Problem 33 - 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 \(2 x-5 y=10\) and passes through (-1,2)

Step-by-Step Solution

Answer

The equations are \(y = \frac{2}{5}x + \frac{12}{5}\) and \(2x - 5y + 12 = 0\).

Step 1: Determine the Slope of the Given Line

The line described is parallel to the line with equation \(2x - 5y = 10\). Lines that are parallel have equal slopes. First, rewrite this equation in slope-intercept form \(y = mx + b\) to find the slope. To do this, solve for \(y\):\[2x - 5y = 10\]Subtract \(2x\) from both sides:\[-5y = -2x + 10\]Divide every term by \(-5\):\[y = \frac{2}{5}x - 2\]Hence, the slope \(m\) of the line is \(\frac{2}{5}\).

Step 2: Write the Equation in Slope-Intercept Form

Using the slope \(m = \frac{2}{5}\) and the point it passes through \((-1, 2)\), substitute these values into the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (-1, 2)\):\[y - 2 = \frac{2}{5}(x + 1)\]Distribute \(\frac{2}{5}\):\[y - 2 = \frac{2}{5}x + \frac{2}{5}\]Add 2 to both sides to solve for \(y\):\[y = \frac{2}{5}x + \frac{2}{5} + 2\]Convert 2 to a fraction with equal denominator:\[y = \frac{2}{5}x + \frac{2}{5} + \frac{10}{5}\]Combine like terms:\[y = \frac{2}{5}x + \frac{12}{5}\]

Step 3: Convert to Standard Form

Now, rewrite the equation \(y = \frac{2}{5}x + \frac{12}{5}\) in standard form, \(Ax + By + C = 0\).First, eliminate the fraction by multiplying the entire equation by 5 to clear the denominators:\[5y = 2x + 12\]Rearrange terms to get all terms on one side:\[2x - 5y + 12 = 0\]Hence, the standard form of the equation is \(2x - 5y + 12 = 0\).

Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never meet; they are always the same distance apart. If two lines are parallel, it means they have the same slope. This is a crucial characteristic that allows us to identify parallel lines mathematically.
When given the equation of a line, you can determine the slope by converting it to the slope-intercept form, which allows us to easily identify the lines that are parallel. For example, if you have a line in standard form like the equation from the exercise, you convert it and find that both lines have the same slope, confirming that they are parallel.

Slope-Intercept Form

The slope-intercept form of a line is one of the most common ways to express the equation of a line. It is written as: \(y = mx + b\). Here, \(m\) represents the slope and \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.
This form is particularly useful because it quickly tells you both the direction and steepness of the line. In our exercise, once we had determined that the slope was \(\frac{2}{5}\), it became straightforward to use the point through which the line passes to find the complete equation. Simply substitute the known values into the equation to solve for \(b\).

Point-Slope Form

The point-slope form of a line is another popular form for writing line equations, expressed as: \(y - y_1 = m(x - x_1)\). This form is particularly helpful when you know a point on the line, \((x_1, y_1)\), and the slope \(m\).
To use this form, plug these values into the formula. For example, using the point \((-1, 2)\) and the slope \(\frac{2}{5}\) from our problem, you directly substitute these into the point-slope form to find the equation. This can be quite handy as it simplifies the process of finding an equation from a point and a slope.

Standard Form of a Line

The standard form of a line is expressed as \(Ax + By + C = 0\). One advantage of this form is its versatility in mathematical computations, such as finding intersections and for use in calculations in higher geometry.
To convert other forms of the equation of a line into standard form, like in our exercise, you typically need to rearrange terms. In this exercise, we first eliminated any fractions and then gathered all terms on one side of the equation. This approach allowed us to derive the standard form \(2x - 5y + 12 = 0\), providing an alternative representation of the line's equation.