Problem 81 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ (-5,7), \quad(-2,7) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is y = 7.

Step 1: 1. Find the slope between the two points

To find the slope between the two given points \((-5,7)\) and \((-2,7)\), we can use the slope formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Plugging the points into the formula, we will get: $$ m = \frac{7 - 7}{-2 - (-5)} $$

Step 2: 2. Calculate the slope

Now we simplify the slope equation to get the value of the slope (m): $$ m = \frac{0}{3} = 0 $$ So, the slope of the line is 0.

Step 3: 3. Use the slope-intercept equation to find the y-intercept

Now that we have the slope, we can use one of the given points, say \((-5,7)\), to find the y-intercept (b) using the slope-intercept equation: $$ y = mx + b $$ Plugging in the values for \(x\), \(y\), and \(m\), we get: $$ 7 = 0(-5) + b $$

Step 4: 4. Solve for the y-intercept

Solving for the y-intercept (b) from the equation in step 3, we find that: $$ b = 7 $$ So, the y-intercept is 7.

Step 5: 5. Write the final equation of the line in slope-intercept form

Now we have both the slope (m) and the y-intercept (b), so we can write the final equation of the line in slope-intercept form: $$ y = 0x + 7 $$ Simplifying, we get the final equation of the line: $$ y = 7 $$

Key Concepts

Linear Equations

Linear equations form the backbone of algebra and are pivotal in solving many real-world problems. They are expressed in the format of a straight line when plotted on a graph. A linear equation in slope-intercept form has the general formula: \[ y = mx + b \]Here, \( y \) and \( x \) represent variables where \( y \) is typically the dependent variable and \( x \) is the independent variable.
The letter \( m \) represents the slope, which is the steepness or inclination of the line, and \( b \) is the y-intercept, which is the point where the line intersects the y-axis.

• This form easily shows the slope and y-intercept, making it user-friendly for both graphing and interpreting linear relationships.
• Linear equations appear as straight, constant-gradient lines, thus illustrating fixed relationships between the two variables.

The primary aim when working with linear equations is often to express relationships clearly and simply so they can be graphed or used in models.

Slope Calculation

Slope calculation is the critical step in understanding linear equations. The slope indicates how much \( y \) changes for a unit change in \( x \). The formula to calculate the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]In our particular example, the two points given are \((-5,7)\) and \((-2,7)\).
By substituting these into the slope formula:

• \( y_2 - y_1 = 7 - 7 = 0 \)
• \( x_2 - x_1 = -2 - (-5) = 3 \)
• Which makes the slope: \( m = \frac{0}{3} = 0 \)

A slope of zero means the line is horizontal, which signifies no change in \( y \) as \( x \) varies. This hints that the equation may be a constant line.

Y-intercept

The y-intercept \( b \) is a core component of the slope-intercept form. It indicates where the line crosses the y-axis. In our example, we discovered that the slope was zero. Given that one of our points is \((-5,7)\), we can use the slope-intercept formula \( y = mx + b \) to find \( b \).
Since the slope \( m \) is zero, our equation simplifies to:

• \( y = 0x + b \), which is simply \( y = b \)
• Plug in the figures from point \((-5,7)\): \( 7 = 0(-5) + b \)
• This gives \( b = 7 \)

Thus, the y-intercept is 7, meaning that across the entire line, every point has the same y-value of 7, characteristic of a horizontal line. This is why the final equation can be simplified simply to \( y = 7 \), showing the line is parallel to the x-axis at a height of 7 on the graph.