Problem 34 - Precalculus

Question

Find an equation for the line that is described. Write the answer in the two forms \(y=m x+b\) and \(A x+B y+C=0\). Is parallel to \(4 x+5 y=20\) and passes through (0,0).

Step-by-Step Solution

Answer

Equations: \(y = -\frac{4}{5}x\) and \(4x + 5y = 0\).

Step 1: Identify the slope of the given line

To find the slope of a line parallel to the given line, we first identify the slope from the given line equation. The line equation \(4x + 5y = 20\) is in standard form \(Ax + By = C\). To find the slope, rewrite it in slope-intercept form \(y = mx + b\). Solving for \(y\):\\[ 5y = -4x + 20 \]\\[ y = -\frac{4}{5}x + 4 \]\The slope \(m\) is \(-\frac{4}{5}\). Parallel lines share the same slope, so the slope of our new line is also \(-\frac{4}{5}\).

Step 2: Write the equation of the new line in slope-intercept form

Using the slope \(m = -\frac{4}{5}\) and the point \((0, 0)\) through which the line passes, apply the point-slope form equation \(y - y_1 = m(x - x_1)\). Here, \((x_1, y_1) = (0, 0)\), so: \\[ y - 0 = -\frac{4}{5}(x - 0) \] \This simplifies to: \\[ y = -\frac{4}{5}x \] \Thus, the equation in slope-intercept form is \(y = -\frac{4}{5}x\).

Step 3: Convert to standard form

To convert the equation from slope-intercept form \(y = -\frac{4}{5}x\) to standard form \(Ax + By + C = 0\), we need to eliminate the fraction and rearrange the terms. Multiply the entire equation by 5 to clear the fraction: \\[ 5y = -4x \] \Rearrange to get \(Ax + By + C = 0\): \\[ 4x + 5y = 0 \] \Now, we have the standard form of the equation \(4x + 5y + 0 = 0\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to write the equation of a line. This form is expressed as \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept. It's easy to use because it tells us the slope directly and shows where the line crosses the y-axis.
For example, if a line has a slope of \(\frac{2}{3}\) and a y-intercept of \(-1\), the equation would be \(y = \frac{2}{3}x - 1\). From here, it’s straightforward to predict the line’s direction and steepness.
Understanding this form is useful when graphing lines or when you need to quickly compare the slopes and positions of different lines in a coordinate plane.

Parallel Lines

Parallel lines are lines in a plane that never meet. They always have the same slope, which means their steepness or angle is identical.
This makes it simple to write an equation for a line parallel to another. If you know the slope of one line, any line parallel to it will have that same slope.
For instance, the line described by \(4x + 5y = 20\) has been converted into slope-intercept form as \(y = -\frac{4}{5}x + 4\). Any line parallel to this one will also have a slope of \(-\frac{4}{5}\). As long as we know a point through which the new line passes, say (0,0), we can use the slope-intercept form to easily find a parallel equation, \(y = -\frac{4}{5}x\).
This characteristic of parallel lines makes them very predictable and easy to work with, whether you're solving algebra problems or designing geometric patterns.

Standard Form

Standard form is another way to write the equation of a line. It is expressed as \(Ax + By + C = 0\), where \(A\), \(B\), and \(C\) are integers. This form is especially useful because it provides a clear way of showcasing the relationship between \(x\) and \(y\).
To convert from slope-intercept form \(y = mx + b\) to standard form, we rearrange the equation and possibly clear fractions to match the \(Ax + By + C = 0\) format.
Take \(y = -\frac{4}{5}x\) which can be rewritten as \(5y = -4x\) when you multiply through by 5 to clear the fraction. Rearranging this gives \(4x + 5y = 0\).

• This form is advantageous when solving systems of equations since it makes the process of elimination much simpler.
• It's also beneficial when plugging values into formulas or checking characteristics of lines, like the perpendicularity of two equations.

Familiarity with both standard form and slope-intercept form allows for flexibility in tackling a wide variety of algebraic problems.