Problem 7 - Applied Calculus

Question

Find an equation for the line that passes through the given points. $$ (-2,1) \text { and }(2,3) $$

Step-by-Step Solution

Answer

The equation of the line is \(y = \frac{1}{2}x + 2\).

Step 1: Determine the Slope

To find the slope of the line that passes through the points \((-2, 1)\) and \((2, 3)\), we use the slope formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substituting the given points, \(y_2 = 3\), \(y_1 = 1\), \(x_2 = 2\), and \(x_1 = -2\):\[m = \frac{3 - 1}{2 - (-2)} = \frac{2}{4} = \frac{1}{2}\]Thus, the slope \(m\) is \(\frac{1}{2}\).

Step 2: Use Point-Slope Form

With the slope \(m = \frac{1}{2}\) and one of the points \((-2, 1)\), we apply the point-slope form of a line: \[y - y_1 = m(x - x_1)\]Substitute \(m = \frac{1}{2}\), \(x_1 = -2\), and \(y_1 = 1\):\[y - 1 = \frac{1}{2}(x - (-2))\]Simplify the equation:\[y - 1 = \frac{1}{2}(x + 2)\]

Step 3: Convert to Slope-Intercept Form

To express the equation in the slope-intercept form \(y = mx + b\), simplify the equation from Step 2:\[y - 1 = \frac{1}{2}(x + 2)\]Distribute the \(\frac{1}{2}\):\[y - 1 = \frac{1}{2}x + 1\]Add 1 to both sides to solve for \(y\):\[y = \frac{1}{2}x + 2\]

Key Concepts

Slope Calculation

To find the equation of a line, you first need to determine its slope. The slope of a line measures its steepness and can be calculated using two points on the line. These are typically labeled \(x_1, y_1\) and \(x_2, y_2\). The formula for slope, usually denoted by \(m\), is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
In simple words, you subtract the y-coordinates of the points to find the rise, and subtract the x-coordinates to find the run. Divide these values, and you’ll get the slope.

• Let's substitute the points \((-2, 1)\) and \(2, 3)\).
• Substitute into the formula: \(y_2 = 3\), \(y_1 = 1\), \(x_2 = 2\), and \(x_1 = -2\).
• Perform the calculations: \[m = \frac{3 - 1}{2 - (-2)} = \frac{2}{4} = \frac{1}{2}\]

Therefore, the slope of the line passing through these points is \(\frac{1}{2}\).

Point-Slope Form

Once the slope \(m\) is known, you can use the point-slope form to write the equation of the line. This is especially useful when you know the slope and one point on the line.
The point-slope form is given by:
\[y - y_1 = m(x - x_1)\]
Let's plug in the slope \(\frac{1}{2}\) and one of the points \((-2, 1)\):

• Substitute \(m = \frac{1}{2}\), \(x_1 = -2\), and \(y_1 = 1\):
• \[y - 1 = \frac{1}{2}(x - (-2))\]
• Simplifying the equation: \[y - 1 = \frac{1}{2}(x + 2)\]

This formula helps you see how the line operates around the point you already have, and it can be easily transformed into other forms like the slope-intercept form.

Slope-Intercept Form

The slope-intercept form is perhaps the most commonly used form of a linear equation because it directly shows the slope and the y-intercept.
This form is expressed as:
\[y = mx + b\]
Where \(m\) is the slope and \(b\) is the y-intercept. Starting from the simplified point-slope form, you can convert it into this form.

• Start with: \[y - 1 = \frac{1}{2}(x + 2)\]
• Distribute the fraction: \[y - 1 = \frac{1}{2}x + 1\]
• To solve for \(y\), add 1 to both sides: \[y = \frac{1}{2}x + 2\]

Now, the equation is in slope-intercept form, and you can clearly see the slope is \(\frac{1}{2}\) and the line crosses the y-axis at 2. This makes it easier to graph and understand the line's characteristics.