Problem 57 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line that passes through the point \((-3,0)\) and is parallel to the line whose equation is \(y=-x+4\)

Step-by-Step Solution

Answer

The equation is \( y = -x - 3 \).

Step 1: Identify the slope of the given line

The equation of the given line is in the slope-intercept form, which is \[ y = -x + 4 \]. Here, the coefficient of \( x \) is the slope. Therefore, the slope (m) of the given line is \( -1 \).

Step 2: Use the slope of the parallel line

Lines that are parallel to each other have the same slope. So the slope of the line we need to find will also be \( -1 \).

Step 3: Use the point-slope form of the equation

The point-slope form of a line's equation is \[ y - y_1 = m(x - x_1) \]. Here, \( (x_1, y_1) \) is the point through which the line passes, and \( m \) is the slope. Using the point \( (-3, 0) \) and the slope \( -1 \), we get: \[ y - 0 = -1(x - (-3)) \].

Step 4: Simplify the equation

Simplify the equation: \[ y = -1(x + 3)\] \[ y = -x - 3 \]. Therefore, the equation of the line that passes through \((-3, 0)\) and is parallel to the line \( y = -x + 4\) is \( y = -x - 3 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to write the equation of a line. It makes it easy to understand both the slope and the y-intercept of the line. The general form is \( y = mx + b \), where:

• \( m \) is the slope of the line
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis

For example, in the equation \( y = -x + 4 \), the slope \( m \) is \( -1 \), and the y-intercept \( b \) is \( 4 \). This form helps in quickly identifying the characteristics of the line, like its steepness and where it intersects the y-axis. When you need to write the equation of a line or understand the relationship between two lines, the slope-intercept form is often the go-to choice.

Point-Slope Form

Another useful way to write the equation of a line is the point-slope form. This form is especially handy when you know one point on the line and the slope. The general form of the point-slope equation is \( y - y_1 = m(x - x_1) \), where:

• \( (x_1, y_1) \) is a specific point on the line
• \( m \) is the slope of the line

In our exercise, we know the point \( (-3, 0) \) and the slope \( -1 \) from the parallel line. Plugging these values into the point-slope form gives us: \[ y - 0 = -1(x - (-3)) \] Simplifying this, we get: \[ y = -x - 3 \] Thus, the point-slope form is a powerful tool for writing the equation of a line when a point and the slope are known.

Parallel Lines

Parallel lines are a fascinating concept in geometry. These are lines that run in the same direction and never intersect. An important property of parallel lines is that they have the same slope. This means if one line has a slope of \( m \), any line that is parallel to it will also have a slope of \( m \).
In our exercise, the original line has an equation \( y = -x + 4 \) with a slope of \( -1 \). Because our new line must be parallel to this, its slope will also be \( -1 \). This makes solving the problem easier since we only need to ensure the new line maintains this slope while passing through the given point \( (-3, 0) \). The resulting parallel line equation is \( y = -x - 3 \). Remember, just keep an eye on the slope and ensure it's the same, and you will correctly identify parallel lines!