappling with algebra and coordinate geometry. This form is expressed as the equation of a line in the format of \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept, the point where the line crosses the y-axis.

To convert an equation into the slope-intercept form, you need to solve for \(y\) such that it is isolated on one side of the equation. For instance, if you start with an equation like \(3x - 2y - 5 = 0\), you'll want to rearrange it to look like \(y = mx + b\). This usually involves moving the \(y\)-term to one side and the other terms to the opposite side, and then dividing by the coefficient of \(y\) to solve for it. The slope-intercept form makes it easy to graph the line and to understand its behavior quickly – you immediately know the steepness of the line (the slope) and where to begin plotting it (the y-intercept).

Parallel Lines

Learning about parallel lines is a key element in geometry. Two lines are parallel if they have the same slope, meaning they rise and run at the same angle and will never cross paths, no matter how far they are extended. This concept has practical implications in real-world design and construction, ensuring features remain equidistant.

When working with equations of lines, if you're told that a new line must be parallel to an existing one, you'll determine the slope of the existing line and apply that same slope to your new line. For instance, if a given line has an equation in slope-intercept form like \(y = 1.5x + 2.5\), any line with a slope of 1.5 will be parallel to this given line. This understanding helps us to generate new equations efficiently, especially when we must find the equation of a line parallel to another that passes through a certain point.

Algebraic Slope Calculation

Calculating the slope for a line is fundamental in algebra and helps us understand how steep a line is. Algebraically, the slope is often represented as \(m\) and can be calculated using the coordinates of two points on the line using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of these two points.

For students, mastering this calculation is important for graphing lines, solving systems of equations, and analyzing linear relationships. It's also crucial when determining the nature of the parallelism between lines – since parallel lines have identical slopes. In exercises, once the slope of one line is known, you can use it to write the equation of any line parallel to it, as they share this key characteristic. The ability to calculate and apply the concept of slope empowers students to tackle a wide range of problems in algebra.