Problem 2 - Algebra A Combined Function

Question

Write an equation of each line with the given slope and containing the given point. Write the equation in the slope-intercept form \(y=m x+b .\) See Example \(1 .\) Slope 4 ; through (5,1)

Step-by-Step Solution

Answer

The equation of the line is \(y = 4x - 19\).

Step 1: Identify Given Information

The slope of the line is given as 4, and the line passes through the point (5, 1). We'll use these in the equation.

Step 2: Use the Point-Slope Formula

The point-slope form is given by the equation \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the point and \(m\) is the slope. Substituting the given slope 4 and point (5, 1), we get: \[ y - 1 = 4(x - 5) \]

Step 3: Simplify the Equation

Simplify the equation to solve for \(y\) in terms of \(x\). Expand and rearrange the equation: \[ y - 1 = 4x - 20 \]Adding 1 to both sides gives:\[ y = 4x - 19 \]

Step 4: Verify the Equation

To verify, substitute \(x = 5\) into the equation and check if \(y = 1\). Calculating gives:\[ y = 4(5) - 19 = 1 \]. This confirms that the line passes through the point (5, 1).

Key Concepts

Equation of a Line

An equation of a line is a mathematical expression that describes all the points along a straight line in a coordinate plane. One of the most common ways to express this is through the slope-intercept form. The slope-intercept form is written as \(y = mx + b\). In this equation,

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

The slope-intercept form is popular because it clearly tells us the line's steepness and where it begins vertically. By knowing these two pieces of information, you can easily plot the line on a graph.

Point-Slope Form

The point-slope form is another way to write the equation of a line. It's especially useful when you know one point on the line and the slope. This form is expressed as \(y - y_1 = m(x - x_1)\), where

• \((x_1, y_1)\) is the known point on the line, and
• \(m\) is the slope.

In problems where you don't directly have the y-intercept but know a specific point and the slope, the point-slope form is convenient.
Once you write the equation in point-slope form, you can easily convert it to slope-intercept form by solving for \(y\). This technique allows you to transition from knowing specific information about a line (like a point and the slope) to a more general formula you can use to plot or analyze it further.

Slope Calculation

The slope of a line is a measure of its steepness or direction. It's calculated as the "rise" over the "run," which represents how much the line goes up or down for each unit it moves left or right. Mathematically, it's given as \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In this formula:

• \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line.

The slope tells us important things:

• If the slope is positive, the line goes upwards as you move from left to right.
• If it's negative, the line goes downwards.
• A slope of zero means the line is perfectly horizontal, and an undefined slope (division by zero) means it's vertical.

Knowing how to calculate the slope is crucial, as it directly influences how you express equations of lines in both point-slope and slope-intercept forms.