Problem 34 - 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. $$ (-4,0) \text { and }(6,-1) $$

Step-by-Step Solution

Answer

The equation is \(x + 10y = -4\).

Step 1: Find the Slope (m)

To find the slope of the line passing through the points \((-4,0)\) and \((6,-1)\), use the formula for slope: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points, we get \(m = \frac{-1 - 0}{6 - (-4)} = \frac{-1}{10}\).

Step 2: Use Point-Slope Form

With the slope found in Step 1, use the point-slope form of the equation \(y - y_1 = m(x - x_1)\). Choose point \((-4,0)\): \(y - 0 = \frac{-1}{10}(x - (-4))\). Simplifying this gives: \(y = \frac{-1}{10}(x + 4)\).

Step 3: Convert to Slope-Intercept Form

Expand the equation from Step 2: \(y = \frac{-1}{10}x - \frac{4}{10}\), which simplifies to \(y = \frac{-1}{10}x - \frac{2}{5}\).

Step 4: Rearrange to General Form

Convert the equation from Step 3 to the form \(Ax + By = C\). Multiply both sides by 10 to eliminate the fractions: \(10y = -x - 4\). Rearrange to get \(x + 10y = -4\).

Key Concepts

Point-Slope Form

Understanding the point-slope form of a linear equation is a great starting point when you're working with linear functions. This form is particularly useful when you know the slope of a line and a point it passes through. The point-slope form is given by the equation \( y - y_1 = m(x - x_1) \). Here, \( m \) represents the slope, and \( (x_1, y_1) \) is a specific point on the line. This formulation allows you to quickly plug in the slope and any known point to find the specific equation of a line.

To illustrate, suppose you have a line passing through the point \((-4, 0)\) with a slope of \(m = -\frac{1}{10}\). Plugging into the point-slope form, you start with \( y - 0 = -\frac{1}{10}(x + 4) \). Solving further can help you identify the line's equation in other forms, depending on your needs for simplification.

Slope-Intercept Form

The slope-intercept form of a linear equation, \( y = mx + b \), is possibly the most well-known way to express the equation of a line. It's especially useful because it directly reveals two key characteristics: the slope \( m \) and the y-intercept \( b \). This makes it easier to quickly understand a line's behavior and draw its graph.

After simplifying the point-slope form \( y = -\frac{1}{10}x - \frac{2}{5} \), we can see that this line has a slope of \(-\frac{1}{10}\) and a y-intercept at \(-\frac{2}{5}\). Whether you're drawing the line or analyzing its position relative to other lines, having this equation in slope-intercept form makes many tasks simpler. The slope tells you how steep the line is and in which direction it leans, while the y-intercept indicates where the line crosses the y-axis.

General Form of a Line

The general form of a line is expressed as \( Ax + By = C \). This form is useful for various analytical purposes and integrating with systems of equations. Unlike the slope-intercept and point-slope forms, the general form doesn’t explicitly display the slope or intercepts, but it can easily interact with other linear equations in the same format.

From our earlier steps, converting \( y = -\frac{1}{10}x - \frac{2}{5} \) to general form involves manipulation to eliminate fractions and rearrange terms, achieving \( x + 10y = -4 \). This essentially standardizes the equation, making it easier to handle algebraically, especially when dealing with multiple lines or combining it with another equation into a system for further problem-solving.