Problem 30 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Line has \(y\) -intercept 3 and \(x\) -intercept 7

Step-by-Step Solution

Answer

The equation is \(y = \frac{-3}{7}x + 3\).

Step 1: Identify Given Points

The line has a y-intercept at 3, which means it passes through the point (0, 3). It also has an x-intercept at 7, meaning it goes through the point (7, 0).

Step 2: Determine the Slope

Use the formula for the slope: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Substitute the points (0, 3) and (7, 0): \(m = \frac{0 - 3}{7 - 0} = \frac{-3}{7}\).

Step 3: Write the Equation in Point-Slope Form

Using the point-slope form equation \(y - y_1 = m(x - x_1)\), with slope \(m = \frac{-3}{7}\) and using the point (0, 3): \(y - 3 = \frac{-3}{7}(x - 0)\).

Step 4: Simplify to Slope-Intercept Form

Convert the equation to slope-intercept form \(y = mx + b\): \(y - 3 = \frac{-3}{7}x\) Add 3 to both sides to solve for y: \(y = \frac{-3}{7}x + 3\).

Key Concepts

slope

The slope of a line describes its steepness and direction. It's represented by the letter **m**. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The formula to find the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For example, in our exercise, the points are (0, 3) and (7, 0).
Using the formula, we get: \[ m = \frac{0 - 3}{7 - 0} = \frac{-3}{7} \]
So, the slope \(m \) is \( -\frac{3}{7} \). This negative value indicates that the line slopes downwards as you move from left to right.

intercepts

Intercepts are points where the line crosses the axes.
The **y-intercept** is where the line crosses the y-axis. This happens when \( x = 0 \). In this exercise, the y-intercept is 3, so the point is (0, 3).
The **x-intercept** is where the line crosses the x-axis. This happens when \( y = 0 \). Here, the x-intercept is 7, so the point is (7, 0).
Both intercepts are key in forming the equation of a line, as they provide two fixed points that help determine its slope and position within the coordinate plane.

slope-intercept form

The slope-intercept form is a way of writing the equation of a line. It has the structure: \ y = mx + b \ where:

• **m** is the slope of the line
• **b** is the y-intercept

In our exercise, we determined that the slope \(m \) is \( -\frac{3}{7} \), and the y-intercept \( b \) is 3.
Substituting these values into the slope-intercept form, we get: \ y = -\frac{3}{7}x + 3 \.
This equation describes a line that crosses the y-axis at 3 and has a slope of \( -\frac{3}{7} \), indicating it falls as we move from left to right on the graph.