Problem 34 - Contemporary Precalculus

Question

Find the equation of the line through the given points. $$(4,3) and (2,-1)$$

Step-by-Step Solution

Answer

Answer: The equation of the line passing through the points (4, 3) and (2, -1) is y = 2x - 5.

Step 1: Calculate the slope of the line

The slope of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula: $$ slope \, (m) = \frac{y_2 - y_1}{x_2 - x_1}$$ Plugging in the given points \((4,3\) and \((2,-1)\), we get: $$m = \frac{-1 - 3}{2 - 4} = \frac{-4}{-2} = 2$$ So, the slope of the line is 2.

Step 2: Find the equation of the line using the point-slope form

The point-slope form of the line equation is given by: $$y - y_1 = m (x - x_1)$$ Using the slope we calculated in Step 1 and one of the given points, for example, \((4,3)\), we get: $$y - 3 = 2(x - 4)$$ Now, we'll simplify the equation to get the equation of the line: $$y - 3 = 2x - 8$$ $$y = 2x - 5$$ So, the equation of the line passing through the given points is: $$y = 2x - 5$$

Key Concepts

Slope Calculation

When dealing with linear equations, finding the slope is a crucial initial step. The slope tells us how steep the line is and in which direction it moves. To calculate the slope between two points \(x_1, y_1\) and \(x_2, y_2\), use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). It represents the change in the y-value divided by the change in the x-value.

This calculation essentially measures the vertical change versus the horizontal change. A positive slope indicates that as we move from left to right on the graph, the line rises. A negative slope means the line falls as we move in the same direction.

In our example, points \(4,3\) and \(2,-1\) gave us a slope of \( m = 2 \). Thus, the line rises sharply as it moves from left to right.

Point-Slope Form

The point-slope form of a line is an effective way to write the equation of a line when you already have a slope and a point on the line. The formula is \( y - y_1 = m(x - x_1) \).

Think of it as a guide showing how each x-value affects the y-value based on the slope. In essence, this format highlights the slope as a constant multiplier and begins its work from a particular point on the line.

For our line with a slope of 2 passing through point \(4,3\), the formula becomes:

• Replace \(m\) with 2.
• Use \(x_1 = 4\) and \(y_1 = 3\).

This transforms our equation into \( y - 3 = 2(x - 4) \). From here, you can simplify further to get the line's complete equation, but this form is both functional and informative at this step.

Equation of a Line

Once you have the slope and a point, the final step is often converting the point-slope form equation into the slope-intercept form, \(y = mx + b\). This conversion helps in easily identifying the slope and y-intercept.

From \(y - 3 = 2(x - 4)\), distribute the 2:

• 2 times \(x\) yields \(2x\).
• 2 times \(-4\) becomes \(-8\).

Thus, you have \(y - 3 = 2x - 8\). Add 3 to both sides to isolate y:

• This results in \(y = 2x - 5\).

The equation \(y = 2x - 5\) reveals that the line crosses the y-axis at \(-5\) and rises 2 units for every 1 unit it travels rightwards. Understanding this form is particularly useful in graphing and analyzing linear relationships efficiently.