Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line passing through the points. \((7,-4),(-7,3)\)

Step-by-Step Solution

Answer

The equation of the line passing through the points (7,-4) and (-7,3) is \(y = -0.5x - 0.5\)

Step 1: Find the slope

First, you need to find the slope (m) of the line that passes through the points. The slope formula is \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). Plugging in the given points, we get \(m = \frac{{3 - (-4)}}{{-7 - 7}} = \frac{7}{-14} = -0.5\)

Step 2: Use point-slope form to find the line

Next, use the point-slope form of the line equation \(y - y_1 = m(x - x_1)\) where (x1, y1) is a point on the line and m is the slope of the line. Using the slope from step one and one of the given points, say (7,-4), we get \(y - (-4) = -0.5(x - 7)\) or \(y + 4 = -0.5x + 3.5\). This simplifies to \(y = -0.5x - 0.5\)

Step 3: Write the final line equation

The final step is to write the equation from Step 2 in the standard form. So, the equation of the line passing through the points (7,-4) and (-7,3) is \(y = -0.5x - 0.5\)

Key Concepts

Slope

The slope of a line is a measure of its steepness. It tells us how much the line goes up or down as it moves from left to right. The slope is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are two different points on the line. Think of it as the 'rise' over the 'run':

• Rise: The change in the vertical direction (difference in y-values).
• Run: The change in the horizontal direction (difference in x-values).

In the example you looked at, the points are (7,-4) and (-7,3), so:

• Rise: \(3 - (-4) = 7\)
• Run: \(-7 - 7 = -14\)

This gave you a slope of \(m = \frac{7}{-14} = -0.5\). A negative slope indicates that the line is going downwards as it moves from left to right.

Point-Slope Form

Once we have the slope of a line, the next step is often to write the equation of the line using the point-slope form. This is a handy form to use when you know one point on the line and its slope. The point-slope form is:\[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope. It's like a template where you simply fill in the blanks:- Slope: What's the steepness of the line?- Point: Which point is on this line?From the previous step, we calculated the slope \(m = -0.5\) and we're given a point, either (7,-4) or (-7,3). Using (7,-4):- Substituting into the point-slope form gives us: \[ y - (-4) = -0.5(x - 7) \]- Simplifying, we get \(y + 4 = -0.5x + 3.5\)- This further simplifies to \(y = -0.5x - 0.5\)This step is often used because it straightforwardly translates the slope and a point directly into an equation, making it a crucial part of understanding linear equations.

Standard Form

The standard form of a linear equation is an alternative way of writing it, usually presented as:\[ Ax + By = C \] Here, A, B, and C are integers, and A should be non-negative. This form can be particularly useful for analyzing and graphing the equation since it displays both the x and y variables on one side of the equation. To convert the equation from point-slope form or slope-intercept form, such as \( y = -0.5x - 0.5 \), to standard form, you need to

• Eliminate fractions, if any, by multiplying through by the least common denominator.
• Move all terms to one side so that \(Ax + By = C\) is achieved.

Following these steps with our line equation:- Multiply the entire equation by 2 to eliminate the fraction: \[ 2y = -x - 1 \]- Rearrange the terms to have both variables on one side: \[ x + 2y = -1 \]This gives us the equation in standard form, which can be useful for various applications, like finding x and y-intercepts or solving systems of equations.