Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Parallel to the line \(2 x-y=-2\); containing the point (0,0)

Short answer

y = 2x

Step-by-step solution

Identify the slope of the given line

The given line is provided in the general form: 2x - y = -2. Convert this to the slope-intercept form (y = mx + b) to identify its slope (m).

Convert to slope-intercept form

Starting from the general form, solve for y: 2x - y = -2. Add y to both sides: 2x = y - 2. Then, add 2 to both sides: y = 2x + 2. Therefore, the slope (m) is 2.

Identify the slope of the parallel line

Since parallel lines have the same slope, the slope of the line we're looking for is also 2.

Use the point-slope form

The equation of a line in point-slope form is: y - y_1 = m(x - x_1), where (x_1, y_1) is a point on the line and m is the slope. Given the point (0,0) and slope 2, substitute these values into the equation: y - 0 = 2(x - 0). This simplifies to y = 2x.

Key concepts

slope-intercept form

The slope-intercept form of a line's equation is very useful for easily identifying the slope and y-intercept of the line. It is written as \(y = mx + b\).
Here, \(m\) represents the slope, and \(b\) represents the y-intercept, which is the point where the line crosses the y-axis. For example, in the equation \(y = 2x + 2\), the slope \(m\) is 2 and the y-intercept \(b\) is 2.
Converting other forms of a line's equation to slope-intercept form helps us quickly recognize these elements. To convert from the general form like \(2x - y = -2\) to the slope-intercept form:
1. Rearrange the equation to solve for \( y\).
2. Here, we get: \(y = 2x + 2\).
Now, we have the equation in slope-intercept form with a slope of 2.
This conversion process is a handy skill for understanding the properties of a line.

parallel lines

Parallel lines are lines in the same plane that never intersect. They always have the same slope.
If you know the slope of one line, you know the slope of any line parallel to it.
For instance, if the equation of one line is \(y = 2x + 2\), and we want to find a parallel line passing through a specific point, the parallel line must also have a slope of 2.
In our case, the line parallel to \(2x - y = -2\) and passing through the point \((0,0)\) has the same slope as the original line. This slope is 2 since we converted the original line to the slope-intercept form and found \( y = 2x + 2\).

point-slope form

The point-slope form of a line's equation is another way to express the equation of a line if you know the slope and a specific point on the line.
This form is written as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
For instance, if you know a line passes through the point \((0,0)\) and has a slope of 2, you can plug these values into the point-slope form equation:
\(y - 0 = 2(x - 0)\).
Simplifying this we get:
\(y = 2x\).
Using the point-slope form allows you to easily write the equation of a line when given a point and the slope.

general form of a line

The general form of the equation of a line is another way to represent a line in the Cartesian plane.
It is often written as \(Ax + By + C = 0\), where \(A, B,\) and \(C\) are constants.
For example,\(2x - y = -2\) is in the general form.
The general form can be converted to the slope-intercept form to reveal the slope and y-intercept.
For instance, converting\(2x - y = -2\) to slope-intercept form:

• Add y to both sides to get \(2x = y - 2\).
• Add 2 to both sides to isolate y: \( y = 2x + 2\).

This reveals that the slope \(m\) is 2 and the y-intercept \(b\) is 2.
Knowing both forms is helpful for different situations, making it easier to understand and work with linear equations.