Problem 20 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((-2,4) ;\) slope \(-1\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find the equation of a line that passes through the point \((-2, 4)\) with a slope of \(-1\).

Step 2: Use Point-Slope Form

The point-slope form of a linear 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. Here, \(m = -1\), \(x_1 = -2\), and \(y_1 = 4\).

Step 3: Substitute Values

Substitute the values into the point-slope form: \(y - 4 = -1(x + 2)\).

Step 4: Simplify the Equation

Distribute the slope on the right side and then simplify: \(y - 4 = -x - 2\).

Step 5: Solve for y

Add 4 to both sides to solve for \(y\): \(y = -x + 2\). This is the equation of the line in slope-intercept form.

Key Concepts

Point-Slope Form of a Linear Equation

The point-slope form is a handy tool when you know one point on a line and the slope. This form is expressed as \( y - y_1 = m(x - x_1) \). Here, \( m \) represents the slope, while \( (x_1, y_1) \) is a point on the line. This format allows you to write an equation easily because you plug in values directly without rearranging terms.

In our example, the problem gives us a slope of -1 and a point (-2, 4). Plug these values into the point-slope formula, replacing \( m \) with -1, \( x_1 \) with -2, and \( y_1 \) with 4. This results in the equation:

• \( y - 4 = -1(x + 2) \)

This setup reflects the line's slope clearly and how it passes through the given point. Simplifying this form leads us toward the more familiar slope-intercept form.

Slope-Intercept Form and Its Advantages

The slope-intercept form, expressed as \( y = mx + b \), is a popular and versatile way of writing equations. Here, \( m \) is the slope of the line, and \( b \) is the y-intercept. This form is straightforward to interpret: it tells us the line's steepness and where it crosses the y-axis.

From our exercise, after simplifying the point-slope equation, we arrive at \( y = -x + 2 \). In this arrangement:

• The slope \( m \) is -1, indicating the line decreases as you move along the x-axis.
• The intercept \( b \) is 2, meaning the line crosses the y-axis at \( y = 2 \).

Use slope-intercept form to quickly graph lines or understand their behavior. It's particularly helpful in identifying parallel or perpendicular lines by comparing slopes.

Algebraic Manipulation to Transition Between Forms

Algebraic manipulation involves changing the equation from one form to another using arithmetic operations. This skill is central in moving from the point-slope form to the slope-intercept form, often making the equation more understandable.

Starting with \( y - 4 = -1(x + 2) \), distribute the -1 to both terms inside the parenthesis:

• \( y - 4 = -x - 2 \)

Next, solve for \( y \) by adding 4 to both sides to isolate \( y \):

• \( y = -x + 2 \)

Each step simplifies the equation, making it easier to work with and analyze. Practicing algebraic manipulation enhances problem-solving skills, as it lets you reshape equations to highlight different features or answer specific questions within an exercise context.