Problem 29 - College Algebra

Question

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. \(f(-5)=-4,\) and \(f(5)=2\)

Step-by-Step Solution

Answer

The linear equation is \(y = \frac{3}{5}x - 1\).

Step 1: Understand the Problem

We're given two points on a straight line: \((-5, -4)\) and \((5, 2)\) and we need to find the linear equation that passes through these points.

Step 2: Determine the Slope

The slope \(m\) of a line through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Plugging the points, \((x_1, y_1) = (-5, -4)\) and \((x_2, y_2) = (5, 2)\), into the formula, we get:\[m = \frac{2 - (-4)}{5 - (-5)} = \frac{6}{10} = \frac{3}{5}\]

Step 3: Use the Point-Slope Form

The point-slope form of a line is:\[y - y_1 = m(x - x_1)\]Using the slope \(m = \frac{3}{5}\) and the point \((-5, -4)\), substitute the values:\[y - (-4) = \frac{3}{5}(x - (-5))\]This simplifies to:\[y + 4 = \frac{3}{5}(x + 5)\]

Step 4: Simplify to Slope-Intercept Form

Distribute \(\frac{3}{5}\):\[y + 4 = \frac{3}{5}x + \frac{15}{5}\]\[y + 4 = \frac{3}{5}x + 3\]Subtract 4 from both sides:\[y = \frac{3}{5}x + 3 - 4\]\[y = \frac{3}{5}x - 1\]This is the linear equation in slope-intercept form \(y = mx + b\) with slope \(m = \frac{3}{5}\) and y-intercept \(b = -1\).

Key Concepts

Slope

In mathematics, the slope of a line is one of the fundamental ideas related to linear equations. Slope describes how steep or flat a line is and is often denoted by the letter \(m\). To find the slope given two points, you can use the slope formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• The numerator, \(y_2 - y_1\), represents the vertical change, also known as the "rise." This measures how much the \(y\) value changes between the two points.
• The denominator, \(x_2 - x_1\), captures the horizontal change, termed the "run," which is the change in the \(x\) values.

Given the example with points \((-5, -4)\) and \((5, 2)\), the slope is calculated as \(\frac{6}{10}\) or \(\frac{3}{5}\). This means for every 5 units you move horizontally, the line moves 3 units vertically.

Point-Slope Form

The point-slope form is a convenient way of writing the equation of a line when you know a point on the line and its slope. The equation is expressed as:\[y - y_1 = m(x - x_1)\]Here, \((x_1, y_1)\) is a specific point on the line, and \(m\) is the slope.

• To use this form, simply substitute the values of the known point and the slope into the equation.
• Point-slope form is particularly useful when you need to quickly write the equation of a line with specific characteristics.

For the example with slope \(\frac{3}{5}\) and point \((-5, -4)\), the point-slope equation becomes:\[y + 4 = \frac{3}{5}(x + 5)\]This form is not simplified for graphing, but it directly shows the relationship between the slope and a point on the line.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common and graph-friendly ways to express a line. It is written as:\[y = mx + b\]In this form:

• \(m\) is the slope, representing the steepness of the line.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

This form makes it easy to graph the line, as you can immediately identify the slope and starting point. To transition from point-slope to slope-intercept form, you simplify the equation.Using our earlier example: After simplifying the point-slope form \(y + 4 = \frac{3}{5}x + 3\), subtracting 4 from both sides yields the slope-intercept form:\[y = \frac{3}{5}x - 1\]Here, the slope \(m\) is \(\frac{3}{5}\), and the y-intercept \(b\) is \(-1\). This clearly shows the rate of change and where the line crosses the y-axis.