Problem 59 - Contemporary Precalculus

Question

Find an equation for the line satisfying the given conditions. Through (2,3) and parallel to \(3 x-2 y=5\).

Step-by-Step Solution

Answer

Question: Given a line with the equation 3x - 2y = 5, and a point (2, 3), find the equation of a line parallel to the given line that passes through the given point. Solution: The equation for the line satisfying the given conditions is \(y=\frac{3}{2}x\).

Step 1: Determine the slope of the given line

The given line is in the standard form of a linear equation, which is \(A x + B y = C\). To determine the slope, we need to rewrite it into slope-intercept form, which is \(y=mx+b\), where \(m\) represents the slope. The given equation is \(3x-2y=5\). We'll rearrange the equation to isolate \(y\) on one side: 1. Add \(2y\) to both sides: \(3x-2y+2y=5+2y\). 2. Simplify the equation: \(3x=5+2y\). 3. Subtract \(5\) from both sides: \(3x-5=2y\). 4. Divide both sides by 2: \(\frac{3x-5}{2}=y\). Now the equation is in slope-intercept form, and the slope is \(\frac{3}{2}\).

Step 2: Write the point-slope form of the equation, substituting the known point and slope

Since the lines are parallel, they have the same slope, so the slope of the desired line is also \(\frac{3}{2}\). Given that the point on the line is (2, 3), we can use the point-slope form of a linear equation, which is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the known point and \(m\) is the slope. Substituting the given values, we get: \(y - 3 = \frac{3}{2}(x - 2)\).

Step 3: Solve for the equation in slope-intercept form

Now, we'll solve the equation for \(y\), to express the line in slope-intercept form: 1. Distribute the slope: \(y - 3 = \frac{3}{2}x - \frac{3}{2}\cdot 2\). 2. Simplify the equation: \(y - 3 = \frac{3}{2}x - 3\). 3. Add 3 to both sides: \(y = \frac{3}{2}x\). Thus, the equation for the line satisfying the given conditions is \(y=\frac{3}{2}x\).

Key Concepts

Slope-Intercept Form

Let's begin by understanding the slope-intercept form of a linear equation, which is written as \(y = mx + b\). This form is widely used because it's very straightforward to interpret:

• \(m\) represents the slope of the line, which indicates how steep the line is and the direction it tilts.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

To convert any linear equation from standard form \(Ax + By = C\) to slope-intercept form, solve for \(y\) so that it is isolated on one side of the equation. The key feature here is the slope \(m\), as it dictates the angle or gradient of the line with respect to the x-axis.
For example, in the equation \(3x-2y=5\), when we rearrange to solve for \(y\), we get \(y=\frac{3}{2}x-\frac{5}{2}\), revealing the slope \(m = \frac{3}{2}\). This tells us that for every two units we move horizontally, the line moves up three units, making it relatively steep.

Point-Slope Form

Point-slope form is another convenient way to define the equation of a line, especially when you know the line's slope and a point on the line. This form is expressed as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point through which the line passes, and \(m\) again represents the slope.
This form is particularly helpful when you want to write the equation of the line quickly from known information without immediately converting to slope-intercept form.

• Start by identifying a point on the line and the slope.
• Plug these values into the point-slope formula.

For the problem at hand, we're given a point \((2, 3)\) and a slope \(\frac{3}{2}\) because our line is parallel to another with the same slope. Thus, using point-slope form, we have \(y - 3 = \frac{3}{2}(x - 2)\), allowing us to clearly and easily depict the line's path through the plane.

Parallel Lines

The concept of parallel lines is fundamental in geometry, referring to two lines in the same plane that never intersect. Regardless of how far they are extended, they maintain a constant distance apart. One crucial feature of parallel lines is that they possess the same slope.

• If two lines have different y-intercepts but identical slopes, they are parallel.
• Parallel lines may look alike but will never meet, maintaining the same directional path.

In our given exercise, the line through the point \((2, 3)\) must be parallel to the line represented by the equation \(3x - 2y = 5\). First, we convert the standard form equation to get the slope \(\frac{3}{2}\). Since they must be parallel, the sought line's slope also equals \(\frac{3}{2}\). This property allows us to derive our new line equation through point-slope form, ensuring the two lines maintain their parallel nature throughout the plane.