Problem 26 - College Algebra

Question

For the following exercises, find the equation of the line using the point- slope formula. Write all the final equations using the slope-intercept form. $$ (-3,10) \text { and }(5,-6) $$

Step-by-Step Solution

Answer

The equation of the line is \(y = -2x + 4\).

Step 1: Identify the Points

We are given two points, \((-3, 10)\) and \((5, -6)\). These points will help us find the slope of the line and then use the point-slope formula.

Step 2: Calculate the Slope

The slope \(m\) is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).Substituting in the values from the points: \(m = \frac{-6 - 10}{5 - (-3)} = \frac{-16}{8} = -2\). So, the slope is \(-2\).

Step 3: Use the Point-Slope Formula

The point-slope formula is \(y - y_1 = m(x - x_1)\). Using the point \((5, -6)\) and the slope \(-2\), we substitute into the formula: \(y + 6 = -2(x - 5)\).

Step 4: Expand the Equation

Distribute \(-2\) through \((x - 5)\):\(y + 6 = -2x + 10\).

Step 5: Rearrange to Slope-Intercept Form

To convert to slope-intercept form \(y = mx + b\), solve for \(y\):Subtract 6 from both sides: \(y = -2x + 4\).

Key Concepts

Slope Calculation

To understand how to find the equation of a line, the first step is to calculate the slope, which is a measure of how steep a line is. The slope is represented by the symbol \(m\). It is determined by how much the line rises (changes in \(y\)) for how much it runs (changes in \(x\)). You can calculate the slope with the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Given the two points \((-3, 10)\) and \((5, -6)\), we plug these into the formula:

• Subtract the \(y\)-coordinates: \(-6 - 10 = -16\)
• Subtract the \(x\)-coordinates: \(5 - (-3) = 8\)
• Divide the differences: \(\frac{-16}{8} = -2\)

So, the slope \(m\) of the line is \(-2\). This value of \(m\) is crucial as it becomes part of the linear equation.

Slope-Intercept Form

The slope-intercept form is a simple yet powerful way to write the equation of a line. It makes it easy to see the slope and the y-intercept, which is where the line crosses the y-axis. The general formula for the slope-intercept form is: \[ y = mx + b \] In this equation, \(m\) is the slope, and \(b\) is the y-intercept. A line in slope-intercept form provides an immediate understanding of how changes in \(x\) impact \(y\). In the exercise provided, after calculating the slope as \(-2\) and performing necessary algebraic manipulations, the line's equation was converted to: \[ y = -2x + 4 \] Here, \(-2\) is the slope and \(4\) is the y-intercept. This means for every unit increase in \(x\), \(y\) decreases by \(2\). The line crosses the y-axis at \(4\).

Equation of a Line

Now, let's focus on finding the equation of a line using the point-slope formula, and then converting it to slope-intercept form. The point-slope formula is particularly useful because it provides a formulaic approach to writing the equation of a line when you have a slope and a single point on the line. The point-slope form is written as: \[ y - y_1 = m(x - x_1) \] Here, \((x_1, y_1)\) is a point on the line, and \(m\) is the slope we've already calculated. Using point \((5, -6)\) and slope \(-2\):

• Start with the point-slope formula: \(y + 6 = -2(x - 5)\)
• Expand through distribution: \(y + 6 = -2x + 10\)
• Isolate \(y\) to find the slope-intercept form: \(y = -2x + 4\)

The resulting equation \(y = -2x + 4\) is now in slope-intercept form, allowing anyone to clearly identify the slope and y-intercept of the line.