Problem 10 - Elementary Algebra

Question

Find the equation of each line given the following information. Use the slope- intercept form as the final form of the equation. $$ \text { The two points }(-7,-1) \text { and }(-4,8) \text { . } $$

Step-by-Step Solution

Answer

Answer: The equation of the line in slope-intercept form is y = 3x + 20.

Step 1: Calculate the slope

Using the two given points, \((-7, -1)\) and \((-4, 8)\), we can calculate the slope using the formula: $$ m = \frac{y_2-y_1}{x_2-x_1} $$ Substitute the coordinates of the two points into the formula: $$ m = \frac{8-(-1)}{-4-(-7)} $$

Step 2: Simplify the slope

Now, we simplify the expression to get the slope: $$ m = \frac{8+1}{-4+7} =\frac{9}{3} =3 $$

Step 3: Find the y-intercept

Now that we have the slope, we can plug it into the slope-intercept formula along with one of the given points. Let's use the point \((-7, -1)\): $$ -1 = 3(-7) + b $$

Step 4: Solve for y-intercept (b)

Now, solve for b in the equation: $$ -1=-21+b \Rightarrow b = -1+21 =20 $$

Step 5: Write the equation in slope-intercept form

Now that we have the slope and y-intercept, we can write the equation of the line in slope-intercept form: $$ y = 3x + 20 $$

Key Concepts

Equation of a Line

The equation of a line is a fundamental concept in geometry and algebra. When you want to express a line mathematically, you often use the slope-intercept form, which is written as \(y = mx + b\). In this expression:

• \(y\) represents the dependent variable or the variable that changes in response to \(x\).
• \(m\) is known as the slope of the line, indicating how steep the line is.
• \(x\) is the independent variable determining the position along the horizontal axis.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

When given two points, you can always use these to derive the equation of a line. Once you have the slope and one point, it becomes straightforward to plug these values into the slope-intercept formula to find the complete equation.

Slope Calculation

The slope of a line is a measure of its steepness. To calculate the slope, you'll use the formula that represents the change in \(y\) over the change in \(x\): \(m = \frac{y_2-y_1}{x_2-x_1}\). This formula comes from the basic principle of rise over run:

• 'Rise' refers to how much the line climbs or descends vertically (\(y_2-y_1\)).
• 'Run' refers to how far the line moves horizontally (\(x_2-x_1\)).

By substituting the given points \((-7, -1)\) and \((-4, 8)\) into the formula, you first find \(y_2 - y_1 = 8 - (-1) = 9\) and \(x_2 - x_1 = -4 - (-7) = 3\). So, the slope \(m\) is \(\frac{9}{3} = 3\). This positive slope indicates that the line rises as it moves from left to right.

Y-Intercept

The y-intercept is where the line crosses the y-axis. It gives you a specific point on the graph of a line where \(x = 0\). Using the equation of the line in slope-intercept form \(y = mx + b\), once you have found the slope \(m\), you can determine the y-intercept \(b\) by using any point on the line.For example, using the slope \(m = 3\) and the point \((-7, -1)\), substitute these values into \(y = mx + b\) to solve for \(b\):- Plug in \(x = -7\) and \(y = -1\) to get: \(-1 = 3(-7) + b\)- Simplify: \(-1 = -21 + b\)- Solve for \(b\) by adding \(21\) to both sides, yielding \(b = 20\).Thus, the y-intercept of this line is \(20\). This means the line crosses the y-axis at the point \((0, 20)\).