Let's denote the price of a fence post as \( p \) and the price of a fence panel as \( q \).
From the problem, we have the following two equations based on Jon's purchases:
- \( 5p + 4q = 122 \) (Equation 1)
- \( 21p + 20q = 570 \) (Equation 2)
We can solve this system of equations to find the values of \( p \) and \( q \).
Step 1: Solve for \( p \) and \( q \)
Let's first manipulate Equation 1 to express one variable in terms of the other. We can solve for \( q \):
\[ 4q = 122 - 5p \] \[ q = \frac{122 - 5p}{4} \tag{Equation 3} \]
Now, substitute Equation 3 into Equation 2:
\[ 21p + 20\left(\frac{122 - 5p}{4}\right) = 570 \]
To eliminate the fraction, multiply the entire equation by 4:
\[ 4 \cdot 21p + 20(122 - 5p) = 4 \cdot 570 \] \[ 84p + 2440 - 100p = 2280 \]
Combine like terms:
\[ -16p + 2440 = 2280 \]
Now, isolate \( p \):
\[ -16p = 2280 - 2440 \] \[ -16p = -160 \] \[ p = 10 \]
Step 2: Substitute \( p \) back to find \( q \)
Now we can substitute \( p = 10 \) back into Equation 3 to find \( q \):
\[ q = \frac{122 - 5(10)}{4} \] \[ q = \frac{122 - 50}{4} \] \[ q = \frac{72}{4} \] \[ q = 18 \]
Step 3: Find the cost of 4 fence posts and 3 fence panels
Now that we have the values for \( p \) and \( q \):
- Price of a fence post \( p = 10 \)
- Price of a fence panel \( q = 18 \)
Now we can calculate the cost for 4 fence posts and 3 fence panels:
\[ \text{Cost} = 4p + 3q \] \[ \text{Cost} = 4(10) + 3(18) \] \[ \text{Cost} = 40 + 54 \] \[ \text{Cost} = 94 \]
Thus, Jon pays $94 for 4 fence posts and 3 fence panels.
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