Problem 36 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in the form \(x=a, y=b\), or \(y=m x+b\). See Examples 5 and \(6 .\) Through (2,-3)\(;\) parallel to \(x-5 y=10\)

Step-by-Step Solution

Answer

The equation is: \[ y = \frac{1}{5}x - \frac{17}{5} \].

Step 1: Identify the form of the given line

The given line equation is in the form of a linear equation: \( x - 5y = 10 \). This can be rewritten in the slope-intercept form \( y = mx + b \).

Step 2: Rewrite the given equation in slope-intercept form

Starting from the equation \( x - 5y = 10 \), we need to convert it to the slope-intercept form.Subtract \(x\) from both sides to get:\[-5y = -x + 10\]Next, divide every term by \(-5\) to solve for \(y\):\[y = \frac{1}{5}x - 2\]Thus, the slope \(m\) of the line is \(\frac{1}{5}\).

Step 3: Use the slope for the parallel line

Lines that are parallel have the same slope. Therefore, the line we are looking for will also have a slope of \(\frac{1}{5}\).

Step 4: Use point-slope form

To find the equation of the line that passes through the point (2, -3) and has a slope of \(\frac{1}{5}\), use the point-slope form of the line equation:\[ y - y_1 = m(x - x_1) \]where \((x_1, y_1)\) is the point (2, -3) and \(m\) is \(\frac{1}{5}\).

Step 5: Substitute the known values into the point-slope equation

Substitute \(x_1 = 2\), \(y_1 = -3\), and \(m = \frac{1}{5}\) into the equation:\[ y - (-3) = \frac{1}{5}(x - 2) \]This simplifies to:\[ y + 3 = \frac{1}{5}x - \frac{2}{5} \]

Step 6: Simplify to slope-intercept form

Solving for \(y\), subtract 3 from both sides of the equation to simplify:\[ y = \frac{1}{5}x - \frac{2}{5} - 3 \]Convert \(-3\) to a fraction with the same denominator:\[-3 = -\frac{15}{5}\]Now combine the fractions:\[ y = \frac{1}{5}x - \frac{2}{5} - \frac{15}{5} \]\[ y = \frac{1}{5}x - \frac{17}{5} \]

Step 7: Write the final equation

Thus, the equation of the line parallel to \( x - 5y = 10 \) and passing through (2, -3) is:\[ y = \frac{1}{5}x - \frac{17}{5} \].

Key Concepts

Slope-Intercept Form

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

• \( m \) is the slope of the line, which represents the steepness or incline. A positive slope means the line rises as it moves from left to right, and a negative slope means it falls.
• \( b \) is the y-intercept, where the line crosses the y-axis.

Converting an equation into slope-intercept form helps identify these two important characteristics of a line immediately. For example, if we start from an equation like \( x - 5y = 10 \), you can rearrange terms to solve for \( y \). This gives you \( y = \frac{1}{5}x - 2 \). Here, the slope \( m \) is \( \frac{1}{5} \) and the y-intercept \( b \) is \( -2 \). These components tell you exactly how the line behaves on a graph.

Point-Slope Form

Another way to express a linear equation is through the point-slope form, which is particularly useful for writing equations when a point on the line and the slope are known. The point-slope formula is \( y - y_1 = m(x - x_1) \), where:

• \((x_1, y_1)\) is a specific point on the line. This point can be any point the line passes through.
• \( m \) is the slope of the line.

Using the point-slope form is a step towards finding an equation when you have different pieces of information about the line. For instance, if you know a line has a slope of \( \frac{1}{5} \) and passes through the point (2, -3), you plug these values into the formula like so: \( y - (-3) = \frac{1}{5}(x - 2) \). This allows you to manipulate the equation further into the slope-intercept form if needed. It's a flexible and intuitive way to build linear equations from known values.

Parallel Lines

Parallel lines are a fascinating concept in geometry, known for maintaining a constant distance apart, never meeting no matter how far they extend. This unique property is mathematically reflected by their slopes.

• Parallel lines have identical slopes. This means if you have a line with a slope \( m \), any line parallel to it will share the same slope \( m \).

Consider the problem of finding a line parallel to \( x - 5y = 10 \). Upon converting it to slope-intercept form, \( y = \frac{1}{5}x - 2 \), you identify the slope as \( \frac{1}{5} \). Consequently, any line parallel to this will also have a slope of \( \frac{1}{5} \). This crucial detail allows you to use either point-slope or slope-intercept form effectively to find equations of lines that share this parallel property. In essence, knowing the slope is the key to unlocking the equation of parallel lines, demonstrating their unyielding consistency and parallelism.