Problem 24 - College Algebra

Question

Find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form. \(x\) -intercept is \(1,\) and \((-2,6)\)

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -2x + 2\).

Step 1: Understanding the Components

The problem provides us with the \(x\)-intercept and a specific point, \((-2, 6)\), on the line. The \(x\)-intercept, being \(1\), means that the point \((1, 0)\) is on the line. These two points, \((1, 0)\) and \((-2, 6)\), will be used to find the slope of the line.

Step 2: Calculate the Slope

To find the slope \(m\), use the formula for the slope of a line passing through two points, \((x_1, y_1)\) and \((x_2, y_2)\), which is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Applying this to our points:\[m = \frac{6 - 0}{-2 - 1} = \frac{6}{-3} = -2.\]So, the slope of the line is \(-2\).

Step 3: Apply the Point-Slope Formula

We use the point-slope formula, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Using the point \((-2, 6)\):\[y - 6 = -2(x + 2).\]

Step 4: Simplify to Slope-Intercept Form

To convert the equation to the slope-intercept form \(y = mx + b\), simplify the equation from Step 3:\[y - 6 = -2(x + 2)\]Distribute the \(-2\):\[y - 6 = -2x - 4\]Add 6 to both sides to solve for \(y\):\[y = -2x + 2\]So, the equation in the slope-intercept form is \(y = -2x + 2\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a very common way to express a linear equation. It is often represented by the formula:

• \[y = mx + b\]

Here, \(m\) stands for the slope of the line, and \(b\) describes the \(y\)-intercept, which is where the line crosses the \(y\)-axis.
This format is particularly useful because it instantly provides two vital pieces of information about the line: its slope and its \(y\)-intercept. This can make graphing and interpreting lines much easier.
In our example, the final equation for the line in this format is \(y = -2x + 2\). In this equation:

• \(-2\) is the slope \((m)\) of the line.
• \(2\) is the \(y\)-intercept \((b)\), meaning the line crosses the \(y\)-axis at \(y = 2\).

Calculating Slope

When calculating the slope of a line, you need two points from the line. The slope tells you how steep the line is, and its formula is:

• \[m = \frac{y_2 - y_1}{x_2 - x_1}\]

This means you subtract the \(y\)-value of the first point from the \(y\)-value of the second point, and divide that by subtracting the \(x\)-value of the first point from the \(x\)-value of the second point.
In our exercise, we use the points \((1, 0)\) and \((-2, 6)\). These points tell us that:

• Change in \(y\) is \(6 - 0 = 6\).
• Change in \(x\) is \(-2 - 1 = -3\).

Thus, the slope \(m\) is \(\frac{6}{-3} = -2\).
This negative slope indicates that the line moves downward from left to right.

Linear Equations

Linear equations represent straight lines. They're called "linear" because their graph forms a straight line. The general format of a linear equation could be any of the following:

• Slope-intercept form: \(y = mx + b\)
• Standard form: \(Ax + By = C\)
• Point-slope form: \(y - y_1 = m(x - x_1)\)

Each form has its own use, but they all describe the same set of equations representing lines on a graph.
To convert from point-slope form to slope-intercept form, like what we did in this exercise, you simply solve for \(y\), rearranging the terms and isolating \(y\) on one side. Point-slope form is especially useful when you're given a point and a slope, making it an excellent starting point for constructing a linear equation.
From here, mathematical manipulations, such as distributing terms and isolating \(y\), turn this format into the slope-intercept form for easier interpretation and graphing.