Problem 28 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-2,0)\) and \((0,2)\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y - 0 = 1(x - (-2))\) or \(y = x + 2\), and in slope-intercept form, it is \(y = x + 2\).

Step 1: Calculate the slope

The formula to calculate the slope between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given as \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Plugging in the given points \((-2,0)\) and \((0,2)\), so we find \(m = \frac{2 - 0}{0 - (-2)} = 1\).

Step 2: Write the point-slope form

The point-slope form of a line's equation is \(y - y_1 = m(x - x_1)\). Plug in the calculated slope \(m = 1\) and one of the given points, for example \((-2,0)\). So we get \(y - 0 = 1(x - (-2)) \rightarrow y = x + 2\).

Step 3: Transform to slope-intercept form

The point-slope form from step 2 \(y = x + 2\) is already in slope-intercept form \(y = mx + b\), with \(m = 1\) as the slope and \(b = 2\) as the y-intercept.

Key Concepts

Point-Slope Form

The point-slope form of an equation is a helpful way to express the equation of a line when you know one point along the line and its slope. It is expressed as:

• \( y - y_1 = m(x - x_1) \)

In this formula,

• \( (x_1, y_1) \) are the coordinates of the known point,
• \( m \) represents the slope of the line.

Let's use the point-slope form with one of the provided points and the calculated slope from the original exercise.We chose

• Point: \((-2,0)\)
• Slope: \(m = 1\)

By plugging these values into the formula, we get:\[ y - 0 = 1(x - (-2)) \]Simplifying the equation, we arrive at:\[ y = x + 2 \] Once it's put this way, it becomes more intuitive to transform it into other equation forms, such as the slope-intercept form.

Slope-Intercept Form

The slope-intercept form is one of the most common and easy ways to express linear equations. Its standard form is:

• \( y = mx + b \)

In this formula,

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

When we obtained \( y = x + 2 \) from the point-slope form, it was already perfectly arranged in the slope-intercept form.In the equation:

• The slope \( m = 1 \), determining how steep the line is.
• The y-intercept \( b = 2 \), showing that the line crosses the y-axis at (0,2).

The slope-intercept form is quite intuitive and allows for easy graphing of the linear relationship. You simply start at the y-intercept and use the slope to determine the line’s direction and steepness. This form is especially useful when you need to quickly sketch the graph of a line or understand its key characteristics.

Slope Calculation

Finding the slope of a line is a fundamental skill in algebra, especially when working with linear equations. The slope is a measure of how steep a line is, and it is influenced by the vertical change relative to the horizontal change between any two points on the line.For any two points

• \((x_1, y_1)\) and \((x_2, y_2)\)

The slope (\(m\)) can be calculated using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In the context of our problem, using points

• \((-2, 0)\) and \((0, 2)\),

we computed the slope as follows:\[m = \frac{2 - 0}{0 - (-2)} = 1\]This tells us that for each unit increase in x, y increases by the same amount, indicating the line is rising evenly.Slope plays a crucial role in determining the angle and direction of the line, allowing for predictions and various analyses of the line’s behavior. Understanding slope is key to mastering linear equations and offers a straightforward method to decode many algebra problems.