Problem 53 - Elementary Algebra

Question

Write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point \((1,3)\) and is parallel to the line \(x+5 y=9\)

Step-by-Step Solution

Answer

The equation is \(x + 5y = 16\).

Step 1: Identify the Slope of the Given Line

The given line is \(x + 5y = 9\). First, we'll convert this to slope-intercept form to easily identify the slope.\ Let us rearrange it to \(y = -\frac{1}{5}x + \frac{9}{5}\). In this form, \(m = -\frac{1}{5}\) is the slope of the given line.

Step 2: Use the Point-Slope Form for the Line

Since the new line is parallel, it will have the same slope, \(m = -\frac{1}{5}\). The point provided is \((1, 3)\). We'll use the point-slope form equation, \(y - y_1 = m(x - x_1)\).Plug in \(m = -\frac{1}{5}\), \(x_1 = 1\), \(y_1 = 3\):\[y - 3 = -\frac{1}{5}(x - 1)\]

Step 3: Simplify the Point-Slope Form Equation

Expand and simplify the equation:\[y - 3 = -\frac{1}{5}x + \frac{1}{5}\]\[y = -\frac{1}{5}x + \frac{1}{5} + 3\]\[y = -\frac{1}{5}x + \frac{16}{5}\]

Step 4: Convert to Standard Form

Convert the equation \(y = -\frac{1}{5}x + \frac{16}{5}\) into standard form, which is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers.Multiply the entire equation by 5 to eliminate fractions:\[5y = -x + 16\]Rearrange to: \[x + 5y = 16\]

Key Concepts

Equation of a Line

The equation of a line is a mathematical expression that represents all the points lying on a straight line in a two-dimensional plane. It is pivotal in geometry and algebra for describing linear relationships. There are various forms of linear equations that offer flexibility depending on the given information, such as a point or slope. Understanding a line's equation helps predict values and solve various real-world problems that involve linear patterns.

Slope-Intercept Form

The slope-intercept form of a line's equation is crucial for both its simplicity and clarity. It is given by the formula:\[y = mx + b\]

• The letter \(m\) represents the slope of the line, which indicates its steepness and direction.
• The letter \(b\) denotes the y-intercept, the point where the line crosses the y-axis.

In our example, converting the line from standard form to slope-intercept form helped identify the slope \(-\frac{1}{5}\). This knowledge is vital, particularly when you need to find parallel lines, as they share the same slope. The ease of interpretation makes this form particularly popular in graphing.

Point-Slope Form

The point-slope form is especially helpful when you know a point on the line and its slope. It is written as:\[y - y_1 = m(x - x_1)\]

• \(m\) represents the slope of the line.
• \((x_1, y_1)\) denotes the known point.

In the scenario where the line goes through the point \((1,3)\) and has a slope of \(-\frac{1}{5}\), this form allows for straightforward substitution leading to:\[y - 3 = -\frac{1}{5}(x - 1)\]This form is advantageous when working directly with specific points and slopes, facilitating the derivation of other forms.

Standard Form

Standard form is a more general presentation for the equation of a line and is expressed as:\[Ax + By = C\]where \(A\), \(B\), and \(C\) are integers. This configuration is beneficial for various applications in algebra like solving systems of equations.

• It always presents a neat integer equation.
• It's particularly useful for mathematical analysis or eliminating fractions.

In our case, after having the equation in the slope-intercept form as \(y = -\frac{1}{5}x + \frac{16}{5}\), transforming it into standard form was achieved by clearing fractions:\[x + 5y = 16\]This form fits better for general presentation and multiple-step calculations.