Question

Short Response Find an equation of the line parallel to the graph of \(y=-3 x+4\) that passes through \((1,-5)\). (Lesson \(4-6)\)

Short answer

The equation of the line is \(y = -3x - 2\).

Step-by-step solution

Identify the Slope of the Given Line

The equation of the given line is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope. For \(y = -3x + 4\), the slope \(m = -3\). Lines that are parallel have the same slope.

Use the Slope for the New Line

Since the new line must be parallel to the given line, it will have the same slope. Therefore, the slope of the line we need to find is \(m = -3\).

Apply the Point-Slope Form

Use the point-slope form of a line equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (1, -5)\) and \(m = -3\). This gives us: \[y - (-5) = -3(x - 1)\] which simplifies to \[y + 5 = -3(x - 1)\].

Simplify to Find the Equation

Expand and simplify the equation from the point-slope form:\[y + 5 = -3x + 3\] Subtract 5 from both sides to isolate \(y\):\[y = -3x + 3 - 5\] \[y = -3x - 2\].

Key concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is incredibly useful for easily identifying the slope and y-intercept of a line. It is written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept—the point where the line crosses the y-axis. This form is especially helpful when graphing a line or understanding its behavior.

• Slope (\(m\)): Describes the steepness of the line. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls.
• Y-intercept (\(b\)): Provides a starting point for graphing. It's where the line crosses the y-axis.

If you're given the slope-intercept form, you can easily read off the slope and y-intercept and sketch the line on a graph. In problems involving parallel lines, having this form is beneficial because parallel lines share the same slope.
For example, given \(y = -3x + 4\), you instantly know the slope is \(-3\), allowing you to find the equation of a line parallel to it that passes through any given point.

Point-Slope Form

The point-slope form of a linear equation is ideal for situations where you know the slope of a line and a single point it passes through. It is written as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a known point on the line, and \(m\) is the line's slope. This form is perfect for constructing an equation quickly when additional details like the y-intercept are unknown. Here's why it's useful:

• Allows flexibility: Start with a given point instead of needing the y-intercept.
• Customizable: Transform the equation into other forms (like slope-intercept) easily through algebraic manipulation.

In the example "Find an equation of the line parallel to the graph of \(y = -3x + 4\) that passes through \((1, -5)\)," we use the point-slope form to adapt to the given point, using the parallel slope \(-3\). This results in the equation \(y + 5 = -3(x - 1)\), perfectly tailoring to the conditions set out in the problem.

Linear Equations

Linear equations are mathematical expressions that represent straight lines when graphed. They are characterized by a constant rate of change, known as the slope. There are several ways to express linear equations, including slope-intercept form, point-slope form, and standard form. Each form has its specific uses and advantages.

• Slope-Intercept Form: Best for graphing and quickly identifying slope and intercepts.
• Point-Slope Form: Useful for deriving an equation from a known point and slope.
• Standard Form (\(Ax + By = C\)): A more general way to write equations that can handle vertical or horizontal lines more flexibly.

Understanding linear equations is essential for solving problems involving parallel and perpendicular lines, intersection points, and other geometric concepts. They form the foundation for many real-world applications like data analysis, budgeting, and projections.
When tasked with finding parallel lines, knowing how to manipulate linear equations quickly and efficiently is crucial. By juggling between these forms, you can tailor solutions to specific scenarios just like transforming the given parallel line to pass through a new point.