Problem 13 - Calculus with Concepts in Calculus

Question

Find an equation of the line described. Then sketch the line. The line through \((2,-1)\) with slope 3

Step-by-Step Solution

Answer

The equation of the line is \(y = 3x - 7\).

Step 1: Identify the Point-Slope Form

To find the equation of the line through the point \((x_1, y_1) = (2, -1)\) with slope \(m = 3\), we use the point-slope form of a linear equation: \[ y - y_1 = m(x - x_1) \]This is a standard method for finding the equation of a line when given a point and a slope.

Step 2: Substitute the Given Values

Substitute the point \((2, -1)\) and the slope \(3\) into the point-slope formula:\[ y - (-1) = 3(x - 2) \]Simplifying the left side, we have:\[ y + 1 = 3(x - 2) \]

Step 3: Simplify to Slope-Intercept Form

Distribute the slope \(3\) on the right side:\[ y + 1 = 3x - 6 \]Subtract \(1\) from both sides to solve for \(y\):\[ y = 3x - 6 - 1 \]Thus, the equation is:\[ y = 3x - 7 \]

Step 4: Sketch the Line

To sketch the line, start by plotting the point \((2, -1)\) on a coordinate plane. Use the slope \(3\), which means "rise over run" (3 up, 1 right), to find another point. From \((2, -1)\), move up 3 units to \(y = 2\) and right 1 unit to \(x = 3\), giving the point \((3, 2)\). Draw a line through these points.

Key Concepts

Point-Slope Form

The point-slope form is a powerful tool in linear equations for a quick start. It helps us to develop the equation of a line when we know a specific point on the line and its slope. The formula is:\[ y - y_1 = m(x - x_1) \]Where:

• \( (x_1, y_1) \) is a point on the line.
• \( m \) is the slope of the line.

When we plug these values into the formula, it gives us a direct path to the line's equation. This form is especially handy when dealing with geometry problems that provide a point and a slope initially. As indicated in our exercise, by using the point \( (2, -1) \) and slope \( 3 \), the calculation started with \( y + 1 = 3(x - 2) \).
From there, we expanded and simplified it to reach the slope-intercept form. This initial setup boosts our progress, guiding us to unlock the line's full equation.

Slope-Intercept Form

The slope-intercept form is excellent for understanding and graphing linear equations. It presents the equation of a line in the format:\[ y = mx + b \]Where:

• \( m \) represents the slope.
• \( b \) is the y-intercept.

This form is a favorite because it clearly shows both vital characteristics of the line: its vertical steepness (slope) and where it crosses the y-axis (y-intercept). In our solution, after deriving the equation using the point-slope form, we converted it to the slope-intercept form: \( y = 3x - 7 \). This means the line rises 3 units for every run of 1 unit right, and it crosses the y-axis at \( y = -7 \).
This clarity makes it painless to plot or interpret the line's route on a graph, becoming a mainstay in algebra lessons and assignments.

Coordinate Geometry

Coordinate geometry, or analytic geometry, connects algebra with geometry through graphs. It uses points on a grid where the position of any given point is determined by coordinates. In our exercise, we deal with lines in the coordinate plane. The goal here is to use algebraic expressions to explain geometric concepts. With linear equations, we can delineate lines by using point coordinates and slopes. To illustrate:

• Start with a known point, like \( (2, -1) \), and use it to plot the line.
• The slope tells us how steep the line is, which helps in plotting additional points.
• Traces like \( (2, -1) \) and \( (3, 2) \)are identified to sketch the full line.

Plotting these elements on a coordinate plane showcases how algebra and geometry harmoniously work together to solve and visualize problems. This dual perspective can lead to a better grasp in understanding spatial relationships and mathematical equations.