Question

Jon buys fencing for his yard
he pays $122 for 5 fence posts and 4 fence panels
he pays $570 for 21 fence posts and 20 fence panels
how much does he pay for 4 fence posts and 3 fence panels

Answer 1

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:

1. \( 5p + 4q = 122 \) (Equation 1)
2. \( 21p + 20q = 570 \) (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.