Problem 39 - Algebra 1

Question

Write the point-slope form of the equation of the line that passes through the point and has the given slope. Then rewrite the equation in slope-intercept form. $$ (1,4), m=2 $$

Step-by-Step Solution

Answer

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

Step 1: Write the point-slope form of the equation

Plug the given point (1,4) and slope 2 into the point-slope form of a linear equation like so: \(y - y_1 = m(x - x_1)\) becomes \(y - 4 = 2(x - 1)\). That simplifies to \(y - 4 = 2x - 2\). So, \(y - 4 = 2x - 2\) is the point-slope form of the equation.

Step 2: Convert to slope-intercept form

Rewrite the equation \(y - 4 = 2x - 2\) from point-slope form into slope-intercept form by isolating \(y\). Add 4 to both sides to get \(y = 2x + 2\). This is the slope-intercept form of the equation.

Key Concepts

Equation of a line

The equation of a line is a fundamental concept in algebra, used to describe straight lines on a coordinate plane. There are different forms of a linear equation, each with its own use and significance. In this context, we'll focus on point-slope form and slope-intercept form. Both serve the purpose of representing a line but do so with different starting information:

• Point-slope form: This form is useful when you know a point on the line and the slope. It's written as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
• Slope-intercept form: This form is convenient when you need to quickly see the slope and the y-intercept of a line. It's given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Understanding these forms allows you to describe linear relationships in equations that reflect real-world situations. By translating a line's characteristics into an equation, you can analyze and predict behaviors tied to linear patterns.

Slope-intercept form

The slope-intercept form is often the most easily recognized linear equation format. It shows at a glance what the slope of the line is, as well as where it crosses the y-axis. To convert an equation from point-slope to slope-intercept, you'll focus on isolating \(y\):

• The formula is \(y = mx + b\).
• \(m\) is the slope — it tells you how steep the line is and its direction.
• \(b\) is the y-intercept — it's the point where the line crosses the y-axis.

To see slope-intercept form in action, take an equation like \(y - 4 = 2(x - 1)\). By adding 4 to both sides, you can transform it to \(y = 2x + 2\). Here, \(2\) is the slope, and \(2\) is the y-intercept. This form provides a clear and quick insight into the behavior of the line.

Linear equations

Linear equations are equations that chart straight lines when graphed on a coordinate plane. They represent constant rates of change and are fundamental in understanding basic algebra concepts. Characteristics of linear equations include:

• They have one or two variables.
• The highest power of the variables is one.
• Linear equations produce a straight line graph.

The reason they are straightforward is due to their consistency; the relationship between any two points on the line is uniform. For example, in a linear equation like \(y = 2x + 2\), the slope of \(2\) means that for every step to the right along the x-axis, the value of \(y\) increases by 2. Understanding linear equations is the key to solving problems involving proportional relationships and interpreting many real-life situations.