Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the general form of the equation of the line that passes through the points. $$(-6,1),(4,-5)$$

Step-by-Step Solution

Answer

The general form of the line that passes through the points (-6,1) and (4,-5) is \(3x + 5y = -13\).

Step 1: Calculate the Slope of the Line

The formula for calculating the slope (m) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is\[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]So, with the points (-6,1) and (4,-5), substitute into the formula and get\[m = \frac{{-5 - 1}}{{4 - (-6)}} = -\frac{6}{10} = -\frac{3}{5}\]

Step 2: Use Point-Slope Form and Convert to General Form

The point-slope form of a line is given by\[y - y_1 = m(x - x_1)\]Here, substitute \((x_1, y_1)\) with one of the given points, let's say (-6,1), and m=-3/5. This gives\[y - 1 = -\frac{3}{5}(x + 6)\]. To convert it to the general form \[Ax + By = C\], clear out the fractions and simplify the equation: \(5y - 5 = -3x - 18\), which simplifies further to \(3x + 5y = -13\).

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is important when working with linear equations. The slope, often denoted as \(m\), represents the steepness or incline of a line. It tells us how far and in what direction the line moves between two points. To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:\[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]Here, the difference in the \(y\)-coordinates (\(y_2 - y_1\)) is divided by the difference in the \(x\)-coordinates (\(x_2 - x_1\)). This ratio gives us the slope.

• In our example, using points \((-6,1)\) and \((4,-5)\), the calculation becomes \(m = \frac{-5 - 1}{4 - (-6)} = -\frac{6}{10} = -\frac{3}{5}\).
• The negative sign indicates the line slopes downwards as we move from left to right.

Point-Slope Form

The point-slope form of a line's equation offers a straightforward way to write the equation when you know a point on the line and its slope. The formula is expressed as:\[y - y_1 = m(x - x_1)\]This is handy because it incorporates a specific point, \((x_1, y_1)\), directly into the equation along with the slope \(m\).

• In our case, we can choose the point \((-6, 1)\) and use the slope \(-\frac{3}{5}\) to produce the equation: \(y - 1 = -\frac{3}{5}(x + 6)\).
• This format is especially useful because of its simplicity and how easily you can rearrange it into other forms, such as the general form.

When you have the point-slope form, you can adjust and transform it into other equations like the general form by following arithmetic and algebraic manipulations.

General Form of Linear Equation

The general form of a linear equation is a more standardized way of representing a line, written as \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are constants.
To convert from point-slope to this form, focus on expanding and simplifying the equation. Consider the point-slope form \(y - 1 = -\frac{3}{5}(x + 6)\).

• First, clear the fraction by multiplying through by 5, giving us \(5(y - 1) = -3(x + 6)\).
• Simplify to find \(5y - 5 = -3x - 18\).
• Rearrange to get all terms involving variables on one side: \(3x + 5y = -13\).

This form is elegant and concise, making it easy to graph and interpret key properties, such as intersection points and direction. By keeping equations in this form, you can readily compare different lines and see how they relate in the greater scheme of a coordinate plane.