$y-5=\frac{4}{3}(x+2)$
The given equation is \(y - 5 = \frac{4}{3}(x + 2)\). This is a linear equation expressed in the point-slope form, which is generally written as \(y - y_1 = m(x - x_1)\). Here, \(m\) represents the slope of the line, and \((x_1, y_1)\) is a specific point through which the line passes. From the given equation, we can immediately identify the slope as \(m = \frac{4}{3}\) and the point \((x_1, y_1)\) as \((-2, 5)\).
To convert the equation into the slope-intercept form, \(y = mx + b\), we start by expanding the right side:
\[y - 5 = \frac{4}{3} \cdot x + \frac{4}{3} \cdot 2\]
\[y - 5 = \frac{4}{3}x + \frac{8}{3}\]
Next, we isolate \(y\) by adding 5 to both sides of the equation:
\[y = \frac{4}{3}x + \frac{8}{3} + 5\]
Since \(5\) can be written as \(\frac{15}{3}\), we have:
\[y = \frac{4}{3}x + \frac{8}{3} + \frac{15}{3}\]
\[y = \frac{4}{3}x + \frac{23}{3}\]
This gives us the equation in the slope-intercept form, where the slope is \(\frac{4}{3}\) and the \(y\)-intercept is \(\frac{23}{3}\).
In real-life scenarios, linear equations like this one are used to describe relationships that change at a constant rate. For example, if you are filling a pool, the amount of water \(y\) (in gallons) at time \(x\) (in minutes) might follow a linear equation. If you know the pool already has 5 gallons and it fills at a rate of \(\frac{4}{3}\) gallons per minute starting from 2 minutes before you began measuring, the equation \(y - 5 = \frac{4}{3}(x + 2)\) accurately models the volume over time. Understanding these equations helps in predicting future values and planning resources effectively.