Question

1. georgias neighbors pay her $20 upfront and $6 per hour to walk their pets. gregs neighbors pay him $10 per hour to mow their lawn. the system below represents the situation.

$$\begin{array} { l } { y = 6 x + 20 } \\ { y = 10 x } end{array}$$

a. graph the system.
b. estimate the solution.
c. explain the meaning of the solution in the context of the situation.

Explanation:

Step1: Graph the equations

For \(y = 6x+20\), when \(x = 0\), \(y=20\) (y - intercept). The slope \(m = 6\).
For \(y = 10x\), when \(x = 0\), \(y = 0\) (passes through the origin). The slope \(m = 10\). Plot these two lines.

Step2: Find the solution

Set \(6x+20=10x\).
Subtract \(6x\) from both sides: \(20=10x - 6x\).
Simplify: \(20 = 4x\).
Solve for \(x\): \(x=\frac{20}{4}=5\).
Substitute \(x = 5\) into \(y = 10x\), \(y=10\times5 = 50\).

Step3: Interpret the solution

In the context of the situation, \(x\) represents the number of hours worked and \(y\) represents the amount of money earned.

Answer:

a. Graph \(y = 6x + 20\) (y - intercept at \((0,20)\), slope \(6\)) and \(y=10x\) (passing through \((0,0)\), slope \(10\)).
b. The solution is \((5,50)\).
c. After \(5\) hours of work, both Georgia and Greg will have earned \(50\) dollars.