Problem 23 - Precalculus Mathematics for Calculus

Question

Find an equation of the line that satisfies the given conditions. Through \((2,1)\) and \((1,6)\)

Step-by-Step Solution

Answer

The equation of the line is \( y = -5x + 11 \).

Step 1: Understand the Slope Formula

To find the equation of a line, we first need to find its slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula gives us the rate at which \( y \) changes with respect to \( x \).

Step 2: Identify Points and Plug into Formula

We have the points \( (x_1, y_1) = (2, 1) \) and \( (x_2, y_2) = (1, 6) \). Plug these into the slope formula: \( m = \frac{6 - 1}{1 - 2} = \frac{5}{-1} = -5 \). The slope of the line is \(-5\).

Step 3: Use the Point-Slope Form

Now use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is one of the points. Let's use the point \((2, 1)\).

Step 4: Plug Values into the Point-Slope Equation

Substitute \( m = -5 \), \( x_1 = 2 \), and \( y_1 = 1 \) into the equation: \( y - 1 = -5(x - 2) \).

Step 5: Simplify the Equation

Distribute \(-5\) in the equation: \( y - 1 = -5x + 10 \). To get the equation in slope-intercept form \( y = mx + c \), add 1 to both sides: \( y = -5x + 11 \).

Key Concepts

Slope Formula

When tackling problems involving linear equations, understanding the slope formula is essential. The slope formula calculates how steep a line is by determining the ratio of the vertical change to the horizontal change between two points on the line.
It's expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \(m\) represents the slope, and \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.

To find the slope, you subtract the first point's \(y\)-coordinate from the second point's \(y\)-coordinate and divide by the difference between their \(x\)-coordinates. This result tells us how much \(y\) changes for each unit change in \(x\).
In our exercise, using this formula with points \((2, 1)\) and \((1, 6)\), the slope \(m\) is calculated as \( \frac{6 - 1}{1 - 2} = -5 \). This means for every 1 unit of increase in \(x\), \(y\) decreases by 5 units.

Point-Slope Form

Once we know the slope, we can express the line's equation using the point-slope form. This form is great for quickly writing the equation when you know a point on the line and the slope. The formula is given as \( y - y_1 = m(x - x_1) \).
Here, \( (x_1, y_1) \) is a point on the line, and \(m\) is the slope. This form highlights the relationship between the coordinates of points on the line and the line's steepness.

Let's apply it to the points from our exercise. We have the slope \(-5\) and point \((2, 1)\). Using these in our formula, we get \( y - 1 = -5(x - 2) \). This equation tells us the coordinates of any point \(x\) on the line can be found if the line passes through \((2, 1)\).

Slope-Intercept Form

Finally, converting the point-slope form to the slope-intercept form makes the equation handy for graphing the line.
The slope-intercept form is \( y = mx + c \). In this formula, \(m\) is the slope, and \(c\) is the \(y\)-intercept, or the point where the line hits the \(y\)-axis.

Starting from our point-slope form equation \( y - 1 = -5(x - 2) \), we expand and simplify to get the slope-intercept form. Distributing the slope, we have \( y - 1 = -5x + 10 \). Adding 1 to both sides gives \( y = -5x + 11 \).
This final equation shows the slope of the line is \(-5\), and it crosses the \(y\)-axis at \(11\). With this form, it's easy to see how the line behaves and where it crosses the axes.