Problem 30 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line. Then rewrite the equation in slope-intercept form. $$ (-2,4), m=3 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y - 4 = 3(x + 2)\) and in slope-intercept form is \(y = 3x + 10\).

Step 1: Write the Point-Slope Form

Using the formula for point-slope form \(y - y1 = m(x - x1)\), and plugging the given point \((-2, 4)\) and slope \(3\) into the formula, the equation becomes: \(y - 4 = 3(x + 2)\).

Step 2: Simplify the Equation

Next, simplify the equation by distributing the slope to the terms in parentheses: \(y - 4 = 3x + 6\). This is the equation in point-slope form.

Step 3: Rewrite the Equation in Slope-Intercept Form

To put the equation in slope-intercept form \(y = mx + c\), solve the equation for \(y\): \(y = 3x + 10\). This is the equation in slope-intercept form.

Key Concepts

Point-Slope Form

To understand the point-slope form of a linear equation, imagine you're drawing a line on a graph. This line can be represented using an equation. The point-slope form is perfect when you know a specific point on the line and the slope (the measure of its steepness). The general formula is:

• \(y - y_1 = m(x - x_1)\)

Here:

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

Let's use the point \((-2, 4)\) and the slope \(m = 3\) given in our exercise. Plugging these values into the formula:

• \(y - 4 = 3(x + 2)\)

This equation tells us how the line behaves by showing the relationship between \(x\) and \(y\). Whenever you have a point and a slope, you can use this form to quickly write the equation of the line.

Slope-Intercept Form

The slope-intercept form is another way to write the equation of a line. It allows you to easily identify the slope and the point where the line crosses the y-axis, which we call the y-intercept. The formula for the slope-intercept form is as follows:

• \(y = mx + b\)

Where:

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

After finding the point-slope form, we simplify it to write it in the slope-intercept form. From the earlier point-slope equation:

• \(y - 4 = 3x + 6\)

Solving for \(y\) gives us: \(y = 3x + 10\). This is our slope-intercept equation, clearly showing that the slope \(m\) is 3 and the y-intercept \(b\) is 10. The beauty of this form is its simplicity in showing how the line ascends and where it cuts the y-axis.

Equation of a Line

The equation of a line represents all the possible points that exist on the line. By using different forms like point-slope and slope-intercept, we get different views and insights about the same line.When constructing an equation:

• Start with what you know: a point and a slope, or a slope and an intercept.
• Choose the form that aligns best with what you have or need to find.

For example, the point-slope form is your go-to if you know a point and the slope, as seen in the exercise with \((-2, 4)\) and \(m = 3\). It gives you the equation \(y - 4 = 3(x + 2)\).From there, you can convert to the slope-intercept form if you need the line's slope at a glance or the y-intercept. The conversion based on our exercise results in \(y = 3x + 10\).This versatility in linear equations helps in simplifying tasks, like graphing lines, calculating intersections, and solving systems of equations. When learning these forms, practice transforming from one form to another to solidify your understanding.