Problem 48 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of line that passes through the given points. \((7,4)\) and \((-3,0)\)

Step-by-Step Solution

Answer

The equation of the line that passes through the points (7,4) and (-3,0) is \(y = 0.4x + 1.2\)

Step 1: Calculate the Slope

The formula for finding the slope (m) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\). For points (7,4) and (-3,0), the slope can be calculated as \(m = \frac{(0 - 4)}{(-3 - 7)} = \frac{-4}{-10} = 0.4\)

Step 2: Find the Y-intercept

The y-intercept (b) can be found using the formula \(b = y - mx\) where (x, y) are the coordinates of any point on the line, and m is the slope. Using point (7,4) and the slope 0.4, the y-intercept can be calculated as \(b = 4 - 0.4*7 = 1.2\)

Step 3: Write the Equation of the Line

Once the slope (m) and the y-intercept (b) have been found, they can be plugged into the slope-intercept form of the equation (y = mx + b) to get the equation of the line. Thus the equation of the line is \(y = 0.4x + 1.2\)

Key Concepts

Calculating Slope

The first step in writing an equation of a line in slope-intercept form is calculating the slope. The slope helps you understand the direction and steepness of a line. It is like determining how steep a hill is! The mathematical formula to find the slope, which is denoted by \( m \), is given by:- \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \) You take two points located on the line, let's call them \((x_1, y_1)\) and \((x_2, y_2)\). First, subtract the y-value of the first point from the y-value of the second point to get the rise. Then, subtract the x-value of the first point from the x-value of the second point to get the run. Finally, divide rise by run to get the slope.
In the example with points (7, 4) and (-3, 0), plugging those coordinates in gives:- \( m = \frac{(0 - 4)}{(-3 - 7)} = \frac{-4}{-10} = 0.4 \)Once calculated, this slope means the line rises 0.4 units for every unit it runs right.

Y-Intercept

Finding the y-intercept is the next crucial step. The y-intercept is simply where the line touches the y-axis. This point is helpful because it provides a place from which the line can originate. In the equation of a line, denoted as \( b \), it's the \( y \)-coordinate when \( x = 0 \).
To find \( b \), we use the formula:- \( b = y - mx \) Here, \( m \) is the slope that you found in the previous step, and \( x, y \) come from any point through which the line passes. Let's say we use the point (7, 4) with slope 0.4:- \( b = 4 - 0.4 \times 7 = 1.2 \)So, \( b \) comes out to be 1.2, which means the line crosses the y-axis at the point (0, 1.2). This is a critical element for forming the complete equation of the line.

Equation of a Line

Once you know both the slope and y-intercept, you can write the line's equation in slope-intercept form, which looks like this:- \( y = mx + b \)
This format is simple and tells you everything you need:- \( y \) is what you'll solve for.- \( m \) is the slope.- \( x \) is any value on the x-axis.- \( b \) is the y-intercept we calculated.
Let's put in the values we have obtained:- \( y = 0.4x + 1.2 \)This equation tells you the rate of change or rise over run for each value of \( x \). Using this, you can predict \( y \) for any given \( x \). It's your complete instruction for drawing the line on a graph, or understanding the relationship between \( x \) and \( y \). With slope-intercept form, you can easily understand and visualize the line's behavior on a coordinate plane.