Graph the system of equations to find its solution. Be sure to show all work.
$y = -x + 6$
$y = \frac{2}{3}x - 9$
The system of equations is given by:
1) \(y = -x + 6\)
2) \(y = \frac{2}{3}x - 9\)
We are asked to find the solution, which is the point \((x, y)\) that lies on both lines. Graphically, this is the intersection point of the two lines in the Cartesian coordinate system. Algebraically, we can find this by setting the expressions for \(y\) equal to each other.
To find the intersection point, set the two expressions equal to each other:
\[-x + 6 = \frac{2}{3}x - 9\]
Add \(x\) to both sides:
\[6 = \frac{2}{3}x + x - 9\]
\[6 = \frac{5}{3}x - 9\]
Add 9 to both sides:
\[15 = \frac{5}{3}x\]
Multiply by 3:
\[45 = 5x\]
Divide by 5:
\[x = 9\]
Now, substitute \(x = 9\) back into the first equation to find \(y\):
\[y = -(9) + 6 = -3\]
Thus, the solution is the point \((9, -3)\).
For the first line \(y = -x + 6\), the y-intercept is at \((0, 6)\) and the slope is \(-1\). Another point on this line is \((6, 0)\).
For the second line \(y = \frac{2}{3}x - 9\), the y-intercept is at \((0, -9)\) and the slope is \(\frac{2}{3}\). Another point on this line is \((3, -7)\).
By plotting these points and drawing lines through them, the lines intersect exactly at \((9, -3)\). Substituting \((9, -3)\) into both equations confirms the solution:
Equation 1: \(-3 = -(9) + 6 \implies -3 = -3\) (True)
Equation 2: \(-3 = \frac{2}{3}(9) - 9 \implies -3 = 6 - 9 = -3\) (True)
Systems of equations are fundamental in real-world scenarios where two variables interact over time. For example, consider two different cell phone plans. Plan A might have a fixed cost with a certain rate per gigabyte of data used, while Plan B has a different fixed cost and a different data rate. By representing these plans as linear equations (where \(y\) is the total cost and \(x\) is the amount of data), you can find the exact usage amount where both plans cost the same, helping you decide which plan is more economical based on your predicted data usage.