Problem 11 - Elementary Algebra

Question

For Problems 1-12, find the equation of the line that contains the given point and has the given slope. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. $$ (-6,-2), m=3 $$

Step-by-Step Solution

Answer

The equation of the line is \(3x - y = -16\).

Step 1: Recall the Point-Slope Formula

To find the equation of a line given a point \(x_1, y_1\) and a slope \(m\), we use the point-slope formula: \(y - y_1 = m(x - x_1)\). In this case, the point is \((-6, -2)\) and the slope \(m = 3\).

Step 2: Substitute the Given Values

Substitute the point \((-6, -2)\) and slope \(m = 3\) into the point-slope formula: \[y + 2 = 3(x + 6)\].

Step 3: Simplify and Rearrange into Slope-Intercept Form

Expand the equation from Step 2: \[y + 2 = 3x + 18\].Subtract 2 from both sides to get the equation in slope-intercept form: \[y = 3x + 16\].

Step 4: Rearrange into Standard Form

The standard form of a line is \(Ax + By = C\).Subtract \(3x\) from both sides of the equation from Step 3: \[-3x + y = 16\].Multiply everything by \(-1\) to make \(A\) positive: \[3x - y = -16\].

Key Concepts

Point-Slope Formula

The point-slope formula is a powerful tool in algebra that allows us to find the equation of a line when we are given a single point on the line and the slope. This formula is expressed as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a specific point on the line and \(m\) is the slope of the line.

- **Example**: If given the point \((-6, -2)\) and slope \(m = 3\), you can substitute these into the formula as follows: \[y + 2 = 3(x + 6)\]
- Start by replacing \(x_1\) and \(y_1\) with \(-6\) and \(-2\), respectively. - Replace \(m\) with \(3\).

This setup provides a simple way to create the line's equation, allowing further manipulation into different forms like slope-intercept and standard forms.

Slope-Intercept Form

The slope-intercept form of a linear equation is regarded as one of the easiest ways to express a line. It is written as \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept, the point where the line crosses the y-axis.

- **From Point-Slope to Slope-Intercept**: - Using the previous example, take \[y + 2 = 3(x + 6)\] and expand it: \[y + 2 = 3x + 18\].
- Simplify by subtracting 2 from both sides to obtain \[y = 3x + 16\].
This format is particularly useful for quickly identifying the slope and y-intercept, providing a clear understanding of the line's behavior in a graph.

- **Applications**: - Quick graphing. - Easy to interpret how the change in \(x\) affects \(y\).

Standard Form

The standard form of a linear equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) should be positive if possible.

- **Converting from Slope-Intercept to Standard Form**: - Begin with the slope-intercept form equation: \[y = 3x + 16\]. - Rearrange terms to place \(x\) and \(y\) on the same side: \[-3x + y = 16\].
- To ensure \(A\) is a positive integer, multiply the whole equation by \(-1\): \[3x - y = -16\].
The standard form is often used for theoretical purposes and simplifying algebraic operations.- **Key Features**: - Offers a clear way to find integer values for \(x\) and \(y\). - Facilitates the computation of intersections with other lines.