Problem 43 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ (0,-4),(5,0) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is $$y = \frac{4}{5}x - 4$$.

Step 1: Find the slope (m) using the given points

Use the given points (0, -4) and (5, 0). The slope (m) can be calculated using the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Substitute the values of the points (0, -4) as (x_1, y_1) and (5, 0) as (x_2, y_2): $$ m = \frac{0 - (-4)}{5 - 0} $$ Simplify and calculate the slope: $$ m = \frac{4}{5} $$

Step 2:Find the y-intercept (b) using one of the given points and the slope

Since we already know the slope (m = 4/5), we can now find the y-intercept (b) using one of the given points. We will use the point (0, -4), because it is the y-intercept: $$ y = mx + b $$ Substitute the values of the point (0, -4) as (x, y) and the slope (4/5) as m: $$ -4 = \frac{4}{5}(0) + b $$ Since the term with x becomes zero, we are left with: $$ b = -4 $$

Step 3: Write the equation in slope-intercept form

Now that we know the slope (m = 4/5) and the y-intercept (b = -4), we can write the equation of the line in slope-intercept form: $$ y = \frac{4}{5}x - 4 $$ This is the equation of the line in slope-intercept form that passes through the given points (0, -4) and (5, 0).

Key Concepts

Finding Slope

The slope of a line is a measure of its steepness or incline. It's expressed as a ratio of the vertical change to the horizontal change between two points on the line. This is often referred to as "rise over run." When you have two points,

• (x₁, y₁)
• (x₂, y₂)

you can calculate the slope (m) using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting specific values from the example, you get:
\[ m = \frac{0 - (-4)}{5 - 0} = \frac{4}{5} \] This means that for every 5 units you move horizontally (run), you move 4 units vertically (rise). When calculated, this gives a positive slope, indicating an upward trend from left to right.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. At this point, the value of x is zero. You can find the y-intercept when you know the slope of the line and at least one point on it. The general equation for a straight line in slope-intercept form is:
\( y = mx + b \)
where

• \( m \) is the slope,
• \( b \) is the y-intercept.

Using the equation of the line, if you substitute one of the given points where x is zero, such as (0, -4), it simplifies the process:
\[ -4 = \frac{4}{5}(0) + b \] Since any number multiplied by zero is zero, the equation simplifies to:
\( b = -4 \). This tells you that the line crosses the y-axis at -4.

Equation of a Line

Once you have both the slope and the y-intercept, you can write the equation of the line using the slope-intercept form. The slope-intercept form of a linear equation is easy to read and very popular for graphing because it clearly shows both the slope and the y-intercept:
\( y = mx + b \)
In the example provided, we've found that:

• The slope \( m = \frac{4}{5} \)
• The y-intercept \( b = -4 \)

Plugging in these values gives the equation:
\[ y = \frac{4}{5}x - 4 \] This equation tells us that for each unit increase in x, y increases by \(\frac{4}{5}\) and the line crosses the y-axis at -4. It's a complete description of the line in a straightforward way.