Problem 22 - Algebra 2

Question

Graph the line passing through the given point with the given slope. $$ (1,2), m=-3 $$

Step-by-Step Solution

Answer

The line with slope -3 through (1,2) is \( y = -3x + 5 \).

Step 1: Write the Point-Slope Formula

To graph a line with a given point and slope, we use the point-slope formula of a line: \[ y - y_1 = m(x - x_1) \]where \((x_1, y_1)\) is the given point and \(m\) is the slope.

Step 2: Substitute the Given Values

Substitute \((x_1, y_1) = (1, 2)\) and \(m = -3\) into the point-slope formula:\[ y - 2 = -3(x - 1) \]

Step 3: Distribute the Slope

Apply the distributive property to the equation:\[ y - 2 = -3x + 3 \]

Step 4: Solve for y

To put the equation in slope-intercept form \(y = mx + b\), solve for \(y\):1. Add 2 to both sides: \[ y = -3x + 3 + 2 \]2. Simplify: \[ y = -3x + 5 \]

Step 5: Identify the y-intercept and Plot the Line

The line equation \(y = -3x + 5\) shows the y-intercept is 5, and the slope is \(-3\), which corresponds to a rise of -3 and a run of 1. Start at the y-intercept (0, 5) on the graph, then move down 3 units and right 1 unit to plot the next point. Draw a line through these points.

Key Concepts

Point-Slope Formula

The point-slope formula is a very useful tool when you need to graph linear equations using a point and a slope. This formula is written as \( y - y_1 = m(x - x_1) \).
Here, \((x_1, y_1)\) represents a specific point on the line, and \(m\) is the slope.
When you have a point and a slope, this formula allows you to quickly put the line equation together:

• Substitute the given values of the point into \((x_1, y_1)\).
• Use the slope \(m\) in the formula.

This formula is the starting point for many graphing problems since it provides a direct link between point locations and how steep the line is. It's especially handy when you have these specific values given in a problem and you need to construct the line's equation.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y = mx + b\).
This format is particularly valuable because it tells you the line's slope and y-intercept directly by just looking at the equation.
In this form:

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

To convert from the point-slope form to the slope-intercept form, you will solve the equation for \(y\). This involves

• Distributing any constants across the parentheses if necessary.
• Isolating \(y\) to one side of the equation.

This form is straightforward and is frequently used for graphing because it provides a clear picture of the line's trajectory.

Y-Intercept

The y-intercept is a crucial element in understanding linear equations. It is the point where the line crosses the y-axis on a graph. In other words, it is the value of \(y\) when \(x = 0\).
This is why in the slope-intercept form \(y = mx + b\), the \(b\) represents the y-intercept.
To find this point in a graph:

• Look at the equation in slope-intercept form to identify \(b\).
• Plot this point on the graph, generally starting as (0, \(b\)).

The y-intercept provides a starting point for drawing the line on the graph. From this point, you can use the slope to determine the direction and angle of the line extending across the graph. Identifying the y-intercept offers a straightforward approach to set up a line for visual representation.

Slope

Slope is a fundamental concept in understanding how a linear equation represents a line. It measures the steepness and direction of the line, showing how much \(y\) changes with a shift in \(x\). This is often referred to as the "rise over run" and is represented by \(m\) in your equations.
To determine the slope:

• Start from one point on the line, calculate how far it rises or falls (change in \(y\)), and how far it runs (change in \(x\)).
• The ratio of these changes \((\text{rise}\,\,\,\text{over}\,\,\,\text{run})\) gives you the slope \(m\).

A positive slope means the line ascends as it moves from left to right, while a negative slope indicates a descending line. The steeper the slope, the more quickly the line rises or falls. Understanding slope allows you to accurately graph the direction and angle of the line, making it a crucial element of linear equations.