Problem 44 - Elementary Algebra

Question

Graph the line that passes through the given point and has the given slope. (Objective 3 ) $$(1,-4), m=-3$$

Step-by-Step Solution

Answer

The line's equation is \(y = -3x - 1\), graph starts at \((0, -1)\) and moves down 3, right 1.

Step 1: Identify the point and slope

We are given the point \((1, -4)\) and a slope \(m = -3\). This means our line passes through the point \((1, -4)\) with a slope of \(-3\).

Step 2: Use the Point-Slope Formula

The general formula for a line in point-slope form is \(y - y_1 = m(x - x_1)\). Here, \((x_1, y_1)\) is the point on the line, and \(m\) is the slope. Substitute \(x_1 = 1\), \(y_1 = -4\), and \(m = -3\) to get \(y - (-4) = -3(x - 1)\). Simplifying gives us the equation \(y + 4 = -3(x - 1)\).

Step 3: Simplify the Equation

Distribute the slope on the right side of the equation: \(y + 4 = -3x + 3\). Subtract 4 from both sides to solve for \(y\): \(y = -3x + 3 - 4\), which simplifies to \(y = -3x - 1\). This is the equation of the line in slope-intercept form.

Step 4: Graph the Equation

To graph the line \(y = -3x - 1\), start by plotting the y-intercept, which is \((0, -1)\). Next, use the slope \(-3\), which means from the y-intercept, move 3 units down and 1 unit right to mark another point on the line. Connect these points with a straight line, extending it across the graph.

Key Concepts

Point-Slope Form

The point-slope form is a powerful tool for graphing linear equations, especially when you know one point on the line and the slope of the line. It is expressed as \( y - y_1 = m(x - x_1) \).
This form is derived from the concept of slope, which represents the steepness and direction of a line.
If you know a point \((x_1, y_1)\) that the line passes through, and the slope \(m\), you can use this formula to find the equation of the line.

• "\(y\)" and "\(x\)" are variables representing any point on the line,
• "\(y_1\)" and "\(x_1\)" are the coordinates of a given point on the line, and
• "\(m\)" is the slope.

This formula is very intuitive because it shows how every point on the line relates to the known point and the slope. When you plug in the values, you instantly see the relationship between the known point and any new point on the line.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. It is written as \( y = mx + b \), where:

• "\(m\)" is the slope of the line,
• "\(b\)" is the y-intercept, which is the point where the line crosses the y-axis, and
• "\(x\)" and "\(y\)" are variables representing points on the line.

This form is very useful when you need to quickly graph a line, as you can directly see the slope and the y-intercept from the equation. You can start graphing by plotting the y-intercept on the y-axis, then using the slope to find other points.
For example, in the solution, we derived the slope-intercept form \(y = -3x - 1\) from the point-slope form. Here, \(-3\) is the slope and \(-1\) is the y-intercept.

Slope of a Line

The slope is a fundamental concept in understanding linear equations. It measures the steepness of a line and is often denoted by \(m\). Mathematically, the slope is defined as the change in \(y\) (vertical change) over the change in \(x\) (horizontal change) between two points on a line.
The formula for slope is given as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

• If the slope \(m\) is positive, the line rises as it moves from left to right.
• If the slope \(m\) is negative, the line falls as it moves from left to right.
• A zero slope means the line is horizontal, and
• An undefined slope means the line is vertical.

Understanding slope gives you the ability to predict how a line behaves and how it will graph. In the solution, the given slope \(-3\) means that for every unit you move to the right along the x-axis, the line moves 3 units down along the y-axis.