Problem 11 - Elementary Algebra

Question

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. (Objective 1a) $$(-6,-2), m=3$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Point-Slope Form

The point-slope form of the equation of a line is given by \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a known point on the line and \(m\) is the slope. Here, the given point is \((-6, -2)\) and the slope is \(m=3\).

Step 2: Substitute the Given Values

Substitute \(x_1 = -6\), \(y_1 = -2\), and \(m = 3\) into the point-slope form equation: \( y - (-2) = 3(x - (-6)) \). Simplifying, we have: \( y + 2 = 3(x + 6) \).

Step 3: Distribute and Simplify

Expand and simplify the equation: \( y + 2 = 3x + 18 \). Then, subtract 2 from both sides to isolate \(y\): \( y = 3x + 16 \).

Step 4: Convert to Standard Form

Convert the equation \(y = 3x + 16\) into the form \(A x + B y = C\). Start by subtracting \(3x\) from both sides: \(-3x + y = 16\). To convert this into \(Ax + By = C\), multiply the entire equation by -1 (to make \(A\) a positive integer): \(3x - y = -16\).

Step 5: Verify the Equation's Form

Ensure the equation is in the form \(Ax + By = C\), with integers \(A, B,\) and \(C\). The equation \(3x - y = -16\) satisfies this requirement with \(A=3, B=-1, \) and \(C=-16\).

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a fundamental concept in mathematics used to describe a straight line on the coordinate plane. This formulation is particularly useful when you have the slope of a line and a point that lies on that line. The point-slope form is given by the equation:\[ y - y_1 = m(x - x_1) \]In this formula:

• \((x_1, y_1)\) is a known point on the line. In our exercise, this is the point \((-6, -2)\).
• \(m\) represents the slope of the line. For our problem, the slope \(m\) is given as \(3\).

The point-slope form is renowned for its simplicity and immediate utility in line equations, as it directly translates the information about a point and slope into an equation form. In our example, substituting the given values into the point-slope formula yields:\[ y - (-2) = 3(x - (-6)) \]Simplifying, we get:\[ y + 2 = 3(x + 6) \]

Slope-Intercept Form

The slope-intercept form is one of the most common ways to represent a linear equation of a line. It is written as:\[ y = mx + b \]Where:

• \(m\) is the slope of the line, identifying how steep the line is.
• \(b\) is the y-intercept of the line, which is the point where the line crosses the y-axis.

To convert an equation from point-slope form, or any other form, to the slope-intercept form, you should rearrange the equation to solve for \(y\). By simplifying the equation from the point-slope form in our exercise, we arrive at:\[ y = 3x + 16 \]Here, the slope \(m\) remains \(3\) and the y-intercept \(b\) is \(16\). The slope-intercept form is particularly handy for quickly identifying both the slope and the intersection point with the y-axis.

Standard Form of a Line

The standard form of a line is another method to express the equation of a line, especially preferred when the equation needs integer coefficients. The standard form is:\[ Ax + By = C \]In which:

• \(A, B,\) and \(C\) are integers.
• Usually, \(A\) is a positive integer.

In our exercise, we started from the slope-intercept form equation \(y = 3x + 16\) and aimed to rewrite it in standard form. By rearranging the terms, first by subtracting \(3x\) from both sides, we achieve:\[ -3x + y = 16 \]To conform with the convention of having a positive \(A\), we multiply the entire equation by \(-1\):\[ 3x - y = -16 \]This transformation respects the standard form, with integers \(A = 3\), \(B = -1\), and \(C = -16\). The standard form is valuable for solving and analyzing equations using algebraic techniques, and often used in various applications dealing with integer operations.