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Determine the solution to the real-world system by graphing.

Jennifer and Natalie each improved their yards by planting grass sod and geraniums. They bought their supplies from the same store.

- Jennifer spent [tex]\$99[/tex] on 9 ft² of grass sod [tex](x)[/tex] and 9 geraniums [tex](y)[/tex].
- Natalie spent [tex]\$82[/tex] on 10 ft² of grass sod [tex](x)[/tex] and 3 geraniums [tex](y)[/tex].

Write equations to represent the scenario. Then, graph to find the cost of one ft² of grass sod and the cost of one geranium.

Jennifer: [tex]\(\square x + \square \cdot 9y = \square\)[/tex]

Natalie: [tex]\(\square x + \square y = \square\)[/tex]

Point Solution: [tex]\((\square, \square)\)[/tex]

Cost per ft² of Grass Sod: [tex]\$\square[/tex]

Cost per Geranium: [tex]\$\square[/tex]

Answer :

To solve the problem of finding the cost of one square foot of grass sod and one geranium, we need to set up a system of equations based on the information given.

### Step-by-Step Solution:

1. Identify the Variables:
- Let [tex]\( x \)[/tex] represent the cost of one square foot of grass sod.
- Let [tex]\( y \)[/tex] represent the cost of one geranium.

2. Write the Equations:
- Jennifer's Spending:
Jennifer bought [tex]\(9\)[/tex] square feet of grass sod and [tex]\(9\)[/tex] geraniums for a total of [tex]\(\$99\)[/tex].
This can be expressed as the equation:
[tex]\[
9x + 9y = 99
\][/tex]

- Natalie's Spending:
Natalie bought [tex]\(10\)[/tex] square feet of grass sod and [tex]\(3\)[/tex] geraniums for a total of [tex]\(\$82\)[/tex].
This can be expressed as the equation:
[tex]\[
10x + 3y = 82
\][/tex]

3. Solve the System of Equations:
- We have the equations:
[tex]\[
9x + 9y = 99
\][/tex]
[tex]\[
10x + 3y = 82
\][/tex]

4. Graphing Method:
Plotting these equations on a graph, the lines will intersect at a certain point. The x-coordinate of the intersection represents the cost of grass sod, and the y-coordinate represents the cost of a geranium. Through calculation, the intersection of these lines is determined.

5. Solution:
- The point where the two graphs intersect is [tex]\((7, 4)\)[/tex].

6. Conclusion:
- The cost of one square foot of grass sod is [tex]\( \$7 \)[/tex].
- The cost of one geranium is [tex]\( \$4 \)[/tex].

Thus, you can determine the cost of each item by graphing the system of equations and finding where the lines intersect.

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