Problem 31

Question

Find an equation for the line that passes through the point $P(2,7)$ and is parallel to the line $3 y-2 x+6=0$

Step-by-Step Solution

Verified

Answer

The equation of the line that passes through the point $P(2,7)$ and is parallel to the line $3 y-2 x+6=0$ is $2x -3y = -5$.

1Step 1: Find the slope of the given line

Rearrange the given equation $3y - 2x + 6 = 0$ into the slope-intercept form $y = mx + c$. This results in $y = 2/3x - 2$. Thus, the slope of the line, $m$ is $2/3$.

2Step 2: Use point-slope form to find the equation of the parallel line

The formula for the point-slope form is $y - y1 = m(x - x1)$, where $m$ is the slope and $(x1, y1)$ is a point on the line. With a slope of $2/3$ and a given point of $P(2,7)$, the equation becomes $y - 7 = 2/3(x - 2)$ which simplify to $y = 2/3x + 5/3$.

3Step 3: Convert to standard form

Equations of lines are often given in standard form: $Ax + By = C$. To convert the equation from slope-intercept form to standard form, simply multiply every term by 3 to get rid of the fractions. This results in $2x - 3y = -5$.

Key Concepts

Slope-Intercept FormPoint-Slope FormStandard Form

Slope-Intercept Form

The slope-intercept form of a linear equation is probably the most common and easily recognizable form. It is expressed as:

$ y = mx + c $

Here, $m$ represents the slope of the line, and $c$ is the y-intercept, where the line crosses the y-axis. This form is crucial because the slope $m$ gives us a direct understanding of the angle at which the line tilts. If the slope is positive, the line rises; if negative, it falls. The y-intercept $c$ helps us find the starting point of the line on the y-axis.

In the existing solution, the line equation $3y - 2x + 6 = 0$ was rearranged into slope-intercept form $y = \frac{2}{3}x - 2$, revealing a slope $m = \frac{2}{3}$. Knowing this slope is key for finding equations of parallel lines since parallel lines share the same slope.

Point-Slope Form

Sometimes, when constructing the equation of a line, you already know a specific point on the line and the slope. The point-slope form of a linear equation is tailor-made for these scenarios, expressed as:

$ y - y_1 = m(x - x_1) $

In this form, $(x_1, y_1)$ represents a known point on the line, and $m$ denotes the slope of the line. This makes it incredibly useful when you have a point and a slope and need to write the equation of a line.

For the current exercise, since the line must be parallel to the given line, it uses the same slope $m = \frac{2}{3}$. Given the point $P(2,7)$, we plug these into the point-slope form to get $y - 7 = \frac{2}{3}(x - 2)$. After simplifying, this can be rewritten in slope-intercept form as $y = \frac{2}{3}x + \frac{5}{3}$.

Standard Form

The standard form of a line is another way of presenting a linear equation, generally noted as:

$ Ax + By = C $

In standard form, $A$, $B$, and $C$ are integers, and $A$ should ideally be positive. It is beneficial when needing a clean integer representation of the line, which can be useful for various types of analyses and graphing.

Converting from slope-intercept form to standard form becomes straightforward by eliminating any fractions through multiplying by the least common multiple. In the solution, the slope-intercept form $y = \frac{2}{3}x + \frac{5}{3}$ was multiplied by 3 to clear the fractions, leading to the standard form: $2x - 3y = -5$. This transformation gives a clear, fraction-free equation, commonly preferred for its simplicity and utility in different mathematical applications.

Problem 31

Problem 32

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Problem 32

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