Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. \(x\) -intercept \(=-4 ; y\) -intercept \(=4\)

Short answer

The equation of the line is \(y = x + 4\).

Step-by-step solution

- Understand the Intercepts

The given intercepts are: \( x \)-intercept = -4 and \( y \)-intercept = 4. This means the line passes through the points \((-4, 0)\) and \((0, 4)\).

- Find the Slope

Use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plug in the points \((-4, 0)\) and \((0, 4)\): \[ m = \frac{4 - 0}{0 + 4} = \frac{4}{4} = 1 \]

- Use the Slope-Intercept Form

The slope-intercept form of a line is \( y = mx + b \). Here, \( m = 1 \) and the \( y \)-intercept \( b = 4 \). Therefore, the equation is: \[y = 1x + 4\] or simply, \[y = x + 4\]

Key concepts

Intercepts

In mathematics, intercepts are critical points where a line crosses the axes on a graph. The two main intercepts are the **x-intercept** and the **y-intercept**. Understanding these concepts is essential for graphing lines and writing their equations.

The **x-intercept** is the point where the line crosses the x-axis. At this point, the value of y is zero. In the given exercise, the x-intercept is -4, which means the line crosses the x-axis at (-4, 0).

The **y-intercept** is the point where the line crosses the y-axis. This is where the value of x is zero. In the given exercise, the y-intercept is 4, meaning the line crosses the y-axis at (0, 4).

These intercepts provide us with two specific points that can be used to find the slope and, eventually, to write the equation of the line.

Slope Calculation

The **slope** of a line indicates its steepness and direction. It is calculated using two points on the line. The slope formula is:
\[m = \frac{y_2 - y_1}{x_2 - x_1} \]

In the given exercise, we use the points (-4, 0) and (0, 4). Let's plug these into the slope formula:
\[m = \frac{4 - 0}{0 + 4} = \frac{4}{4} = 1 \]

The slope, denoted as 'm,' turns out to be 1. This means that for every unit increase in x, y increases by the same amount. If the slope were negative, it would indicate that the line descends as we move from left to right.

Slope-Intercept Form

The **slope-intercept form** of a line's equation is a very user-friendly way to express linear equations. It is written as:
\[y = mx + b \]

Here, 'm' represents the slope of the line, and 'b' denotes the y-intercept.

We have already calculated the slope (m = 1) and identified the y-intercept (b = 4) in the given exercise.

By plugging these values into the slope-intercept form equation, we get:
\[y = 1x + 4\]

Or more simply:
\[y = x + 4\]

This is the final equation of the line in the slope-intercept form. It easily tells us that the line has a slope of 1 and crosses the y-axis at 4.