Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-32: Find a general form of an equation of the line through the point \(A\) that satisfies the given condition. $$ A(4,-5) ; \quad \text { through } B(-3,6) $$

Step-by-Step Solution

Answer

The general form of the line is \(11x + 7y = 9\).

Step 1: Identify the form of the line equation

We need to find the equation of the line that passes through two given points, \(A(4, -5)\) and \(B(-3, 6)\). The general form of the equation of a line is \(Ax + By = C\).

Step 2: Calculate the slope

The slope \(m\) of a line passing through points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the given points to find the slope: \(m = \frac{6 - (-5)}{-3 - 4} = \frac{11}{-7} = -\frac{11}{7}\).

Step 3: Use point-slope form

The point-slope form of the equation of a line is \(y - y_1 = m(x - x_1)\). Use point \(A(4, -5)\) and the calculated slope \(m = -\frac{11}{7}\) to get \(y + 5 = -\frac{11}{7}(x - 4)\).

Step 4: Simplify to slope-intercept form

Distribute the slope and simplify the equation from Step 3: \(y + 5 = -\frac{11}{7}x + \frac{44}{7}\). Isolate \(y\) by subtracting 5 (or \(\frac{35}{7}\)) from both sides to get \(y = -\frac{11}{7}x + \frac{44}{7} - \frac{35}{7} = -\frac{11}{7}x + \frac{9}{7}\).

Step 5: Convert to general form

Rewrite the equation in the general form \(Ax + By = C\). Start from \(y = -\frac{11}{7}x + \frac{9}{7}\) and multiply every term by 7 to eliminate the fractions: \(7y = -11x + 9\). Rearrange it to obtain \(11x + 7y = 9\), which is the general form of the line equation.

Key Concepts

Slope Calculation

To find the equation of a line, understanding the concept of slope is essential. The slope describes how steep a line is. It is the ratio of the vertical change to the horizontal change between two points on a line. You can calculate it using the slope formula:

• Pick two points on the line,
• Label them (x₁, y₁) and (x₂, y₂)
.

To find the slope (m), use the formula:
\[ m = \frac{y₂ - y₁}{x₂ - x₁} \]Substituting the points from our exercise, A(4, -5) and B(-3, 6), gives us:
\[ m = \frac{6 - (-5)}{-3 - 4} = \frac{11}{-7} = -\frac{11}{7} \]This negative slope indicates that the line decreases from left to right, starting from point A to B.

Point-Slope Form

Once you have the slope, you can use the point-slope form of the equation of a line. This form is particularly useful because it directly uses the slope and a point on the line. The point-slope form is represented as:
\[ y - y₁ = m(x - x₁) \]where

• m is the slope,
• (x₁, y₁) is a point on the line.

In our example, using point A(4, -5) and our calculated slope of \(-\frac{11}{7}\), we substitute these values into the formula:
\[ y + 5 = -\frac{11}{7}(x - 4) \]This tells us the equation of the line using the slope from point A.

Slope-Intercept Form

Transforming equations into slope-intercept form can make them easier to understand and work with. It gives the equation of the line as:
\[ y = mx + b \]In this equation:

• m is the slope,
• b is the y-intercept, the value where the line crosses the y-axis.

To convert our point-slope equation \(y + 5 = -\frac{11}{7}(x - 4)\) into slope-intercept form, distribute the slope and simplify:
\[ y + 5 = -\frac{11}{7}x + \frac{44}{7} \]Subtract 5 (which is equivalent to \(\frac{35}{7}\)) from both sides:
\[ y = -\frac{11}{7}x + \frac{44}{7} - \frac{35}{7} = -\frac{11}{7}x + \frac{9}{7} \]Thus, the slope-intercept form is \(y = -\frac{11}{7}x + \frac{9}{7}\).

General Form of a Line

The general form of a line equation is another way to represent the line, typically expressed as:
\[ Ax + By = C \]Here, A, B, and C are integers, and it provides a more standard representation of linear equations. To derive this form from the slope-intercept form \(y = -\frac{11}{7}x + \frac{9}{7}\), you need to clear the fractions by multiplying through by 7:
\[ 7y = -11x + 9 \]Rearrange this equation to bring all terms to one side:
\[ 11x + 7y = 9 \]This is the general form of our line equation, providing a neat representation without fractions, making it easy to understand at a glance.