Problem 85 - Algebra for College Students

Question

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((-3,-1)\) and is parallel to \(y=2 x+6\).

Step-by-Step Solution

Answer

The equation of the line in standard form is \2x - y + 7 = 0\.

Step 1 - Identify the slope of the given line

The line is given by the equation \(y = 2x + 6\). The slope of this line is the coefficient of \(x\), which is \(2\). Since line \l\ is parallel to this line, it will have the same slope.

Step 2 - Use the point-slope form

Line \(l\) goes through the point \((-3, -1)\) and has a slope of \(2\). Use the point-slope form of the equation \((y - y_1 = m(x - x_1))\) where \(m\) is the slope and \( (x_1, y_1) \) is the point: \[ y + 1 = 2(x + 3) \].

Step 3 - Simplify the equation

Distribute the \(2\) on the right side: \{y + 1 = 2x + 6 + 6\}\. Then, simplify the equation to \[ y + 1 = 2x + 6 \].

Step 4 - Convert to standard form

To convert to the standard form \(Ax + By + C = 0\) and ensure integer coefficients, rearrange the equation: \[ 2x - y + 7 = 0 \].

Key Concepts

parallel lines

Parallel lines are lines in the same plane that never intersect, no matter how far they extend. They always have the same slope. If two lines are parallel, this means their steepness is identical. For example, if one line has the equation of the form [y = 2x + 6], any line parallel to it will also have a slope of [2]. In our exercise, we needed to find the equation of a line parallel to [y = 2x + 6]. Since it’s parallel, we know its slope is also [2]. Working with parallel lines is fundamental because it simplifies finding the new line’s equation once the slope is known.

point-slope form

The point-slope form of a line is very useful when you know a point on the line and the slope. The general formula is: [y - y_1 = m(x - x_1)], where:

• [y_1] is the y-coordinate of the known point.
• [x_1] is the x-coordinate of the known point.
• m is the slope of the line.

To find the equation of our line, we used the point [(-3,-1)] and the slope [2]. We substituted these values into the point-slope formula: [y - (-1) = 2(x + 3)]. Simplifying this gave us the intermediate equation. Knowing how to use the point-slope form makes working with equations of lines straightforward once you have a slope and a point.

standard form

Standard form is another way to represent line equations and is particularly handy for certain types of algebra problems. The standard form is [Ax + By + C = 0], where:

• A, B, and C are integers.
• Typically, A, should be a non-negative integer.

In our exercise, after simplifying the equation using the point-slope form, we converted it to standard form. Going from [y + 1 = 2(x + 3)] to [2x - y + 7 = 0] involved distributing and rearranging terms so that [A], [B], and [C] met the criteria needed for standard form. Understanding standard form is crucial because it makes certain algebraic processes, like finding intercepts, easier.