Problem 35 - Algebra 1

Question

Write an equation of the line that is parallel to the given line and passes through the given point. $$y=-3 x+1,(4,2)$$

Step-by-Step Solution

Answer

The equation of the line that's parallel to y=-3x+1 and passes the point (4,2) is y=-3x+14.

Step 1: Identify the Given Values

First, identify the slope, m, from the given equation as -3. The point through which the line passes is (4,2), where 4 is the x-coordinate and 2 is the y-coordinate.

Step 2: Substitute the Values

Substitute these values into the equation of a line (y=mx+b). This gives us the equation: 2 = -3 * 4 + b.

Step 3: Solve for b

Solving for b (the y-intercept), you’ll get b = 2 + 12 = 14.

Step 4: Write the Equation of the Parallel Line

Now, you'll write the equation of the line that's parallel to y=-3x+1 and passes the point (4,2). This becomes: y = -3x + 14.

Key Concepts

Parallel Lines

Parallel lines are an important concept in geometry and algebra. They have the distinct feature of never intersecting, no matter how far they are extended.
Mathematically, parallel lines share the same slope. Suppose you have a line given by the equation \( y = mx + b \). If another line is parallel to it, it will also have a slope of \( m \).
This means that the only difference between two parallel lines is their y-intercept. When determining the equation of a parallel line, keep the slope constant and change the y-intercept to suit the conditions provided.

Slope-Intercept Form

The slope-intercept form is a widely used format for linear equations. It is expressed as \( y = mx + b \). This equation format makes it easy to identify two critical aspects of a line: the slope \( m \) and the y-intercept \( b \).

• The slope \( m \) indicates the steepness and direction of the line. A positive slope points upward, moving from left to right, while a negative slope points downward.
• The y-intercept \( b \) is simply the point where the line crosses the y-axis.

Understanding the slope-intercept form is crucial for graphing lines and examining their properties.
It becomes especially handy when identifying parallel lines since the task primarily involves locating the slope.

Y-Intercept

The y-intercept is a fundamental concept in understanding the position of a line on a graph. It refers to the specific point where a line crosses the y-axis. This point is represented by the coordinate \( (0, b) \), where \( b \) is the actual y-intercept value.

• To find the y-intercept for a given line, set \( x = 0 \) in the equation \( y = mx + b \). This gives you \( y = b \), indicating that the y-value at this point is \( b \).
• In practical terms, the y-intercept tells you where the line "starts" or passes when drawn on a graph.

In the context of parallel lines, altering the y-intercept allows you to derive new, yet parallel, lines from the original equation. Understanding and manipulating the y-intercept is thus a key part of line equations.