Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible. $$(2,-3), \quad m=-\frac{1}{2}$$

Step-by-Step Solution

Answer

The equation of the line is \(y = -\frac{1}{2}x - 2\). After sketching the line by hand, the line can be confirmed accurate by using a graphing utility.

Step 1: Use the point-slope form

The point-slope form of an equation is given by \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. In this case, the point is (2, -3) and the slope (\(m\)) is -1/2. Substitute these values into the equation: \(y - (-3) = -\frac{1}{2}(x - 2)\).

Step 2: Simplify the equation

Simplify the equation by doing the arithmetic: \(y + 3 = -\frac{1}{2}x + 1\). Then isolate \(y\) to put the equation in slope-intercept form (\(y = mx + b\)): \(y = -\frac{1}{2}x + 1 - 3\), which simplifies to \(y = -\frac{1}{2}x - 2\).

Step 3: Sketch the line

The slope of the line is -1/2, which means the line goes down 1 unit for every 2 units to the right. The y-intercept is -2, which is where the line crosses the y-axis. Start from the point (0, -2) and, following the slope, mark the point (2, -3), which means you should move 2 units to the right and 1 unit down from the y-intercept, and draw the line that passes through these points.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a standard way to write a linear equation. It looks like this: \( y = mx + b \). Here, \( m \) represents the slope of the line, while \( b \) indicates the y-intercept—the point where the line crosses the y-axis. This form is particularly useful because it readily reveals both the slope and the y-intercept, making it easier to understand the line's behavior.To convert an equation into slope-intercept form, ensure it looks like \( y = mx + b \). Start with your equation, rearrange terms if necessary, and perform any arithmetic required to isolate \( y \) on one side. Once in slope-intercept form, you can immediately identify both the slope and y-intercept, aiding in easy graphing and understanding of the line's movement.

Graphing Lines

Graphing lines involves plotting a line on a coordinate plane using its equation. With the slope-intercept form \( y = mx + b \), you can graph a line by identifying its slope and y-intercept.

• Y-intercept: Start your graphing at \( b \), the y-intercept, by placing a point where the line will intersect the y-axis.
• Slope: Use the slope \( m \), which is rise over run, to determine the direction and steepness of the line. For example, a slope of -\( \frac{1}{2} \) means that for every 2 units moved to the right, the line descends 1 unit.

Begin drawing from the y-intercept and use the slope to locate another point. Connect these points with a straight line to complete the graph. Double-check your plot by ensuring it passes through all points predicted by the equation.

Linear Equations

Linear equations are the equations of straight lines. They describe a line's relationship in terms of its slope and positional constants. In a basic linear form, the equation can be structured in either point-slope or slope-intercept format to best describe the line's relationship to coordinate points.- **Point-Slope Form:** This used when you know one point on the line and its slope. The form \( y - y_1 = m(x - x_1) \) tells you how the line behaves around that fixed point.- **Slope-Intercept Form:** This is ideal when you need to derive or directly use a line's slope and y-intercept. It's easy to graph from this form since it tells you where the line crosses the y-axis (the intercept), and how steep it is (the slope).Linear equations are fundamental in mathematics and appear in various applications. From calculating interest to determining speed, understanding how to interpret and manipulate these equations can be quite powerful in both academic and real-world scenarios.