Graph the line that represents this equation:
$3 x-4 y=8$
To graph the line defined by the linear equation \(3x - 4y = 8\), we first identify key features of the line, such as its intercepts, which allow us to plot it accurately on a coordinate plane. An intercept is a point where the line crosses one of the axes. We can find these by setting one of the variables to zero and solving for the other.
To find the x-intercept, we set \(y = 0\) in the given equation and solve for \(x\):
\[3x - 4(0) = 8\]
\[3x = 8\]
\[x = \frac{8}{3} \approx 2.67.\]
Thus, the x-intercept is at the point \((\frac{8}{3}, 0)\).
To find the y-intercept, we set \(x = 0\) and solve for \(y\):
\[3(0) - 4y = 8\]
\[-4y = 8\]
\[y = -2.\]
Thus, the y-intercept is at the point \((0, -2)\).
We can also write the equation in slope-intercept form, \(y = mx + b\), to better understand the line's steepness and vertical position. Starting with \(3x - 4y = 8\), we rearrange for \(y\):
\[-4y = -3x + 8\]
\[y = \frac{-3x + 8}{-4}\]
\[y = \frac{3}{4}x - 2.\]
Here, the slope \(m = \frac{3}{4}\) and the y-intercept \(b = -2\). This confirms our y-intercept calculation and provides a slope of \(\frac{3}{4}\), meaning that for every 4 units we move to the right, the line rises by 3 units. Starting from \((0, -2)\), moving right by 4 gives \(x = 0+4=4\) and \(y = -2+3=1\), confirming that the point \((4, 1)\) also lies on the line.
With the points \((0, -2)\), \((\frac{8}{3}, 0)\), and \((4, 1)\) identified, we have sufficient information to draw the line. By plotting these points on a coordinate plane and connecting them with a straight line, we accurately represent the equation \(3x - 4y = 8\). The final line passes through \((0, -2)\) and \((4, 1)\).
In real life, linear equations like this are fundamental in fields such as economics and engineering to represent relationships between two changing quantities. For example, if you are planning a budget, this equation could represent a scenario where you have a total cost constraint of \(\$8\) for two items with prices related by the coefficients \(3\) and \(4\). By graphing this relationship, you can visually determine all the possible combinations of those items you can afford, helping you make balanced decisions.