Problem 35 - Algebra A Combined Function

Question

Find an equation of the line passing through each pair of points. Write the equation in the form $A x+B y=C. $$ (2,3) \text { and }(-1,-1) $$

Step-by-Step Solution

Answer

The equation of the line is \(4x - 3y = -1\).

Step 1: Find the Slope

The slope of a line passing through two points, \((x_1, y_1)\) and \((x_2, y_2)\), can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]For our points \((2, 3)\) and \((-1, -1)\), this becomes: \[ m = \frac{-1 - 3}{-1 - 2} = \frac{-4}{-3} = \frac{4}{3} \]

Step 2: Use Point-Slope Form

The point-slope form of a line equation is: \[ y - y_1 = m(x - x_1) \]Using the point \((2, 3)\) and the slope \(m = \frac{4}{3}\), we have: \[ y - 3 = \frac{4}{3}(x - 2) \]

Step 3: Simplify to Slope-Intercept Form

Expand and simplify the equation from the previous step:\[ y - 3 = \frac{4}{3}x - \frac{8}{3} \]Add 3 to both sides:\[ y = \frac{4}{3}x - \frac{8}{3} + \frac{9}{3} \]\[ y = \frac{4}{3}x + \frac{1}{3} \]

Step 4: Convert to Standard Form \(Ax + By = C\)

Multiply the entire equation by 3 to eliminate fractions:\[ 3y = 4x + 1 \]Rearrange to standard form:\[ 4x - 3y = -1 \]

Key Concepts

Slope of a Line

The slope of a line is a numerical measure of the line's steepness. It tells us how much the line rises or falls as it moves from left to right.To find the slope, we use two points on the line, let's say \((x_1, y_1)\) and \((x_2, y_2)\). The slope \(m\) is calculated with the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula gives you a fraction which describes the vertical change (rise) over the horizontal change (run).

• If the slope is positive, the line rises as it moves to the right.
• If the slope is negative, the line falls as it moves to the right.
• A slope of zero means the line is horizontal.
• An undefined slope occurs for vertical lines.

In our example, for the points \((2, 3)\) and \((-1, -1)\), the slope is \(\frac{4}{3}\). This tells us the line rises 4 units for every 3 units it moves to the right.

Point-Slope Form

The point-slope form of a linear equation is particularly useful when you know a point on the line and the slope.It is written as: \[y - y_1 = m (x - x_1)\]Here, \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope.This form is helpful for quickly writing the equation of a line when you only know one point and the slope. After deriving the slope, you can substitute these values into the formula.In our case:

• Point: \((2, 3)\)
• Slope: \(\frac{4}{3}\)

Substituting into the point-slope form, you get: \[y - 3 = \frac{4}{3}(x - 2)\]This equation can now be further simplified if required.

Slope-Intercept Form

The slope-intercept form of a line is one of the most common ways to express a linear equation. It is easy to read because it shows both the slope and the y-intercept of the line at a glance. The slope-intercept form is: \[y = mx + b\]Where:

• \(m\) is the slope.
• \(b\) is the y-intercept (where the line crosses the y-axis).

To convert from point-slope form to slope-intercept form, you simply solve for \(y\). In the exercise, from:\[y - 3 = \frac{4}{3}x - \frac{8}{3}\]Adding 3, you get:\[y = \frac{4}{3}x + \frac{1}{3}\]Now the equation is in slope-intercept form, showing a slope of \(\frac{4}{3}\) and a y-intercept of \(\frac{1}{3}\).

Standard Form of a Line

The standard form of a line is another way to express the equation of a line, written more generally as: \(Ax + By = C\).In this form, integers \(A\), \(B\), and \(C\) have specific roles, with \(A\) typically a positive integer. This form is especially useful for finding intercepts quickly and is often used in algebraic applications.To convert a line equation into standard form, you may have to manipulate the equation by:

• Eliminating fractions if they exist.
• Rearranging terms to get all variables and constants on one side.

In the example, starting from the slope-intercept form:\[y = \frac{4}{3}x + \frac{1}{3}\]We eliminate fractions by multiplying through by 3, giving:\[3y = 4x + 1\]Finally, rearrange terms:\[4x - 3y = -1\]Now, this equation is in standard form, ready for various applications!