Problem 30 - Algebra 1: Concepts and Skills

Question

Write in standard form an equation of the line that passes through the two points. Use integer coefficients. \((4,0),(0,5)\)

Step-by-Step Solution

Answer

The equation of the line that passes through the two points (4,0) and (0,5) in standard form is \(5x + 4y = 20\).

Step 1: Calculate the slope of the line

The formula to find the slope (m) between two points (x1, y1) and (x2, y2) is \(m = \frac{y2 - y1}{x2 - x1}\). Substituting the given points (4,0) and (0,5) into this formula gives the slope \(m = \frac{5 - 0}{0 - 4} = -\frac{5}{4}\).

Step 2: Find the equation in slope-intercept form

The formula for the slope-intercept form is \(y = mx + b\), where m is the slope and b is the y-intercept. From Step 1, we know m is -5/4. The point (4,0) lies on the line, so if we substitute these coordinates in for x and y in the slope intercept equation, we find \(0 = -\frac{5}{4} * 4 + b\), hence \(b = 5\). So, the equation of the line in slope-intercept form is \(y = -\frac{5}{4}x + 5\).

Step 3: Convert to standard form

Finally, we need to convert this equation into standard form, \(Ax + By = C\), where A, B and C are integers and A is non-negative. Multiplying through by 4 to eliminate the fraction gives \(4y = -5x + 20\). Reorganizing the terms to put it in standard form gives \(5x + 4y = 20\).

Key Concepts

Standard Form

The standard form of a linear equation is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers. This form has a couple of important characteristics:

• It allows you to immediately identify the intercepts of the line.
• It is useful for solving systems of linear equations, especially when using the elimination method.
• \(A\) should be non-negative.

To convert an equation from slope-intercept form to standard form, you should clear any fractions by multiplying through by the least common denominator. Then, rearrange the terms so that the \(x\) and \(y\) variables are on one side of the equation, and the constant term is on the other. Using integer coefficients is crucial as it makes the equation cleaner and simpler to analyze. Moving the variables around may require you to add or subtract terms to organize them as needed. In our example, moving the terms of \(y = -\frac{5}{4}x + 5\) to get \(5x + 4y = 20\) puts it nicely in standard form.

Linear Equations

Linear equations produce straight lines when graphed. They represent relationships where each input (or \(x\)) corresponds to exactly one output (or \(y\)). Characteristics of linear equations include:

• Constant rate of change, also known as the slope.
• The graph is a straight line.
• They can be represented in various forms such as standard, slope-intercept, and point-slope.

Understanding linear equations is foundational in algebra because they describe a direct relationship between two variables. In our exercise, we found the linear equation in multiple forms. Initially we calculated the slope from two points, which is the rate at which \(y\) changes for a unit change in \(x\). This slope remains consistent across the entire length of the line, underlining the key concept of linearity.

Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\). Here, \(m\) represents the slope, which is the rate of change, and \(b\) is the y-intercept, where the line crosses the y-axis. Using this form gives an intuitive way to visualize and understand a line:

• You can easily see the slope of the line, informing you how steep or flat it is.
• The y-intercept \(b\) tells you the starting point of the line on the y-axis.

When deriving the slope-intercept form, knowing the slope \(m\) and a point on the line allows you to solve for \(b\). In our example, after calculating the slope as \(-\frac{5}{4}\), substituting the point \((4,0)\) into the equation \(y = mx + b\) helps find \(b = 5\). This culminates in the complete slope-intercept form equation \(y = -\frac{5}{4}x + 5\). Visual graphing becomes straightforward once all these components are clear.